%------------------------------------------------------------------------------
% File     : ITP221_2 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Definitions 00493_022399
% 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    : 0063_VEBT_Definitions_00493_022399 [Des22]

% Status   : Theorem
% Rating   : 0.50 v8.1.0
% Syntax   : Number of formulae    : 11249 (2552 unt;1442 typ;   0 def)
%            Number of atoms       : 27938 (8255 equ)
%            Maximal formula atoms :   73 (   2 avg)
%            Number of connectives : 20187 (2056   ~; 333   |;2322   &)
%                                         (2088 <=>;13388  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   6 avg)
%            Maximal term depth    :   31 (   2 avg)
%            Number of types       :   13 (  12 usr)
%            Number of type conns  : 1202 (1007   >; 195   *;   0   +;   0  <<)
%            Number of predicates  :  228 ( 225 usr;   2 prp; 0-7 aty)
%            Number of functors    : 1205 (1205 usr;  51 con; 0-8 aty)
%            Number of variables   : 32175 (29095   !; 866   ?;32175   :)
%                                         (2214  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TF1_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 17:39:10.671
%------------------------------------------------------------------------------
% Could-be-implicit typings (18)
tff(ty_t_VEBT__Definitions_OVEBT,type,
    vEBT_VEBT: $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_Extended__Nat_Oenat,type,
    extended_enat: $tType ).

tff(ty_t_Complex_Ocomplex,type,
    complex: $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_HOL_Obool,type,
    bool: $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_fun,type,
    fun: ( $tType * $tType ) > $tType ).

% Explicit typings (1424)
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_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_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_Obounded__lattice,type,
    bounded_lattice: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_cl_Topological__Spaces_Ot3__space,type,
    topological_t3_space: 
      !>[A: $tType] : $o ).

tff(sy_cl_Topological__Spaces_Ot4__space,type,
    topological_t4_space: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

tff(sy_cl_Real__Vector__Spaces_Oreal__field,type,
    real_V7773925162809079976_field: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Oring__1__no__zero__divisors,type,
    ring_15535105094025558882visors: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
    cancel2418104881723323429up_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Olinordered__ab__group__add,type,
    linord5086331880401160121up_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Oordered__comm__monoid__add,type,
    ordere6911136660526730532id_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Limits_Otopological__monoid__mult,type,
    topolo1898628316856586783d_mult: 
      !>[A: $tType] : $o ).

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

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

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

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

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

tff(sy_cl_Topological__Spaces_Ouniform__space,type,
    topolo7287701948861334536_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_Real__Vector__Spaces_Ouniformity__dist,type,
    real_V768167426530841204y_dist: 
      !>[A: $tType] : $o ).

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

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

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

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

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

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

tff(sy_cl_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_Countable__Complete__Lattices_Ocountable__complete__distrib__lattice,type,
    counta4013691401010221786attice: 
      !>[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] : fun(A,B) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__oj____,type,
    aTP_Lamp_oj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

tff(sy_c_ATP_058Lamp__om____,type,
    aTP_Lamp_om: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__on____,type,
    aTP_Lamp_on: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oo____,type,
    aTP_Lamp_oo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__op____,type,
    aTP_Lamp_op: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oq____,type,
    aTP_Lamp_oq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__or____,type,
    aTP_Lamp_or: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__os____,type,
    aTP_Lamp_os: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_BNF__Def_OGr,type,
    bNF_Gr: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) ) > set(product_prod(A,B)) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
    bNF_Greatest_image2: 
      !>[C: $tType,A: $tType,B: $tType] : ( ( set(C) * fun(C,A) * fun(C,B) ) > set(product_prod(A,B)) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
    bNF_Gr4221423524335903396lImage: 
      !>[B: $tType,A: $tType] : ( ( set(product_prod(B,B)) * fun(B,A) ) > set(product_prod(A,A)) ) ).

tff(sy_c_BNF__Greatest__Fixpoint_OrelInvImage,type,
    bNF_Gr7122648621184425601vImage: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(product_prod(B,B)) * fun(A,B) ) > set(product_prod(A,A)) ) ).

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

tff(sy_c_BNF__Wellorder__Constructions_OFunc__map,type,
    bNF_We4925052301507509544nc_map: 
      !>[B: $tType,C: $tType,A: $tType,D: $tType] : ( ( set(B) * fun(C,A) * fun(B,D) ) > fun(fun(D,C),fun(B,A)) ) ).

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

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

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

tff(sy_c_Bit__Operations_Oand__not__num__rel,type,
    bit_and_not_num_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).

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

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

tff(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
    bit_or3848514188828904588eg_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).

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

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

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

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

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

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

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

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

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_Oand__num,type,
    bit_un1837492267222099188nd_num: fun(num,fun(num,option(num))) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oand__num__rel,type,
    bit_un5425074673868309765um_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num,type,
    bit_un6178654185764691216or_num: fun(num,fun(num,option(num))) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_Oxor__num__rel,type,
    bit_un3595099601533988841um_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
    bit_un7362597486090784418nd_num: fun(num,fun(num,option(num))) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
    bit_un4731106466462545111um_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
    bit_un2480387367778600638or_num: fun(num,fun(num,option(num))) ).

tff(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
    bit_un2901131394128224187um_rel: fun(product_prod(num,num),fun(product_prod(num,num),bool)) ).

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_COMBB,type,
    combb: 
      !>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).

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

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

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

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

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

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

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

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

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

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

tff(sy_c_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_Complete__Partial__Order_Ochain,type,
    comple1602240252501008431_chain: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * set(A) ) > $o ) ).

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_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_Divides_Oadjust__div,type,
    adjust_div: product_prod(int,int) > int ).

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_Filter_Ofinite__subsets__at__top,type,
    finite5375528669736107172at_top: 
      !>[A: $tType] : ( set(A) > filter(set(A)) ) ).

tff(sy_c_Filter_Omap__filter__on,type,
    map_filter_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * filter(A) ) > filter(B) ) ).

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

tff(sy_c_Finite__Set_OFpow,type,
    finite_Fpow: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

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

tff(sy_c_Finite__Set_Ocomp__fun__commute__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),bool) ).

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_Ofold__graph,type,
    finite_fold_graph: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) * B ) > $o ) ).

tff(sy_c_Finite__Set_Ofolding__idem__on,type,
    finite1890593828518410140dem_on: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,fun(B,B)) ) > $o ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_If,type,
    if: 
      !>[A: $tType] : ( ( bool * A * A ) > A ) ).

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

tff(sy_c_Int_ORep__Integ,type,
    rep_Integ: int > product_prod(nat,nat) ).

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

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

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

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

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

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

tff(sy_c_Lattices_Osemilattice__neutr__order,type,
    semila1105856199041335345_order: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * A * fun(A,fun(A,bool)) * fun(A,fun(A,bool)) ) > $o ) ).

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

tff(sy_c_Lattices__Big_Olinorder_OMax,type,
    lattices_Max: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),A) ) ).

tff(sy_c_Lattices__Big_Olinorder_OMin,type,
    lattices_Min: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(set(A),A) ) ).

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

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

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

tff(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
    lattic7752659483105999362nf_fin: 
      !>[A: $tType] : fun(set(A),A) ).

tff(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
    lattic5882676163264333800up_fin: 
      !>[A: $tType] : fun(set(A),A) ).

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

tff(sy_c_Limits_Oat__infinity,type,
    at_infinity: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_List_OBleast,type,
    bleast: 
      !>[A: $tType] : ( ( set(A) * fun(A,bool) ) > A ) ).

tff(sy_c_List_Oabort__Bleast,type,
    abort_Bleast: 
      !>[A: $tType] : ( ( set(A) * fun(A,bool) ) > A ) ).

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

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

tff(sy_c_List_Oarg__min__list__rel,type,
    arg_min_list_rel: 
      !>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),list(A)),fun(product_prod(fun(A,B),list(A)),bool)) ).

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_Odrop,type,
    drop: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(A) ) ).

tff(sy_c_List_OdropWhile,type,
    dropWhile: 
      !>[A: $tType] : ( ( fun(A,bool) * 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,bool) * list(A) ) > option(product_prod(list(A),product_prod(A,list(A)))) ) ).

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

tff(sy_c_List_Ofind,type,
    find: 
      !>[A: $tType] : ( ( fun(A,bool) * 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,bool)) * fun(A,fun(A,bool)) * 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_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)) > fun(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_Olexordp,type,
    lexordp: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) * list(A) ) > $o ) ).

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

tff(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
    sorted8670434370408473282of_set: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(A,bool)) > fun(fun(B,A),fun(set(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__key__list__of__set,type,
    linord144544945434240204of_set: 
      !>[B: $tType,A: $tType] : ( fun(B,A) > fun(set(B),list(B)) ) ).

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

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

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

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

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

tff(sy_c_List_Olist_Olist__all2,type,
    list_all2: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(list(A),fun(list(B),bool)) ) ).

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

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

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

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

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_Olistrel1p,type,
    listrel1p: 
      !>[A: $tType] : ( ( fun(A,fun(A,bool)) * list(A) * list(A) ) > $o ) ).

tff(sy_c_List_Olistrelp,type,
    listrelp: 
      !>[A: $tType,B: $tType] : ( fun(A,fun(B,bool)) > fun(list(A),fun(list(B),bool)) ) ).

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

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

tff(sy_c_List_Omap__filter,type,
    map_filter: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) ) > list(B) ) ).

tff(sy_c_List_Omin__list,type,
    min_list: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_List_Omin__list__rel,type,
    min_list_rel: 
      !>[A: $tType] : fun(list(A),fun(list(A),bool)) ).

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_Opartition,type,
    partition: 
      !>[A: $tType] : ( ( fun(A,bool) * 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_Oremdups__adj__rel,type,
    remdups_adj_rel: 
      !>[A: $tType] : fun(list(A),fun(list(A),bool)) ).

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

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

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

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

tff(sy_c_List_Oshuffles__rel,type,
    shuffles_rel: 
      !>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),bool)) ).

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

tff(sy_c_List_Osorted__wrt__rel,type,
    sorted_wrt_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,fun(A,bool)),list(A)),fun(product_prod(fun(A,fun(A,bool)),list(A)),bool)) ).

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

tff(sy_c_List_Osplice__rel,type,
    splice_rel: 
      !>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),bool)) ).

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,bool) * list(A) ) > list(A) ) ).

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

tff(sy_c_List_Otranspose__rel,type,
    transpose_rel: 
      !>[A: $tType] : fun(list(list(A)),fun(list(list(A)),bool)) ).

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),bool)) ).

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,bool) ) ).

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_Olist__encode,type,
    nat_list_encode: list(nat) > nat ).

tff(sy_c_Nat__Bijection_Olist__encode__rel,type,
    nat_list_encode_rel: fun(list(nat),fun(list(nat),bool)) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux,type,
    nat_prod_decode_aux: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
    nat_pr5047031295181774490ux_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),bool)) ).

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_Otriangle,type,
    nat_triangle: nat > nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: nat > fun(real,real) ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun(real,real) ).

tff(sy_c_Num_OBitM,type,
    bitM: num > num ).

tff(sy_c_Num_Oinc,type,
    inc: num > num ).

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

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

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

tff(sy_c_Num_Oneg__numeral__class_Osub,type,
    neg_numeral_sub: 
      !>[A: $tType] : ( ( num * num ) > A ) ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: fun(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] : ( ( A * fun(num,A) * fun(num,A) * 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: 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_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_Omap__option,type,
    map_option: 
      !>[A: $tType,Aa: $tType] : ( fun(A,Aa) > fun(option(A),option(Aa)) ) ).

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

tff(sy_c_Option_Othese,type,
    these: 
      !>[A: $tType] : ( set(option(A)) > set(A) ) ).

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

tff(sy_c_Orderings_Oord_Omax,type,
    max: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,A)) ) ).

tff(sy_c_Orderings_Oord_Omin,type,
    min: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,fun(A,A)) ) ).

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

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

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

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

tff(sy_c_Orderings_Oorder__class_OGreatest,type,
    order_Greatest: 
      !>[A: $tType] : ( fun(A,bool) > 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_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)) ) > fun(A,fun(nat,A)) ) ).

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

tff(sy_c_Product__Type_OPair,type,
    product_Pair: 
      !>[A: $tType,B: $tType] : ( 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(B,C) * 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_Rat_OFract,type,
    fract: ( 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_Opositive,type,
    positive: fun(rat,bool) ).

tff(sy_c_Real__Vector__Spaces_OReals,type,
    real_Vector_Reals: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Real__Vector__Spaces_Obounded__linear,type,
    real_V3181309239436604168linear: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Real__Vector__Spaces_Obounded__linear__axioms,type,
    real_V4916620083959148203axioms: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
    real_V557655796197034286t_dist: 
      !>[A: $tType] : ( ( A * A ) > real ) ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
    real_V7770717601297561774m_norm: 
      !>[A: $tType] : ( A > real ) ).

tff(sy_c_Real__Vector__Spaces_Oof__real,type,
    real_Vector_of_real: 
      !>[A: $tType] : ( real > A ) ).

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
    real_V8093663219630862766scaleR: 
      !>[A: $tType] : ( real > fun(A,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_Oirrefl,type,
    irrefl: 
      !>[A: $tType] : ( set(product_prod(A,A)) > $o ) ).

tff(sy_c_Relation_Orefl__on,type,
    refl_on: 
      !>[A: $tType] : ( ( set(A) * set(product_prod(A,A)) ) > $o ) ).

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_Rings_Odivide__class_Odivide,type,
    divide_divide: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Rings_Odvd__class_Odvd,type,
    dvd_dvd: 
      !>[A: $tType] : ( ( A * A ) > bool ) ).

tff(sy_c_Rings_Omodulo__class_Omodulo,type,
    modulo_modulo: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
    zero_neq_one_of_bool: 
      !>[A: $tType] : fun(bool,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,bool) > set(A) ) ).

tff(sy_c_Set_OPow,type,
    pow2: 
      !>[A: $tType] : ( set(A) > set(set(A)) ) ).

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

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

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

tff(sy_c_Set_Othe__elem,type,
    the_elem: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
    set_fo6178422350223883121st_nat: 
      !>[A: $tType] : ( ( fun(nat,fun(A,A)) * nat * nat * A ) > A ) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
    set_fo1817059534552279752at_rel: 
      !>[A: $tType] : fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool)) ).

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

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

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

tff(sy_c_String_Ochar_OChar,type,
    char2: ( bool * bool * bool * bool * bool * bool * bool * bool ) > char ).

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

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

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

tff(sy_c_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_Oclosed,type,
    topolo7761053866217962861closed: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Ocompact,type,
    topolo2193935891317330818ompact: 
      !>[A: $tType] : ( set(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_Topological__Spaces_Ouniform__space__class_Ocauchy__filter,type,
    topolo6773858410816713723filter: 
      !>[A: $tType] : ( filter(A) > $o ) ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_Ocomplete,type,
    topolo2479028161051973599mplete: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type,
    topolo6688025880775521714ounded: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity,type,
    topolo7806501430040627800ormity: 
      !>[A: $tType] : filter(product_prod(A,A)) ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun(real,real) ).

tff(sy_c_Transcendental_Oarcosh,type,
    arcosh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun(real,real) ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun(real,real) ).

tff(sy_c_Transcendental_Oarsinh,type,
    arsinh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Oartanh,type,
    artanh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Ocos,type,
    cos: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: nat > real ).

tff(sy_c_Transcendental_Ocosh,type,
    cosh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Ocot,type,
    cot: 
      !>[A: $tType] : fun(A,A) ).

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

tff(sy_c_Transcendental_Oexp,type,
    exp: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Oln__class_Oln,type,
    ln_ln: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Olog,type,
    log: real > fun(real,real) ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr,type,
    powr: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Transcendental_Osin,type,
    sin: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: nat > real ).

tff(sy_c_Transcendental_Osinh,type,
    sinh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Otan,type,
    tan: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Otanh,type,
    tanh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transitive__Closure_Ontrancl,type,
    transitive_ntrancl: 
      !>[A: $tType] : ( ( nat * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_Transitive__Closure_Ortrancl,type,
    transitive_rtrancl: 
      !>[A: $tType] : ( 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: ( bool * bool ) > 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(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))) * fun(bool,fun(bool,A)) * 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: fun(vEBT_VEBT,fun(nat,bool)) ).

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),bool)) ).

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),bool)) ).

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),bool)) ).

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,bool)) ).

tff(sy_c_Wellfounded_Oaccp,type,
    accp: 
      !>[A: $tType] : ( fun(A,fun(A,bool)) > fun(A,bool) ) ).

tff(sy_c_Wellfounded_Ofinite__psubset,type,
    finite_psubset: 
      !>[A: $tType] : set(product_prod(set(A),set(A))) ).

tff(sy_c_Wellfounded_Omax__ext,type,
    max_ext: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_Wellfounded_Omeasure,type,
    measure: 
      !>[A: $tType] : ( fun(A,nat) > set(product_prod(A,A)) ) ).

tff(sy_c_Wellfounded_Omin__ext,type,
    min_ext: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(set(A),set(A))) ) ).

tff(sy_c_Wellfounded_Omlex__prod,type,
    mlex_prod: 
      !>[A: $tType] : ( ( fun(A,nat) * set(product_prod(A,A)) ) > set(product_prod(A,A)) ) ).

tff(sy_c_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_fAll,type,
    fAll: 
      !>[A: $tType] : ( fun(A,bool) > bool ) ).

tff(sy_c_fChoice,type,
    fChoice: 
      !>[A: $tType] : ( fun(A,bool) > A ) ).

tff(sy_c_fEx,type,
    fEx: 
      !>[A: $tType] : fun(fun(A,bool),bool) ).

tff(sy_c_fFalse,type,
    fFalse: bool ).

tff(sy_c_fNot,type,
    fNot: fun(bool,bool) ).

tff(sy_c_fTrue,type,
    fTrue: bool ).

tff(sy_c_fconj,type,
    fconj: ( bool * bool ) > bool ).

tff(sy_c_fdisj,type,
    fdisj: ( bool * bool ) > bool ).

tff(sy_c_fequal,type,
    fequal: 
      !>[A: $tType] : fun(A,fun(A,bool)) ).

tff(sy_c_fimplies,type,
    fimplies: fun(bool,fun(bool,bool)) ).

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

tff(sy_c_pp,type,
    pp: bool > $o ).

tff(sy_v_deg,type,
    deg: nat ).

tff(sy_v_info,type,
    info: option(product_prod(nat,nat)) ).

tff(sy_v_m____,type,
    m: nat ).

tff(sy_v_ma____,type,
    ma: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_n,type,
    n: 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_x,type,
    x: nat ).

% Relevant facts (8999)
tff(fact_0_both__member__options__def,axiom,
    ! [T2: vEBT_VEBT,X: nat] :
      ( pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2),X))
    <=> ( vEBT_V5719532721284313246member(T2,X)
        | vEBT_VEBT_membermima(T2,X) ) ) ).

% both_member_options_def
tff(fact_1__092_060open_0621_A_060_Adeg_092_060close_062,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),deg)) ).

% \<open>1 < deg\<close>
tff(fact_2_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_3__C5_Ohyps_C_I3_J,axiom,
    ( ( summary = vEBT_Node(info,deg,treeList,summary) )
   => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_Node(info,deg,treeList,summary)),x)) ) ).

% "5.hyps"(3)
tff(fact_4__C5_Ohyps_C_I2_J,axiom,
    vEBT_invar_vebt(summary,m) ).

% "5.hyps"(2)
tff(fact_5_assms_I1_J,axiom,
    vEBT_invar_vebt(vEBT_Node(info,deg,treeList,summary),n) ).

% assms(1)
tff(fact_6_assms_I2_J,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),x),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),deg))) ).

% assms(2)
tff(fact_7__092_060open_062naive__member_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_A_Ilow_Ax_A_Ideg_Adiv_A2_J_J_A_092_060Longrightarrow_062_Anaive__member_A_INode_Ainfo_Adeg_AtreeList_Asummary_J_Ax_092_060close_062,axiom,
    ( vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(x,divide_divide(nat,deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(x,divide_divide(nat,deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
   => vEBT_V5719532721284313246member(vEBT_Node(info,deg,treeList,summary),x) ) ).

% \<open>naive_member (treeList ! high x (deg div 2)) (low x (deg div 2)) \<Longrightarrow> naive_member (Node info deg treeList summary) x\<close>
tff(fact_8__092_060open_062membermima_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_A_Ilow_Ax_A_Ideg_Adiv_A2_J_J_A_092_060Longrightarrow_062_Amembermima_A_INode_Ainfo_Adeg_AtreeList_Asummary_J_Ax_092_060close_062,axiom,
    ( vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(x,divide_divide(nat,deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(x,divide_divide(nat,deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
   => vEBT_VEBT_membermima(vEBT_Node(info,deg,treeList,summary),x) ) ).

% \<open>membermima (treeList ! high x (deg div 2)) (low x (deg div 2)) \<Longrightarrow> membermima (Node info deg treeList summary) x\<close>
tff(fact_9__092_060open_062high_Ax_A_Ideg_Adiv_A2_J_A_060_A2_A_094_Am_092_060close_062,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(x,divide_divide(nat,deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m))) ).

% \<open>high x (deg div 2) < 2 ^ m\<close>
tff(fact_10__092_060open_062membermima_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_A_Ilow_Ax_A_Ideg_Adiv_A2_J_J_A_092_060or_062_Anaive__member_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_A_Ilow_Ax_A_Ideg_Adiv_A2_J_J_092_060close_062,axiom,
    ( vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(x,divide_divide(nat,deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(x,divide_divide(nat,deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
    | vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(x,divide_divide(nat,deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(x,divide_divide(nat,deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% \<open>membermima (treeList ! high x (deg div 2)) (low x (deg div 2)) \<or> naive_member (treeList ! high x (deg div 2)) (low x (deg div 2))\<close>
tff(fact_11_assms_I3_J,axiom,
    pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),vEBT_VEBT_high(x,divide_divide(nat,deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),vEBT_VEBT_low(x,divide_divide(nat,deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% assms(3)
tff(fact_12__C5_Ohyps_C_I1_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList)))
     => ( vEBT_invar_vebt(X2,na)
        & ( ( X2 = vEBT_Node(info,deg,treeList,summary) )
         => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_Node(info,deg,treeList,summary)),x)) ) ) ) ).

% "5.hyps"(1)
tff(fact_13__C5_Ohyps_C_I7_J,axiom,
    ! [I: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m)))
     => ( ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I)),X_1))
      <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,summary),I)) ) ) ).

% "5.hyps"(7)
tff(fact_14__C5_Ohyps_C_I6_J,axiom,
    deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),na),m) ).

% "5.hyps"(6)
tff(fact_15__C5_Ohyps_C_I10_J,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),ma),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),deg))) ).

% "5.hyps"(10)
tff(fact_16_VEBT_Osimps_I5_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),F2: fun(bool,fun(bool,A)),X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : vEBT_case_VEBT(A,F1,F2,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_17_high__def,axiom,
    ! [X: nat,N: nat] : vEBT_VEBT_high(X,N) = divide_divide(nat,X,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ).

% high_def
tff(fact_18_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Ma,N)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M))) ) ).

% high_bound_aux
tff(fact_19__C5_Ohyps_C_I8_J,axiom,
    ( ( mi = ma )
   => ! [X2: vEBT_VEBT] :
        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList)))
       => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),X_12)) ) ) ).

% "5.hyps"(8)
tff(fact_20__C5_Ohyps_C_I5_J,axiom,
    m = aa(nat,nat,suc,na) ).

% "5.hyps"(5)
tff(fact_21__C5_Ohyps_C_I4_J,axiom,
    aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),treeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m) ).

% "5.hyps"(4)
tff(fact_22_in__children__def,axiom,
    ! [N: nat,TreeList: list(vEBT_VEBT),X: nat] :
      ( vEBT_V5917875025757280293ildren(N,TreeList,X)
    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,N))),vEBT_VEBT_low(X,N))) ) ).

% in_children_def
tff(fact_23_add__self__div__2,axiom,
    ! [M: nat] : divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = M ).

% add_self_div_2
tff(fact_24__C5_Ohyps_C_I11_J,axiom,
    ( ( mi != ma )
   => ! [I: nat] :
        ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),m)))
       => ( ( ( vEBT_VEBT_high(ma,na) = I )
           => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I)),vEBT_VEBT_low(ma,na))) )
          & ! [X2: nat] :
              ( ( ( vEBT_VEBT_high(X2,na) = I )
                & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),I)),vEBT_VEBT_low(X2,na))) )
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),mi),X2))
                & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X2),ma)) ) ) ) ) ) ).

% "5.hyps"(11)
tff(fact_25_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),aa(num,num,bit0,one2)) ) ) ).

% one_add_one
tff(fact_26_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,X: nat,Y: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,B2),X)),aa(nat,A,power_power(A,B2),Y)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y)) ) ) ) ).

% power_strict_increasing_iff
tff(fact_27_numeral__plus__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ) ).

% numeral_plus_one
tff(fact_28_one__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N)) ) ).

% one_plus_numeral
tff(fact_29_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one2),N)) ) ) ).

% one_less_numeral_iff
tff(fact_30_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => ( ( aa(nat,A,power_power(A,A2),M) = aa(nat,A,power_power(A,A2),N) )
          <=> ( M = N ) ) ) ) ).

% power_inject_exp
tff(fact_31_numeral__eq__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: num] :
          ( ( aa(num,A,numeral_numeral(A),N) = one_one(A) )
        <=> ( N = one2 ) ) ) ).

% numeral_eq_one_iff
tff(fact_32_one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),N) )
        <=> ( one2 = N ) ) ) ).

% one_eq_numeral_iff
tff(fact_33_div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,M: nat,N: nat] : divide_divide(A,divide_divide(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = divide_divide(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ).

% div_exp_eq
tff(fact_34_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N2: nat] : X != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),N2)
     => ~ ! [N2: nat] : X != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,suc,N2)) ) ).

% even_odd_cases
tff(fact_35__C5_Ohyps_C_I9_J,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),mi),ma)) ).

% "5.hyps"(9)
tff(fact_36_inthall,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),N: nat] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X3)) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N))) ) ) ).

% inthall
tff(fact_37_numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: num,N: num] :
          ( ( aa(num,A,numeral_numeral(A),M) = aa(num,A,numeral_numeral(A),N) )
        <=> ( M = N ) ) ) ).

% numeral_eq_iff
tff(fact_38_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ) ).

% numeral_le_iff
tff(fact_39_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_40_power__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [N: nat] : aa(nat,A,power_power(A,one_one(A)),N) = one_one(A) ) ).

% power_one
tff(fact_41_power__one__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,power_power(A,A2),one_one(nat)) = A2 ) ).

% power_one_right
tff(fact_42_set__n__deg__not__0,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,M: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
         => vEBT_invar_vebt(X3,N) )
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N)) ) ) ).

% set_n_deg_not_0
tff(fact_43_add__numeral__left,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [V: num,W: num,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W)),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),W))),Z) ) ).

% add_numeral_left
tff(fact_44_mem__Collect__eq,axiom,
    ! [A: $tType,A2: A,P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),collect(A,P)))
    <=> pp(aa(A,bool,P,A2)) ) ).

% mem_Collect_eq
tff(fact_45_Collect__mem__eq,axiom,
    ! [A: $tType,A3: set(A)] : collect(A,aTP_Lamp_a(set(A),fun(A,bool),A3)) = A3 ).

% Collect_mem_eq
tff(fact_46_Collect__cong,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P,X3))
        <=> pp(aa(A,bool,Q,X3)) )
     => ( collect(A,P) = collect(A,Q) ) ) ).

% Collect_cong
tff(fact_47_ext,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),G: fun(A,B)] :
      ( ! [X3: A] : aa(A,B,F3,X3) = aa(A,B,G,X3)
     => ( F3 = G ) ) ).

% ext
tff(fact_48_numeral__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ) ).

% numeral_plus_numeral
tff(fact_49_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ) ).

% numeral_less_iff
tff(fact_50_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),one_one(A)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),one2)) ) ) ).

% numeral_le_one_iff
tff(fact_51_Suc__numeral,axiom,
    ! [N: num] : aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),N)) = aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ).

% Suc_numeral
tff(fact_52_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,X: nat,Y: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,B2),X)),aa(nat,A,power_power(A,B2),Y)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),Y)) ) ) ) ).

% power_increasing_iff
tff(fact_53_add__2__eq__Suc_H,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).

% add_2_eq_Suc'
tff(fact_54_add__2__eq__Suc,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).

% add_2_eq_Suc
tff(fact_55_Suc__1,axiom,
    aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ).

% Suc_1
tff(fact_56_div2__Suc__Suc,axiom,
    ! [M: nat] : divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,M)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,suc,divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ).

% div2_Suc_Suc
tff(fact_57_Suc__div__le__mono,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),divide_divide(nat,M,N)),divide_divide(nat,aa(nat,nat,suc,M),N))) ).

% Suc_div_le_mono
tff(fact_58_add__One__commute,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N) = aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2) ).

% add_One_commute
tff(fact_59_le__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),one_one(A))) ) ).

% le_numeral_extra(4)
tff(fact_60_div__le__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),divide_divide(nat,M,N)),M)) ).

% div_le_dividend
tff(fact_61_div__le__mono,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),divide_divide(nat,M,K)),divide_divide(nat,N,K))) ) ).

% div_le_mono
tff(fact_62_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N3: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A2),N)),aa(nat,A,power_power(A,A2),N3))) ) ) ) ).

% power_increasing
tff(fact_63_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A2),M)),aa(nat,A,power_power(A,A2),N)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).

% power_le_imp_le_exp
tff(fact_64_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),one2))),N) ).

% Suc_nat_number_of_add
tff(fact_65_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% one_le_numeral
tff(fact_66_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(nat,A,power_power(A,A2),N))) ) ) ).

% one_le_power
tff(fact_67_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(nat,A,power_power(A,A2),aa(nat,nat,suc,N)))) ) ) ).

% power_gt1
tff(fact_68_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,power_power(nat,M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% power2_nat_le_imp_le
tff(fact_69_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,power_power(nat,M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,power_power(nat,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% power2_nat_le_eq_le
tff(fact_70_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,power_power(nat,K),M))) ) ).

% self_le_ge2_pow
tff(fact_71_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_72_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
       => ? [N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,power_power(nat,B2),N2)),K))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(nat,nat,power_power(nat,B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat))))) ) ) ) ).

% ex_power_ivl2
tff(fact_73_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),K))
       => ? [N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,power_power(nat,B2),N2)),K))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(nat,nat,power_power(nat,B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat))))) ) ) ) ).

% ex_power_ivl1
tff(fact_74_less__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),one_one(A))) ) ).

% less_numeral_extra(4)
tff(fact_75_power__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,N: nat] : aa(nat,A,power_power(A,divide_divide(A,A2,B2)),N) = divide_divide(A,aa(nat,A,power_power(A,A2),N),aa(nat,A,power_power(A,B2),N)) ) ).

% power_divide
tff(fact_76_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N)),one_one(A))) ) ).

% not_numeral_less_one
tff(fact_77_one__plus__numeral__commute,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) ) ).

% one_plus_numeral_commute
tff(fact_78_numeral__Bit0,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N)) ) ).

% numeral_Bit0
tff(fact_79_numeral__One,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).

% numeral_One
tff(fact_80_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_81_power__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,N: nat] : aa(nat,A,power_power(A,divide_divide(A,one_one(A),A2)),N) = divide_divide(A,one_one(A),aa(nat,A,power_power(A,A2),N)) ) ).

% power_one_over
tff(fact_82_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_83_numeral__Bit0__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: num] : divide_divide(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(num,A,numeral_numeral(A),N) ) ).

% numeral_Bit0_div_2
tff(fact_84_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N3: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A2),N)),aa(nat,A,power_power(A,A2),N3))) ) ) ) ).

% power_strict_increasing
tff(fact_85_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A2),M)),aa(nat,A,power_power(A,A2),N)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ) ).

% power_less_imp_less_exp
tff(fact_86_one__power2,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,power_power(A,one_one(A)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(A) ) ) ).

% one_power2
tff(fact_87_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),aa(num,num,bit0,one2)) ).

% nat_1_add_1
tff(fact_88_less__exp,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% less_exp
tff(fact_89_semiring__norm_I76_J,axiom,
    ! [N: num] : pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one2),aa(num,num,bit0,N))) ).

% semiring_norm(76)
tff(fact_90_semiring__norm_I2_J,axiom,
    aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),one2) = aa(num,num,bit0,one2) ).

% semiring_norm(2)
tff(fact_91_field__less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ) ).

% field_less_half_sum
tff(fact_92_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),one2)) ).

% semiring_norm(75)
tff(fact_93_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit0,M)),aa(num,num,bit0,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ).

% semiring_norm(78)
tff(fact_94_semiring__norm_I6_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit0,M)),aa(num,num,bit0,N)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).

% semiring_norm(6)
tff(fact_95_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% nat_add_left_cancel_le
tff(fact_96_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% nat_add_left_cancel_less
tff(fact_97_add__Suc__right,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% add_Suc_right
tff(fact_98_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N))) ) ).

% enat_ord_number(2)
tff(fact_99_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(nat,nat,suc,M)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ).

% Suc_le_mono
tff(fact_100_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_less_eq
tff(fact_101_semiring__norm_I87_J,axiom,
    ! [M: num,N: num] :
      ( ( aa(num,num,bit0,M) = aa(num,num,bit0,N) )
    <=> ( M = N ) ) ).

% semiring_norm(87)
tff(fact_102_nat_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( aa(nat,nat,suc,X22) = aa(nat,nat,suc,Y2) )
    <=> ( X22 = Y2 ) ) ).

% nat.inject
tff(fact_103_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_104_semiring__norm_I85_J,axiom,
    ! [M: num] : aa(num,num,bit0,M) != one2 ).

% semiring_norm(85)
tff(fact_105_semiring__norm_I83_J,axiom,
    ! [N: num] : one2 != aa(num,num,bit0,N) ).

% semiring_norm(83)
tff(fact_106_lessI,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,N))) ).

% lessI
tff(fact_107_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N))) ) ).

% Suc_mono
tff(fact_108_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit0,M)),aa(num,num,bit0,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ).

% semiring_norm(71)
tff(fact_109_semiring__norm_I68_J,axiom,
    ! [N: num] : pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),one2),N)) ).

% semiring_norm(68)
tff(fact_110_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit0,M)),one2)) ).

% semiring_norm(69)
tff(fact_111_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less_eq(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N))) ) ).

% enat_ord_number(1)
tff(fact_112_enat__less__induct,axiom,
    ! [P: fun(extended_enat,bool),N: extended_enat] :
      ( ! [N2: extended_enat] :
          ( ! [M2: extended_enat] :
              ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),M2),N2))
             => pp(aa(extended_enat,bool,P,M2)) )
         => pp(aa(extended_enat,bool,P,N2)) )
     => pp(aa(extended_enat,bool,P,N)) ) ).

% enat_less_induct
tff(fact_113_le__num__One__iff,axiom,
    ! [X: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),X),one2))
    <=> ( X = one2 ) ) ).

% le_num_One_iff
tff(fact_114_two__realpow__ge__one,axiom,
    ! [N: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),N))) ).

% two_realpow_ge_one
tff(fact_115_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F3: fun(A,B),P: fun(A,bool),A2: A] :
          ( ! [X3: A] :
              ( ! [Y3: A] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,Y3)),aa(A,B,F3,X3)))
                 => pp(aa(A,bool,P,Y3)) )
             => pp(aa(A,bool,P,X3)) )
         => pp(aa(A,bool,P,A2)) ) ) ).

% measure_induct
tff(fact_116_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F3: fun(A,B),P: fun(A,bool),A2: A] :
          ( ! [X3: A] :
              ( ! [Y3: A] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,Y3)),aa(A,B,F3,X3)))
                 => pp(aa(A,bool,P,Y3)) )
             => pp(aa(A,bool,P,X3)) )
         => pp(aa(A,bool,P,A2)) ) ) ).

% measure_induct_rule
tff(fact_117_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa(nat,nat,suc,X) = aa(nat,nat,suc,Y) )
     => ( X = Y ) ) ).

% Suc_inject
tff(fact_118_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa(nat,nat,suc,N) ).

% n_not_Suc_n
tff(fact_119_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ).

% nat_neq_iff
tff(fact_120_less__not__refl,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).

% less_not_refl
tff(fact_121_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( M != N ) ) ).

% less_not_refl2
tff(fact_122_less__not__refl3,axiom,
    ! [S: nat,T2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),S),T2))
     => ( S != T2 ) ) ).

% less_not_refl3
tff(fact_123_less__irrefl__nat,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N)) ).

% less_irrefl_nat
tff(fact_124_nat__less__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N2))
             => pp(aa(nat,bool,P,M2)) )
         => pp(aa(nat,bool,P,N2)) )
     => pp(aa(nat,bool,P,N)) ) ).

% nat_less_induct
tff(fact_125_infinite__descent,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( ! [N2: nat] :
          ( ~ pp(aa(nat,bool,P,N2))
         => ? [M2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N2))
              & ~ pp(aa(nat,bool,P,M2)) ) )
     => pp(aa(nat,bool,P,N)) ) ).

% infinite_descent
tff(fact_126_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),X)) ) ) ).

% linorder_neqE_nat
tff(fact_127_infinite__descent__measure,axiom,
    ! [A: $tType,P: fun(A,bool),V2: fun(A,nat),X: A] :
      ( ! [X3: A] :
          ( ~ pp(aa(A,bool,P,X3))
         => ? [Y3: A] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,V2,Y3)),aa(A,nat,V2,X3)))
              & ~ pp(aa(A,bool,P,Y3)) ) )
     => pp(aa(A,bool,P,X)) ) ).

% infinite_descent_measure
tff(fact_128_le__refl,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N)) ).

% le_refl
tff(fact_129_le__trans,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),K)) ) ) ).

% le_trans
tff(fact_130_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% eq_imp_le
tff(fact_131_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
       => ( M = N ) ) ) ).

% le_antisym
tff(fact_132_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
      | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ).

% nat_le_linear
tff(fact_133_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun(nat,bool),K: nat,B2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [Y4: nat] :
            ( pp(aa(nat,bool,P,Y4))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y4),B2)) )
       => ? [X3: nat] :
            ( pp(aa(nat,bool,P,X3))
            & ! [Y3: nat] :
                ( pp(aa(nat,bool,P,Y3))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y3),X3)) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_134_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,nat,size_size(A),X) != aa(A,nat,size_size(A),Y) )
         => ( X != Y ) ) ) ).

% size_neq_size_imp_neq
tff(fact_135_Nat_OlessE,axiom,
    ! [I2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),K))
     => ( ( K != aa(nat,nat,suc,I2) )
       => ~ ! [J2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
             => ( K != aa(nat,nat,suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_136_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_lessD
tff(fact_137_Suc__lessE,axiom,
    ! [I2: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),K))
     => ~ ! [J2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J2))
           => ( K != aa(nat,nat,suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_138_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( ( aa(nat,nat,suc,M) != N )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),N)) ) ) ).

% Suc_lessI
tff(fact_139_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ( M = N ) ) ) ).

% less_SucE
tff(fact_140_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N))) ) ).

% less_SucI
tff(fact_141_Ex__less__Suc,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ? [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,suc,N)))
          & pp(aa(nat,bool,P,I3)) )
    <=> ( pp(aa(nat,bool,P,N))
        | ? [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
            & pp(aa(nat,bool,P,I3)) ) ) ) ).

% Ex_less_Suc
tff(fact_142_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | ( M = N ) ) ) ).

% less_Suc_eq
tff(fact_143_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M))) ) ).

% not_less_eq
tff(fact_144_All__less__Suc,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,suc,N)))
         => pp(aa(nat,bool,P,I3)) )
    <=> ( pp(aa(nat,bool,P,N))
        & ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
           => pp(aa(nat,bool,P,I3)) ) ) ) ).

% All_less_Suc
tff(fact_145_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
    <=> ? [M3: nat] :
          ( ( M = aa(nat,nat,suc,M3) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M3)) ) ) ).

% Suc_less_eq2
tff(fact_146_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M)))
       => ( M = N ) ) ) ).

% less_antisym
tff(fact_147_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_less_SucD
tff(fact_148_less__trans__Suc,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),K)) ) ) ).

% less_trans_Suc
tff(fact_149_less__Suc__induct,axiom,
    ! [I2: nat,J: nat,P: fun(nat,fun(nat,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => ( ! [I4: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I4),aa(nat,nat,suc,I4)))
       => ( ! [I4: nat,J2: nat,K2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),J2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),K2))
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I4),J2))
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,J2),K2))
                   => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I4),K2)) ) ) ) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,I2),J)) ) ) ) ).

% less_Suc_induct
tff(fact_150_strict__inc__induct,axiom,
    ! [I2: nat,J: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => ( ! [I4: nat] :
            ( ( J = aa(nat,nat,suc,I4) )
           => pp(aa(nat,bool,P,I4)) )
       => ( ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),J))
             => ( pp(aa(nat,bool,P,aa(nat,nat,suc,I4)))
               => pp(aa(nat,bool,P,I4)) ) )
         => pp(aa(nat,bool,P,I2)) ) ) ) ).

% strict_inc_induct
tff(fact_151_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M)))
      <=> ( N = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_152_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% Suc_leD
tff(fact_153_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( M = aa(nat,nat,suc,N) ) ) ) ).

% le_SucE
tff(fact_154_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N))) ) ).

% le_SucI
tff(fact_155_Suc__le__D,axiom,
    ! [N: nat,M4: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M4))
     => ? [M5: nat] : M4 = aa(nat,nat,suc,M5) ) ).

% Suc_le_D
tff(fact_156_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
        | ( M = aa(nat,nat,suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_157_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),N)) ).

% Suc_n_not_le_n
tff(fact_158_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M)) ) ).

% not_less_eq_eq
tff(fact_159_full__nat__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M2)),N2))
             => pp(aa(nat,bool,P,M2)) )
         => pp(aa(nat,bool,P,N2)) )
     => pp(aa(nat,bool,P,N)) ) ).

% full_nat_induct
tff(fact_160_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,P,M))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N2))
             => ( pp(aa(nat,bool,P,N2))
               => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_induct_at_least
tff(fact_161_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: fun(nat,fun(nat,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( ! [X3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),R,X3),X3))
       => ( ! [X3: nat,Y4: nat,Z2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),R,X3),Y4))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),R,Y4),Z2))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),R,X3),Z2)) ) )
         => ( ! [N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),R,N2),aa(nat,nat,suc,N2)))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),R,M),N)) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_162_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_163_add__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% add_Suc
tff(fact_164_add__Suc__shift,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,N)) ).

% add_Suc_shift
tff(fact_165_nat__less__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
        & ( M != N ) ) ) ).

% nat_less_le
tff(fact_166_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% less_imp_le_nat
tff(fact_167_le__eq__less__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | ( M = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_168_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | ( M = N ) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% less_or_eq_imp_le
tff(fact_169_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( ( M != N )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% le_neq_implies_less
tff(fact_170_less__mono__imp__le__mono,axiom,
    ! [F3: fun(nat,nat),I2: nat,J: nat] :
      ( ! [I4: nat,J2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),J2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,F3,I4)),aa(nat,nat,F3,J2))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,F3,I2)),aa(nat,nat,F3,J))) ) ) ).

% less_mono_imp_le_mono
tff(fact_171_add__lessD1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)),K))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),K)) ) ).

% add_lessD1
tff(fact_172_add__less__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L))) ) ) ).

% add_less_mono
tff(fact_173_not__add__less1,axiom,
    ! [I2: nat,J: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)),I2)) ).

% not_add_less1
tff(fact_174_not__add__less2,axiom,
    ! [J: nat,I2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I2)),I2)) ).

% not_add_less2
tff(fact_175_add__less__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ).

% add_less_mono1
tff(fact_176_trans__less__add1,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M))) ) ).

% trans_less_add1
tff(fact_177_trans__less__add2,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J))) ) ).

% trans_less_add2
tff(fact_178_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),L))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% less_add_eq_less
tff(fact_179_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N))
     => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ) ).

% add_leE
tff(fact_180_le__add1,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))) ).

% le_add1
tff(fact_181_le__add2,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ).

% le_add2
tff(fact_182_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% add_leD1
tff(fact_183_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ).

% add_leD2
tff(fact_184_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),L))
     => ? [N2: nat] : L = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N2) ) ).

% le_Suc_ex
tff(fact_185_add__le__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L))) ) ) ).

% add_le_mono
tff(fact_186_add__le__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ).

% add_le_mono1
tff(fact_187_trans__le__add1,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M))) ) ).

% trans_le_add1
tff(fact_188_trans__le__add2,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J))) ) ).

% trans_le_add2
tff(fact_189_nat__le__iff__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
    <=> ? [K3: nat] : N = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3) ) ).

% nat_le_iff_add
tff(fact_190_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),N: nat,N4: nat] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N2)),aa(nat,A,F3,aa(nat,nat,suc,N2))))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N4))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N)),aa(nat,A,F3,N4))) ) ) ) ).

% lift_Suc_mono_less
tff(fact_191_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),N: nat,M: nat] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N2)),aa(nat,A,F3,aa(nat,nat,suc,N2))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N)),aa(nat,A,F3,M)))
          <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_192_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),N: nat,N4: nat] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N2)),aa(nat,A,F3,aa(nat,nat,suc,N2))))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N4))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N)),aa(nat,A,F3,N4))) ) ) ) ).

% lift_Suc_mono_le
tff(fact_193_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),N: nat,N4: nat] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,aa(nat,nat,suc,N2))),aa(nat,A,F3,N2)))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N4))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N4)),aa(nat,A,F3,N))) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_194_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N))) ) ).

% le_imp_less_Suc
tff(fact_195_less__eq__Suc__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),M)) ) ).

% less_eq_Suc_le
tff(fact_196_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% less_Suc_eq_le
tff(fact_197_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M)))
      <=> ( N = M ) ) ) ).

% le_less_Suc_eq
tff(fact_198_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_le_lessD
tff(fact_199_inc__induct,axiom,
    ! [I2: nat,J: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( pp(aa(nat,bool,P,J))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),J))
               => ( pp(aa(nat,bool,P,aa(nat,nat,suc,N2)))
                 => pp(aa(nat,bool,P,N2)) ) ) )
         => pp(aa(nat,bool,P,I2)) ) ) ) ).

% inc_induct
tff(fact_200_dec__induct,axiom,
    ! [I2: nat,J: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( pp(aa(nat,bool,P,I2))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),J))
               => ( pp(aa(nat,bool,P,N2))
                 => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) ) )
         => pp(aa(nat,bool,P,J)) ) ) ) ).

% dec_induct
tff(fact_201_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_le_eq
tff(fact_202_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N)) ) ).

% Suc_leI
tff(fact_203_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ~ ! [Q2: nat] : N != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q2)) ) ).

% less_natE
tff(fact_204_less__add__Suc1,axiom,
    ! [I2: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),M)))) ).

% less_add_Suc1
tff(fact_205_less__add__Suc2,axiom,
    ! [I2: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I2)))) ).

% less_add_Suc2
tff(fact_206_less__iff__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
    <=> ? [K3: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3)) ) ).

% less_iff_Suc_add
tff(fact_207_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ? [K2: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2)) ) ).

% less_imp_Suc_add
tff(fact_208_mono__nat__linear__lb,axiom,
    ! [F3: fun(nat,nat),M: nat,K: nat] :
      ( ! [M5: nat,N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N2))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,F3,M5)),aa(nat,nat,F3,N2))) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,F3,M)),K)),aa(nat,nat,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)))) ) ).

% mono_nat_linear_lb
tff(fact_209_Suc__eq__plus1,axiom,
    ! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) ).

% Suc_eq_plus1
tff(fact_210_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_211_Suc__eq__plus1__left,axiom,
    ! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),N) ).

% Suc_eq_plus1_left
tff(fact_212_field__sum__of__halves,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,X,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),divide_divide(A,X,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = X ) ).

% field_sum_of_halves
tff(fact_213_div__by__1,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : divide_divide(A,A2,one_one(A)) = A2 ) ).

% div_by_1
tff(fact_214_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% add_less_cancel_left
tff(fact_215_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% add_less_cancel_right
tff(fact_216_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% add_le_cancel_left
tff(fact_217_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% add_le_cancel_right
tff(fact_218_nth__mem,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,nth(A,Xs),N)),aa(list(A),set(A),set2(A),Xs))) ) ).

% nth_mem
tff(fact_219_list__ball__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A),P: fun(A,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
           => pp(aa(A,bool,P,X3)) )
       => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N))) ) ) ).

% list_ball_nth
tff(fact_220_in__set__conv__nth,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
    <=> ? [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
          & ( aa(nat,A,nth(A,Xs),I3) = X ) ) ) ).

% in_set_conv_nth
tff(fact_221_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),X: A] :
      ( ! [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I4))) )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => pp(aa(A,bool,P,X)) ) ) ).

% all_nth_imp_all_set
tff(fact_222_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X4)) )
    <=> ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I3))) ) ) ).

% all_set_conv_all_nth
tff(fact_223_gt__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_224_less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_225_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_226_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_227_real__arch__pow,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => ? [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),aa(nat,real,power_power(real,X),N2))) ) ).

% real_arch_pow
tff(fact_228_less__eq__real__def,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y))
    <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
        | ( X = Y ) ) ) ).

% less_eq_real_def
tff(fact_229_complete__real,axiom,
    ! [S2: set(real)] :
      ( ? [X2: real] : pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X2),S2))
     => ( ? [Z3: real] :
          ! [X3: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),S2))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),Z3)) )
       => ? [Y4: real] :
            ( ! [X2: real] :
                ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X2),S2))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),Y4)) )
            & ! [Z3: real] :
                ( ! [X3: real] :
                    ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),S2))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),Z3)) )
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),Z3)) ) ) ) ) ).

% complete_real
tff(fact_230_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_231_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X2: A] :
        ? [X_13: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),X_13)) ) ).

% linordered_field_no_ub
tff(fact_232_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X2: A] :
        ? [Y4: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X2)) ) ).

% linordered_field_no_lb
tff(fact_233_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [X: A] :
          ( ( one_one(A) = X )
        <=> ( X = one_one(A) ) ) ) ).

% one_reorient
tff(fact_234_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_235_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_236_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_237_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_238_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_239_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_240_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_241_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_242_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_243_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( ( I2 = J )
            & ( K = L ) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L) ) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_244_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_245_subset__code_I1_J,axiom,
    ! [A: $tType,Xs: list(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),B3))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),B3)) ) ) ).

% subset_code(1)
tff(fact_246_neq__if__length__neq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) != aa(list(A),nat,size_size(list(A)),Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_247_Ex__list__of__length,axiom,
    ! [A: $tType,N: nat] :
    ? [Xs2: list(A)] : aa(list(A),nat,size_size(list(A)),Xs2) = N ).

% Ex_list_of_length
tff(fact_248_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% add_le_imp_le_right
tff(fact_249_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% add_le_imp_le_left
tff(fact_250_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ? [C3: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3) ) ) ).

% le_iff_add
tff(fact_251_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_252_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ~ ! [C4: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C4) ) ) ).

% less_eqE
tff(fact_253_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_254_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2))) ) ) ) ).

% add_mono
tff(fact_255_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),J))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_256_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( ( I2 = J )
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_257_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),J))
            & ( K = L ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_258_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% add_less_imp_less_right
tff(fact_259_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% add_less_imp_less_left
tff(fact_260_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_261_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_262_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2))) ) ) ) ).

% add_strict_mono
tff(fact_263_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),J))
            & ( K = L ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_264_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( ( I2 = J )
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_265_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),J))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_266_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_267_length__induct,axiom,
    ! [A: $tType,P: fun(list(A),bool),Xs: list(A)] :
      ( ! [Xs2: list(A)] :
          ( ! [Ys2: list(A)] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Ys2)),aa(list(A),nat,size_size(list(A)),Xs2)))
             => pp(aa(list(A),bool,P,Ys2)) )
         => pp(aa(list(A),bool,P,Xs2)) )
     => pp(aa(list(A),bool,P,Xs)) ) ).

% length_induct
tff(fact_268_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2))) ) ) ) ).

% add_less_le_mono
tff(fact_269_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2))) ) ) ) ).

% add_le_less_mono
tff(fact_270_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),J))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_271_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I2: A,J: A,K: A,L: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),J))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L))) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_272_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),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_273_less__add__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)))) ) ).

% less_add_one
tff(fact_274_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys),I3) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_275_Skolem__list__nth,axiom,
    ! [A: $tType,K: nat,P: fun(nat,fun(A,bool))] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),K))
         => ? [X_1: A] : pp(aa(A,bool,aa(nat,fun(A,bool),P,I3),X_1)) )
    <=> ? [Xs3: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs3) = K )
          & ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),K))
             => pp(aa(A,bool,aa(nat,fun(A,bool),P,I3),aa(nat,A,nth(A,Xs3),I3))) ) ) ) ).

% Skolem_list_nth
tff(fact_276_nth__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
     => ( ! [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Ys),I4) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_277_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),B2)) ) ) ).

% discrete
tff(fact_278_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] : ~ vEBT_V5719532721284313246member(vEBT_vebt_buildup(N),X) ).

% buildup_nothing_in_leaf
tff(fact_279_low__def,axiom,
    ! [X: nat,N: nat] : vEBT_VEBT_low(X,N) = modulo_modulo(nat,X,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ).

% low_def
tff(fact_280_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X: nat] : ~ vEBT_VEBT_membermima(vEBT_vebt_buildup(N),X) ).

% buildup_nothing_in_min_max
tff(fact_281_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),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,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M) )
         => ( ( M = aa(nat,nat,suc,N) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
             => ( ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),X_13))
               => ( ! [X3: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                     => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_13)) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_282_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),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,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M) )
         => ( ( M = N )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
             => ( ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),X_13))
               => ( ! [X3: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                     => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_13)) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_283_dbl__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl(A,one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)) ) ) ).

% dbl_simps(3)
tff(fact_284_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,F3: fun(A,nat),N: nat] :
      ( pp(aa(A,bool,P,K))
     => ( ! [X3: A] :
            ( pp(aa(A,bool,P,X3))
           => ? [Y3: A] :
                ( pp(aa(A,bool,P,Y3))
                & ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,Y3)),aa(A,nat,F3,X3))) ) )
       => ? [Y4: A] :
            ( pp(aa(A,bool,P,Y4))
            & ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,Y4)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F3,K)),N))) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_285_power__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num,L: num] : aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),K)),aa(num,nat,numeral_numeral(nat),L)) = aa(num,A,numeral_numeral(A),pow(K,L)) ) ).

% power_numeral
tff(fact_286_valid__eq2,axiom,
    ! [T2: vEBT_VEBT,D2: nat] :
      ( vEBT_VEBT_valid(T2,D2)
     => vEBT_invar_vebt(T2,D2) ) ).

% valid_eq2
tff(fact_287_valid__eq,axiom,
    ! [T2: vEBT_VEBT,D2: nat] :
      ( vEBT_VEBT_valid(T2,D2)
    <=> vEBT_invar_vebt(T2,D2) ) ).

% valid_eq
tff(fact_288_valid__eq1,axiom,
    ! [T2: vEBT_VEBT,D2: nat] :
      ( vEBT_invar_vebt(T2,D2)
     => vEBT_VEBT_valid(T2,D2) ) ).

% valid_eq1
tff(fact_289_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : divide_divide(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,V)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W))) = divide_divide(int,aa(num,int,numeral_numeral(int),V),aa(num,int,numeral_numeral(int),W)) ).

% zdiv_numeral_Bit0
tff(fact_290_mod__mod__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,modulo_modulo(A,A2,B2),B2) = modulo_modulo(A,A2,B2) ) ).

% mod_mod_trivial
tff(fact_291_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_292_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_293_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( modulo_modulo(nat,M,N) = M ) ) ).

% mod_less
tff(fact_294_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),aa(num,num,bit0,K)) ) ).

% dbl_simps(5)
tff(fact_295_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),aa(num,num,bit0,one2))) = one_one(A) ) ) ).

% bits_one_mod_two_eq_one
tff(fact_296_one__mod__two__eq__one,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ).

% one_mod_two_eq_one
tff(fact_297_mod2__Suc__Suc,axiom,
    ! [M: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,M)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% mod2_Suc_Suc
tff(fact_298_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_299_mod__add__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,A4: A,B2: A,B4: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A4,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B4,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),A4),B4),C2) ) ) ) ) ).

% mod_add_cong
tff(fact_300_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_301_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_302_power__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,N: nat] : modulo_modulo(A,aa(nat,A,power_power(A,modulo_modulo(A,A2,B2)),N),B2) = modulo_modulo(A,aa(nat,A,power_power(A,A2),N),B2) ) ).

% power_mod
tff(fact_303_mod__Suc__eq,axiom,
    ! [M: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,modulo_modulo(nat,M,N)),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% mod_Suc_eq
tff(fact_304_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,modulo_modulo(nat,M,N))),N) = modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,M)),N) ).

% mod_Suc_Suc_eq
tff(fact_305_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),modulo_modulo(nat,M,N)),M)) ).

% mod_less_eq_dividend
tff(fact_306_cong__exp__iff__simps_I9_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q3: num,N: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,M)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Q3)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q3)) ) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_307_cong__exp__iff__simps_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),one2)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),one2)) ) ).

% cong_exp_iff_simps(4)
tff(fact_308_cong__exp__iff__simps_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q3: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,M)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) ) ).

% cong_exp_iff_simps(8)
tff(fact_309_cong__exp__iff__simps_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q3: num,N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) ) ).

% cong_exp_iff_simps(6)
tff(fact_310_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_311_mod__induct,axiom,
    ! [P: fun(nat,bool),N: nat,P2: nat,M: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),P2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),P2))
         => ( ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),P2))
               => ( pp(aa(nat,bool,P,N2))
                 => pp(aa(nat,bool,P,modulo_modulo(nat,aa(nat,nat,suc,N2),P2))) ) )
           => pp(aa(nat,bool,P,M)) ) ) ) ) ).

% mod_induct
tff(fact_312_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),modulo_modulo(nat,M,aa(nat,nat,suc,N))),N)) ).

% mod_Suc_le_divisor
tff(fact_313_dbl__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl(A,X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X) ) ).

% dbl_def
tff(fact_314_pow_Osimps_I1_J,axiom,
    ! [X: num] : pow(X,one2) = X ).

% pow.simps(1)
tff(fact_315_bounded__Max__nat,axiom,
    ! [P: fun(nat,bool),X: nat,M6: nat] :
      ( pp(aa(nat,bool,P,X))
     => ( ! [X3: nat] :
            ( pp(aa(nat,bool,P,X3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),M6)) )
       => ~ ! [M5: nat] :
              ( pp(aa(nat,bool,P,M5))
             => ~ ! [X2: nat] :
                    ( pp(aa(nat,bool,P,X2))
                   => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X2),M5)) ) ) ) ) ).

% bounded_Max_nat
tff(fact_316_ex__has__least__nat,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,M: fun(A,nat)] :
      ( pp(aa(A,bool,P,K))
     => ? [X3: A] :
          ( pp(aa(A,bool,P,X3))
          & ! [Y3: A] :
              ( pp(aa(A,bool,P,Y3))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,M,X3)),aa(A,nat,M,Y3))) ) ) ) ).

% ex_has_least_nat
tff(fact_317_div__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat,M: nat] : modulo_modulo(A,divide_divide(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) = divide_divide(A,modulo_modulo(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) ) ).

% div_exp_mod_exp_eq
tff(fact_318_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P: fun(A,bool),K: A,F3: fun(A,nat),B2: nat] :
      ( pp(aa(A,bool,P,K))
     => ( ! [Y4: A] :
            ( pp(aa(A,bool,P,Y4))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,Y4)),B2)) )
       => ? [X3: A] :
            ( pp(aa(A,bool,P,X3))
            & ! [Y3: A] :
                ( pp(aa(A,bool,P,Y3))
               => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,Y3)),aa(A,nat,F3,X3))) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_319_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => vEBT_invar_vebt(vEBT_vebt_buildup(N),N) ) ).

% buildup_gives_valid
tff(fact_320_psubsetI,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( ( A3 != B3 )
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3)) ) ) ).

% psubsetI
tff(fact_321_subset__antisym,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
       => ( A3 = B3 ) ) ) ).

% subset_antisym
tff(fact_322_subsetI,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B3)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ).

% subsetI
tff(fact_323_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y2: num] :
      ( ( aa(num,num,bit0,X22) = aa(num,num,bit0,Y2) )
    <=> ( X22 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_324_order__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X)) ) ).

% order_refl
tff(fact_325_dual__order_Orefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),A2)) ) ).

% dual_order.refl
tff(fact_326_mod2__gr__0,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
    <=> ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_327_length__subseqs,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),subseqs(A,Xs)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_subseqs
tff(fact_328_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),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,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,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] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)))
                   => ( ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),X_1))
                    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),I4)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                       => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_13)) ) )
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),Ma))
                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I4: nat] :
                              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)))
                             => ( ( ( vEBT_VEBT_high(Ma,N) = I4 )
                                 => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),vEBT_VEBT_low(Ma,N))) )
                                & ! [X3: nat] :
                                    ( ( ( vEBT_VEBT_high(X3,N) = I4 )
                                      & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),vEBT_VEBT_low(X3,N))) )
                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X3))
                                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),Ma)) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_329_deg__not__0,axiom,
    ! [T2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(T2,N)
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% deg_not_0
tff(fact_330__C5_Ohyps_C_I12_J,axiom,
    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),ma)) = info ).

% "5.hyps"(12)
tff(fact_331_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),zero_zero(A)))
        <=> ( N = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_332_not__gr__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N))
        <=> ( N = zero_zero(A) ) ) ) ).

% not_gr_zero
tff(fact_333_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_334_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Y: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_335_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_336_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_337_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_338_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_339_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_340_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_341_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_342_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_343_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_344_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_345_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_346_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_347_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : divide_divide(A,zero_zero(A),A2) = zero_zero(A) ) ).

% div_0
tff(fact_348_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_349_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_350_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_351_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,zero_zero(A),A2) = zero_zero(A) ) ).

% mod_0
tff(fact_352_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,zero_zero(A)) = A2 ) ).

% mod_by_0
tff(fact_353_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : modulo_modulo(A,A2,A2) = zero_zero(A) ) ).

% mod_self
tff(fact_354_less__nat__zero__code,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

% less_nat_zero_code
tff(fact_355_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% neq0_conv
tff(fact_356_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),A2)) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_357_le0,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).

% le0
tff(fact_358_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),A2)) ).

% bot_nat_0.extremum
tff(fact_359_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),zero_zero(nat)) = M ).

% Nat.add_0_right
tff(fact_360_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        & ( N = zero_zero(nat) ) ) ) ).

% add_is_0
tff(fact_361_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_362_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,N: 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,Mi),Ma)),Deg,TreeList,Summary),N)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),Ma))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg))) ) ) ).

% mi_ma_2_deg
tff(fact_363_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% add_le_same_cancel1
tff(fact_364_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% add_le_same_cancel2
tff(fact_365_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ).

% le_add_same_cancel1
tff(fact_366_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ).

% le_add_same_cancel2
tff(fact_367_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_368_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_369_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_370_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_371_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ).

% less_add_same_cancel2
tff(fact_372_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ).

% less_add_same_cancel1
tff(fact_373_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% add_less_same_cancel2
tff(fact_374_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% add_less_same_cancel1
tff(fact_375_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_376_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_377_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_378_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_379_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( ( A2 = zero_zero(A) )
           => ( divide_divide(A,A2,A2) = zero_zero(A) ) )
          & ( ( A2 != zero_zero(A) )
           => ( divide_divide(A,A2,A2) = one_one(A) ) ) ) ) ).

% divide_self_if
tff(fact_380_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_381_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_382_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_383_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_384_power__0__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,power_power(A,zero_zero(A)),aa(nat,nat,suc,N)) = zero_zero(A) ) ).

% power_0_Suc
tff(fact_385_power__zero__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K: num] : aa(nat,A,power_power(A,zero_zero(A)),aa(num,nat,numeral_numeral(nat),K)) = zero_zero(A) ) ).

% power_zero_numeral
tff(fact_386_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_387_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_388_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_389_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_390_power__Suc0__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,power_power(A,A2),aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% power_Suc0_right
tff(fact_391_less__Suc0,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,zero_zero(nat))))
    <=> ( N = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_392_zero__less__Suc,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,suc,N))) ).

% zero_less_Suc
tff(fact_393_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% add_gr_0
tff(fact_394_less__one,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),one_one(nat)))
    <=> ( N = zero_zero(nat) ) ) ).

% less_one
tff(fact_395_div__by__Suc__0,axiom,
    ! [M: nat] : divide_divide(nat,M,aa(nat,nat,suc,zero_zero(nat))) = M ).

% div_by_Suc_0
tff(fact_396_div__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( divide_divide(nat,M,N) = zero_zero(nat) ) ) ).

% div_less
tff(fact_397_power__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,power_power(nat,aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,suc,zero_zero(nat)) ).

% power_Suc_0
tff(fact_398_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M: nat] :
      ( ( aa(nat,nat,power_power(nat,X),M) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = zero_zero(nat) )
        | ( X = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_399_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,power_power(nat,X),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X))
        | ( N = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_400_mod__by__Suc__0,axiom,
    ! [M: nat] : modulo_modulo(nat,M,aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% mod_by_Suc_0
tff(fact_401_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,one_one(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% zero_le_divide_1_iff
tff(fact_402_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,one_one(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% divide_le_0_1_iff
tff(fact_403_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,one_one(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% zero_less_divide_1_iff
tff(fact_404_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),divide_divide(A,B2,A2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_405_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),divide_divide(A,B2,A2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_406_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,A2)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_407_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,A2)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_408_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,one_one(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% divide_less_0_1_iff
tff(fact_409_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,N: nat] :
          ( ( aa(nat,A,power_power(A,A2),N) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% power_eq_0_iff
tff(fact_410_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_411_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_412_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,A2)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_413_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,A2)),one_one(A)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_414_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,B2),M)),aa(nat,A,power_power(A,B2),N)))
            <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M)) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_415_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A2),N)),aa(nat,A,power_power(A,B2),N)))
              <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ) ) ).

% power_mono_iff
tff(fact_416_zero__eq__power2,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A] :
          ( ( aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% zero_eq_power2
tff(fact_417_one__div__two__eq__zero,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ).

% one_div_two_eq_zero
tff(fact_418_bits__1__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ).

% bits_1_div_2
tff(fact_419_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,B2),M)),aa(nat,A,power_power(A,B2),N)))
            <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)) ) ) ) ) ).

% power_decreasing_iff
tff(fact_420_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => ( ( aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
            <=> ( X = Y ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_421_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),zero_zero(A)))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% power2_less_eq_zero_iff
tff(fact_422_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% zero_less_power2
tff(fact_423_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_424_add__self__mod__2,axiom,
    ! [M: nat] : modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),M),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = zero_zero(nat) ).

% add_self_mod_2
tff(fact_425_psubset__trans,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),B3),C5))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),C5)) ) ) ).

% psubset_trans
tff(fact_426_psubsetD,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B3)) ) ) ).

% psubsetD
tff(fact_427_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [X: A] :
          ( ( zero_zero(A) = X )
        <=> ( X = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_428_subseqs__refl,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Xs),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs)))) ).

% subseqs_refl
tff(fact_429_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_430_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list(vEBT_VEBT),Vb: vEBT_VEBT,X: 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,Mi),Ma)),zero_zero(nat),Va,Vb),X)
    <=> ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_431_power__0__left,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(nat,A,power_power(A,zero_zero(A)),N) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,A,power_power(A,zero_zero(A)),N) = zero_zero(A) ) ) ) ) ).

% power_0_left
tff(fact_432_zero__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(nat,A,power_power(A,zero_zero(A)),N) = zero_zero(A) ) ) ) ).

% zero_power
tff(fact_433_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X)) ) ).

% zero_le
tff(fact_434_le__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),zero_zero(A))) ) ).

% le_numeral_extra(3)
tff(fact_435_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N))
        <=> ( N != zero_zero(A) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_436_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [M: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),N))
         => ( N != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_437_not__less__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N),zero_zero(A))) ) ).

% not_less_zero
tff(fact_438_gr__zeroI,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( ( N != zero_zero(A) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),N)) ) ) ).

% gr_zeroI
tff(fact_439_less__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),zero_zero(A))) ) ).

% less_numeral_extra(3)
tff(fact_440_field__lbound__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D1: A,D22: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D1))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D22))
           => ? [E: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),E),D1))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),E),D22)) ) ) ) ) ).

% field_lbound_gt_zero
tff(fact_441_zero__neq__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: num] : zero_zero(A) != aa(num,A,numeral_numeral(A),N) ) ).

% zero_neq_numeral
tff(fact_442_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_443_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_444_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_445_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_446_power__not__zero,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,N: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(nat,A,power_power(A,A2),N) != zero_zero(A) ) ) ) ).

% power_not_zero
tff(fact_447_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_448_vebt__buildup_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero(nat) )
     => ( ( X != aa(nat,nat,suc,zero_zero(nat)) )
       => ~ ! [Va2: nat] : X != aa(nat,nat,suc,aa(nat,nat,suc,Va2)) ) ) ).

% vebt_buildup.cases
tff(fact_449_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => ? [M5: nat] : N = aa(nat,nat,suc,M5) ) ).

% not0_implies_Suc
tff(fact_450_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_not_Suc
tff(fact_451_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_neq_Suc
tff(fact_452_Suc__neq__Zero,axiom,
    ! [M: nat] : aa(nat,nat,suc,M) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_453_zero__induct,axiom,
    ! [P: fun(nat,bool),K: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,P,aa(nat,nat,suc,N2)))
           => pp(aa(nat,bool,P,N2)) )
       => pp(aa(nat,bool,P,zero_zero(nat))) ) ) ).

% zero_induct
tff(fact_454_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,bool)),M: nat,N: nat] :
      ( ! [X3: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,X3),zero_zero(nat)))
     => ( ! [Y4: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,zero_zero(nat)),aa(nat,nat,suc,Y4)))
       => ( ! [X3: nat,Y4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,X3),Y4))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,aa(nat,nat,suc,X3)),aa(nat,nat,suc,Y4))) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M),N)) ) ) ) ).

% diff_induct
tff(fact_455_nat__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,P,N2))
           => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) )
       => pp(aa(nat,bool,P,N)) ) ) ).

% nat_induct
tff(fact_456_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_457_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa(nat,nat,suc,X22) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_458_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_459_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_460_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero(nat) != aa(nat,nat,suc,X22) ).

% nat.distinct(1)
tff(fact_461_infinite__descent0__measure,axiom,
    ! [A: $tType,V2: fun(A,nat),P: fun(A,bool),X: A] :
      ( ! [X3: A] :
          ( ( aa(A,nat,V2,X3) = zero_zero(nat) )
         => pp(aa(A,bool,P,X3)) )
     => ( ! [X3: A] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,V2,X3)))
           => ( ~ pp(aa(A,bool,P,X3))
             => ? [Y3: A] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,V2,Y3)),aa(A,nat,V2,X3)))
                  & ~ pp(aa(A,bool,P,Y3)) ) ) )
       => pp(aa(A,bool,P,X)) ) ) ).

% infinite_descent0_measure
tff(fact_462_infinite__descent0,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
           => ( ~ pp(aa(nat,bool,P,N2))
             => ? [M2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N2))
                  & ~ pp(aa(nat,bool,P,M2)) ) ) )
       => pp(aa(nat,bool,P,N)) ) ) ).

% infinite_descent0
tff(fact_463_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( N != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_464_less__zeroE,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

% less_zeroE
tff(fact_465_not__less0,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),zero_zero(nat))) ).

% not_less0
tff(fact_466_not__gr0,axiom,
    ! [N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
    <=> ( N = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_467_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% gr0I
tff(fact_468_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] : ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A2),zero_zero(nat))) ).

% bot_nat_0.extremum_strict
tff(fact_469_le__0__eq,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),zero_zero(nat)))
    <=> ( N = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_470_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),zero_zero(nat)))
     => ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_471_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),zero_zero(nat)))
    <=> ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_472_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),zero_zero(nat)),N)) ).

% less_eq_nat.simps(1)
tff(fact_473_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = M )
     => ( N = zero_zero(nat) ) ) ).

% add_eq_self_zero
tff(fact_474_plus__nat_Oadd__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),N) = N ).

% plus_nat.add_0
tff(fact_475_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat,B2: A] :
          ( ( aa(nat,A,power_power(A,A2),N) = aa(nat,A,power_power(A,B2),N) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
               => ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_476_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,A2: A,B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( ( aa(nat,A,power_power(A,A2),N) = aa(nat,A,power_power(A,B2),N) )
              <=> ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_477_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A2),N)),aa(nat,A,power_power(A,B2),N))) ) ) ) ) ).

% power_strict_mono
tff(fact_478_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% zero_le_numeral
tff(fact_479_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),zero_zero(A))) ) ).

% not_numeral_le_zero
tff(fact_480_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N)),zero_zero(A))) ) ).

% not_numeral_less_zero
tff(fact_481_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),N))) ) ).

% zero_less_numeral
tff(fact_482_zero__less__one__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) ) ).

% zero_less_one_class.zero_le_one
tff(fact_483_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),one_one(A))) ) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_484_not__one__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),zero_zero(A))) ) ).

% not_one_le_zero
tff(fact_485_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2)) ) ) ) ).

% add_decreasing
tff(fact_486_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2))) ) ) ) ).

% add_increasing
tff(fact_487_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2)) ) ) ) ).

% add_decreasing2
tff(fact_488_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2))) ) ) ) ).

% add_increasing2
tff(fact_489_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_490_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_491_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_492_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),zero_zero(A)))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y) = zero_zero(A) )
            <=> ( ( X = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_493_not__one__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),zero_zero(A))) ) ).

% not_one_less_zero
tff(fact_494_zero__less__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) ) ).

% zero_less_one
tff(fact_495_less__numeral__extra_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),one_one(A))) ) ).

% less_numeral_extra(1)
tff(fact_496_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2))) ) ) ) ).

% pos_add_strict
tff(fact_497_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ~ ! [C4: A] :
                ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C4) )
               => ( C4 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_498_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ) ) ).

% add_pos_pos
tff(fact_499_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A))) ) ) ) ).

% add_neg_neg
tff(fact_500_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A))) ) ) ) ).

% add_less_zeroD
tff(fact_501_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,A2,B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ) ) ).

% divide_le_0_iff
tff(fact_502_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2))) ) ) ) ).

% divide_right_mono
tff(fact_503_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,A2,B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_le_divide_iff
tff(fact_504_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,X,Y))) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_505_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Y)),zero_zero(A))) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_506_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Y)),zero_zero(A))) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_507_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,X,Y))) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_508_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C2)),divide_divide(A,A2,C2))) ) ) ) ).

% divide_right_mono_neg
tff(fact_509_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A2),N)),aa(nat,A,power_power(A,B2),N))) ) ) ) ).

% power_mono
tff(fact_510_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),N))) ) ) ).

% zero_le_power
tff(fact_511_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2))) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_512_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2))) ) ) ) ).

% divide_strict_right_mono
tff(fact_513_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,A2,B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_less_divide_iff
tff(fact_514_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) )
            & ( C2 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_515_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,A2,B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ) ).

% divide_less_0_iff
tff(fact_516_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,X,Y))) ) ) ) ).

% divide_pos_pos
tff(fact_517_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,X,Y)),zero_zero(A))) ) ) ) ).

% divide_pos_neg
tff(fact_518_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,X,Y)),zero_zero(A))) ) ) ) ).

% divide_neg_pos
tff(fact_519_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,X,Y))) ) ) ) ).

% divide_neg_neg
tff(fact_520_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,A2),N))) ) ) ).

% zero_less_power
tff(fact_521_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_522_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),modulo_modulo(A,A2,B2)),A2)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_523_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),modulo_modulo(A,A2,B2)),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_524_power__0,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A2: A] : aa(nat,A,power_power(A,A2),zero_zero(nat)) = one_one(A) ) ).

% power_0
tff(fact_525_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_526_Ex__less__Suc2,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ? [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,suc,N)))
          & pp(aa(nat,bool,P,I3)) )
    <=> ( pp(aa(nat,bool,P,zero_zero(nat)))
        | ? [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
            & pp(aa(nat,bool,P,aa(nat,nat,suc,I3))) ) ) ) ).

% Ex_less_Suc2
tff(fact_527_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
    <=> ? [M7: nat] : N = aa(nat,nat,suc,M7) ) ).

% gr0_conv_Suc
tff(fact_528_All__less__Suc2,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,suc,N)))
         => pp(aa(nat,bool,P,I3)) )
    <=> ( pp(aa(nat,bool,P,zero_zero(nat)))
        & ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
           => pp(aa(nat,bool,P,aa(nat,nat,suc,I3))) ) ) ) ).

% All_less_Suc2
tff(fact_529_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ? [M5: nat] : N = aa(nat,nat,suc,M5) ) ).

% gr0_implies_Suc
tff(fact_530_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,suc,N)))
    <=> ( ( M = zero_zero(nat) )
        | ? [J3: nat] :
            ( ( M = aa(nat,nat,suc,J3) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N)) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_531_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% add_is_1
tff(fact_532_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% one_is_add
tff(fact_533_ex__least__nat__le,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
       => ? [K2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
            & ! [I: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),K2))
               => ~ pp(aa(nat,bool,P,I)) )
            & pp(aa(nat,bool,P,K2)) ) ) ) ).

% ex_least_nat_le
tff(fact_534_less__imp__add__positive,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => ? [K2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K2))
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_535_div__less__mono,axiom,
    ! [A3: nat,B3: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( ( modulo_modulo(nat,A3,N) = zero_zero(nat) )
         => ( ( modulo_modulo(nat,B3,N) = zero_zero(nat) )
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),divide_divide(nat,A3,N)),divide_divide(nat,B3,N))) ) ) ) ) ).

% div_less_mono
tff(fact_536_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_537_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( divide_divide(nat,M,N) = zero_zero(nat) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | ( N = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_538_nat__power__less__imp__less,axiom,
    ! [I2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),I2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,power_power(nat,I2),M)),aa(nat,nat,power_power(nat,I2),N)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% nat_power_less_imp_less
tff(fact_539_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ( aa(nat,nat,suc,modulo_modulo(nat,M,N)) = N )
       => ( modulo_modulo(nat,aa(nat,nat,suc,M),N) = zero_zero(nat) ) )
      & ( ( aa(nat,nat,suc,modulo_modulo(nat,M,N)) != N )
       => ( modulo_modulo(nat,aa(nat,nat,suc,M),N) = aa(nat,nat,suc,modulo_modulo(nat,M,N)) ) ) ) ).

% mod_Suc
tff(fact_540_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),modulo_modulo(nat,M,N)),N)) ) ).

% mod_less_divisor
tff(fact_541_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_542_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
          <=> ( X = Y ) ) ) ) ).

% order_antisym_conv
tff(fact_543_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% linorder_le_cases
tff(fact_544_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F3: fun(A,B),C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( ( aa(A,B,F3,B2) = C2 )
           => ( ! [X3: A,Y4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y4))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y4))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,A2)),C2)) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_545_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C2: B] :
          ( ( A2 = aa(B,A,F3,B2) )
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C2))
           => ( ! [X3: B,Y4: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X3),Y4))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,F3,Y4))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(B,A,F3,C2))) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_546_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% linorder_linear
tff(fact_547_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
          | ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
          | ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% verit_la_disequality
tff(fact_548_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% order_eq_refl
tff(fact_549_order__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A2: A,B2: A,F3: fun(A,C),C2: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F3,B2)),C2))
           => ( ! [X3: A,Y4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y4))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F3,X3)),aa(A,C,F3,Y4))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F3,A2)),C2)) ) ) ) ) ).

% order_subst2
tff(fact_550_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(B,A,F3,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C2))
           => ( ! [X3: B,Y4: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X3),Y4))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,F3,Y4))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(B,A,F3,C2))) ) ) ) ) ).

% order_subst1
tff(fact_551_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_552_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G: fun(A,B)] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F3),G))
        <=> ! [X4: A] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,G,X4))) ) ) ).

% le_fun_def
tff(fact_553_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G: fun(A,B)] :
          ( ! [X3: A] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,G,X3)))
         => pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F3),G)) ) ) ).

% le_funI
tff(fact_554_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G: fun(A,B),X: A] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F3),G))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,G,X))) ) ) ).

% le_funE
tff(fact_555_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G: fun(A,B),X: A] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F3),G))
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,G,X))) ) ) ).

% le_funD
tff(fact_556_antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
           => ( A2 = B2 ) ) ) ) ).

% antisym
tff(fact_557_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2)) ) ) ) ).

% dual_order.trans
tff(fact_558_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( A2 = B2 ) ) ) ) ).

% dual_order.antisym
tff(fact_559_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% dual_order.eq_iff
tff(fact_560_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,bool)),A2: A,B2: A] :
          ( ! [A5: A,B5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),B5))
             => pp(aa(A,bool,aa(A,fun(A,bool),P,A5),B5)) )
         => ( ! [A5: A,B5: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),P,B5),A5))
               => pp(aa(A,bool,aa(A,fun(A,bool),P,A5),B5)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),P,A2),B2)) ) ) ) ).

% linorder_wlog
tff(fact_561_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z)) ) ) ) ).

% order_trans
tff(fact_562_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2)) ) ) ) ).

% order.trans
tff(fact_563_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
           => ( X = Y ) ) ) ) ).

% order_antisym
tff(fact_564_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( ( B2 = C2 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2)) ) ) ) ).

% ord_le_eq_trans
tff(fact_565_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = B2 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2)) ) ) ) ).

% ord_eq_le_trans
tff(fact_566_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ) ).

% order_class.order_eq_iff
tff(fact_567_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z)) )
         => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z))
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),Y)) )
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),Y))
                 => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) )
               => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z))
                   => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X)) )
                 => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X))
                     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_568_nle__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            & ( B2 != A2 ) ) ) ) ).

% nle_le
tff(fact_569_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),A2)) ) ).

% verit_comp_simplify1(2)
tff(fact_570_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% order_less_imp_not_less
tff(fact_571_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( Y != X ) ) ) ).

% order_less_imp_not_eq2
tff(fact_572_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( X != Y ) ) ) ).

% order_less_imp_not_eq
tff(fact_573_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
          | ( X = Y )
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% linorder_less_linear
tff(fact_574_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,P: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
           => pp(P) ) ) ) ).

% order_less_imp_triv
tff(fact_575_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% order_less_not_sym
tff(fact_576_order__less__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A2: A,B2: A,F3: fun(A,C),C2: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,B2)),C2))
           => ( ! [X3: A,Y4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y4))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,X3)),aa(A,C,F3,Y4))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,A2)),C2)) ) ) ) ) ).

% order_less_subst2
tff(fact_577_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F3,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C2))
           => ( ! [X3: B,Y4: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),Y4))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X3)),aa(B,A,F3,Y4))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F3,C2))) ) ) ) ) ).

% order_less_subst1
tff(fact_578_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X)) ) ).

% order_less_irrefl
tff(fact_579_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F3: fun(A,B),C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( ( aa(A,B,F3,B2) = C2 )
           => ( ! [X3: A,Y4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y4))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X3)),aa(A,B,F3,Y4))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,A2)),C2)) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_580_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C2: B] :
          ( ( A2 = aa(B,A,F3,B2) )
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C2))
           => ( ! [X3: B,Y4: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),Y4))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X3)),aa(B,A,F3,Y4))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F3,C2))) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_581_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z)) ) ) ) ).

% order_less_trans
tff(fact_582_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ).

% order_less_asym'
tff(fact_583_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).

% linorder_neq_iff
tff(fact_584_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% order_less_asym
tff(fact_585_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).

% linorder_neqE
tff(fact_586_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( A2 != B2 ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_587_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( A2 != B2 ) ) ) ).

% order.strict_implies_not_eq
tff(fact_588_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A2)) ) ) ) ).

% dual_order.strict_trans
tff(fact_589_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
            | ( X = Y ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_590_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2)) ) ) ) ).

% order.strict_trans
tff(fact_591_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,bool)),A2: A,B2: A] :
          ( ! [A5: A,B5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A5),B5))
             => pp(aa(A,bool,aa(A,fun(A,bool),P,A5),B5)) )
         => ( ! [A5: A] : pp(aa(A,bool,aa(A,fun(A,bool),P,A5),A5))
           => ( ! [A5: A,B5: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),P,B5),A5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),P,A5),B5)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),P,A2),B2)) ) ) ) ) ).

% linorder_less_wlog
tff(fact_592_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool)] :
          ( ? [X_1: A] : pp(aa(A,bool,P,X_1))
        <=> ? [N5: A] :
              ( pp(aa(A,bool,P,N5))
              & ! [M7: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M7),N5))
                 => ~ pp(aa(A,bool,P,M7)) ) ) ) ) ).

% exists_least_iff
tff(fact_593_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),A2)) ) ).

% dual_order.irrefl
tff(fact_594_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% dual_order.asym
tff(fact_595_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( ( X != Y )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ) ).

% linorder_cases
tff(fact_596_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
          <=> ( X = Y ) ) ) ) ).

% antisym_conv3
tff(fact_597_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,bool),A2: A] :
          ( ! [X3: A] :
              ( ! [Y3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y3),X3))
                 => pp(aa(A,bool,P,Y3)) )
             => pp(aa(A,bool,P,X3)) )
         => pp(aa(A,bool,P,A2)) ) ) ).

% less_induct
tff(fact_598_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( ( B2 = C2 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2)) ) ) ) ).

% ord_less_eq_trans
tff(fact_599_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = B2 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2)) ) ) ) ).

% ord_eq_less_trans
tff(fact_600_order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ).

% order.asym
tff(fact_601_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( X != Y ) ) ) ).

% less_imp_neq
tff(fact_602_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ? [Z2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),Y)) ) ) ) ).

% dense
tff(fact_603_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [X: A] :
        ? [X_13: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X_13)) ) ).

% gt_ex
tff(fact_604_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [X: A] :
        ? [Y4: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y4),X)) ) ).

% lt_ex
tff(fact_605_verit__comp__simplify1_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),A2)) ) ).

% verit_comp_simplify1(1)
tff(fact_606_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2))) ) ) ) ).

% add_strict_increasing2
tff(fact_607_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2))) ) ) ) ).

% add_strict_increasing
tff(fact_608_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ) ) ).

% add_pos_nonneg
tff(fact_609_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A))) ) ) ) ).

% add_nonpos_neg
tff(fact_610_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ) ) ).

% add_nonneg_pos
tff(fact_611_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A))) ) ) ) ).

% add_neg_nonpos
tff(fact_612_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( ! [E: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),E))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% field_le_epsilon
tff(fact_613_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Y)),zero_zero(A))) ) ) ) ).

% divide_nonpos_pos
tff(fact_614_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,X,Y))) ) ) ) ).

% divide_nonpos_neg
tff(fact_615_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,X,Y))) ) ) ) ).

% divide_nonneg_pos
tff(fact_616_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Y)),zero_zero(A))) ) ) ) ).

% divide_nonneg_neg
tff(fact_617_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,A2,C2)),divide_divide(A,B2,C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ) ).

% divide_le_cancel
tff(fact_618_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,W: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W),Z))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,X,Z)),divide_divide(A,Y,W))) ) ) ) ) ) ).

% frac_less2
tff(fact_619_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,W: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W),Z))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,X,Z)),divide_divide(A,Y,W))) ) ) ) ) ) ).

% frac_less
tff(fact_620_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,W: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),W))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W),Z))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Z)),divide_divide(A,Y,W))) ) ) ) ) ) ).

% frac_le
tff(fact_621_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,A2,B2))) ) ) ) ).

% div_positive
tff(fact_622_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( divide_divide(A,A2,B2) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_623_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A2),N)),aa(nat,A,power_power(A,B2),N)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).

% power_less_imp_less_base
tff(fact_624_zero__less__two,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),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_625_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A2),N)),one_one(A))) ) ) ) ).

% power_le_one
tff(fact_626_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),divide_divide(A,B2,A2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ) ).

% less_divide_eq_1
tff(fact_627_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,A2)),one_one(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_628_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_629_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_630_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat,B2: A] :
          ( ( aa(nat,A,power_power(A,A2),aa(nat,nat,suc,N)) = aa(nat,A,power_power(A,B2),aa(nat,nat,suc,N)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( A2 = B2 ) ) ) ) ) ).

% power_inject_base
tff(fact_631_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A2),aa(nat,nat,suc,N))),aa(nat,A,power_power(A,B2),aa(nat,nat,suc,N))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% power_le_imp_le_base
tff(fact_632_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( modulo_modulo(A,A2,B2) = A2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_633_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),modulo_modulo(A,A2,B2))) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_634_in__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),X: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B3)) ) ) ).

% in_mono
tff(fact_635_subsetD,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),A3))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),C2),B3)) ) ) ).

% subsetD
tff(fact_636_equalityE,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( A3 = B3 )
     => ~ ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
         => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3)) ) ) ).

% equalityE
tff(fact_637_subset__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),B3)) ) ) ).

% subset_eq
tff(fact_638_equalityD1,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( A3 = B3 )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ).

% equalityD1
tff(fact_639_equalityD2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( A3 = B3 )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3)) ) ).

% equalityD2
tff(fact_640_subset__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
    <=> ! [T3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),T3),A3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),T3),B3)) ) ) ).

% subset_iff
tff(fact_641_subset__refl,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),A3)) ).

% subset_refl
tff(fact_642_Collect__mono,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P,X3))
         => pp(aa(A,bool,Q,X3)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,P)),collect(A,Q))) ) ).

% Collect_mono
tff(fact_643_subset__trans,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C5))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C5)) ) ) ).

% subset_trans
tff(fact_644_set__eq__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( A3 = B3 )
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3)) ) ) ).

% set_eq_subset
tff(fact_645_Collect__mono__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,P)),collect(A,Q)))
    <=> ! [X4: A] :
          ( pp(aa(A,bool,P,X4))
         => pp(aa(A,bool,Q,X4)) ) ) ).

% Collect_mono_iff
tff(fact_646_subset__iff__psubset__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
        | ( A3 = B3 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_647_subset__psubset__trans,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),B3),C5))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),C5)) ) ) ).

% subset_psubset_trans
tff(fact_648_subset__not__subset__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
        & ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3)) ) ) ).

% subset_not_subset_eq
tff(fact_649_psubset__subset__trans,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C5))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),C5)) ) ) ).

% psubset_subset_trans
tff(fact_650_psubset__imp__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ).

% psubset_imp_subset
tff(fact_651_psubset__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
        & ( A3 != B3 ) ) ) ).

% psubset_eq
tff(fact_652_psubsetE,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
     => ~ ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3)) ) ) ).

% psubsetE
tff(fact_653_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num,Q3: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) = zero_zero(A) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q3)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(2)
tff(fact_654_cong__exp__iff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),one2)) = zero_zero(A) ) ).

% cong_exp_iff_simps(1)
tff(fact_655_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_656_num_Osize_I5_J,axiom,
    ! [X22: num] : aa(num,nat,size_size(num),aa(num,num,bit0,X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(5)
tff(fact_657_ex__least__nat__less,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,N))
     => ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
       => ? [K2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),N))
            & ! [I: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I),K2))
               => ~ pp(aa(nat,bool,P,I)) )
            & pp(aa(nat,bool,P,aa(nat,nat,suc,K2))) ) ) ) ).

% ex_least_nat_less
tff(fact_658_length__pos__if__in__set,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% length_pos_if_in_set
tff(fact_659_nat__induct__non__zero,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,P,one_one(nat)))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
             => ( pp(aa(nat,bool,P,N2))
               => pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_induct_non_zero
tff(fact_660_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(nat,nat,power_power(nat,N),K))) ) ).

% power_gt_expt
tff(fact_661_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),divide_divide(nat,M,N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% div_greater_zero_iff
tff(fact_662_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),divide_divide(nat,K,N)),divide_divide(nat,K,M))) ) ) ).

% div_le_mono2
tff(fact_663_nat__one__le__power,axiom,
    ! [I2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),I2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,power_power(nat,I2),N))) ) ).

% nat_one_le_power
tff(fact_664_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),modulo_modulo(nat,M,N)),N)) ) ).

% mod_le_divisor
tff(fact_665_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( ( divide_divide(nat,M,N) = M )
      <=> ( N = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_666_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),divide_divide(nat,M,N)),M)) ) ) ).

% div_less_dividend
tff(fact_667_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_668_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),divide_divide(A,B2,A2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ) ).

% le_divide_eq_1
tff(fact_669_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,A2)),one_one(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_670_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A2),aa(nat,nat,suc,N))),A2)) ) ) ) ).

% power_Suc_le_self
tff(fact_671_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A2),aa(nat,nat,suc,N))),one_one(A))) ) ) ) ).

% power_Suc_less_one
tff(fact_672_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N3: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A2),N3)),aa(nat,A,power_power(A,A2),N))) ) ) ) ) ).

% power_strict_decreasing
tff(fact_673_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N3: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A2),N3)),aa(nat,A,power_power(A,A2),N))) ) ) ) ) ).

% power_decreasing
tff(fact_674_zero__power2,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,power_power(A,zero_zero(A)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ).

% zero_power2
tff(fact_675_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(nat,A,power_power(A,A2),N))) ) ) ) ).

% self_le_power
tff(fact_676_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(nat,A,power_power(A,A2),N))) ) ) ) ).

% one_less_power
tff(fact_677_numeral__2__eq__2,axiom,
    aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% numeral_2_eq_2
tff(fact_678_verit__le__mono__div,axiom,
    ! [A3: nat,B3: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A3),B3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),divide_divide(nat,A3,N)),if(nat,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),modulo_modulo(nat,B3,N)),zero_zero(nat)),one_one(nat),zero_zero(nat)))),divide_divide(nat,B3,N))) ) ) ).

% verit_le_mono_div
tff(fact_679_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% half_gt_zero_iff
tff(fact_680_half__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ) ).

% half_gt_zero
tff(fact_681_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% power2_le_imp_le
tff(fact_682_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( ( aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
             => ( X = Y ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_683_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% zero_le_power2
tff(fact_684_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),zero_zero(A))) ) ).

% power2_less_0
tff(fact_685_exp__add__not__zero__imp__right,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) != zero_zero(A) )
         => ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_right
tff(fact_686_exp__add__not__zero__imp__left,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) != zero_zero(A) )
         => ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_left
tff(fact_687_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
    <=> ( ( N = zero_zero(nat) )
        | ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases_iff
tff(fact_688_less__2__cases,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))
     => ( ( N = zero_zero(nat) )
        | ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases
tff(fact_689_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).

% power2_less_imp_less
tff(fact_690_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),zero_zero(A)))
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_691_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% sum_power2_ge_zero
tff(fact_692_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))
        <=> ( ( X != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_693_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),zero_zero(A))) ) ).

% not_sum_power2_lt_zero
tff(fact_694_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( ( divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2)))) = zero_zero(A) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_695_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
            | ( X = Y ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_696_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% linorder_le_less_linear
tff(fact_697_order__less__le__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A2: A,B2: A,F3: fun(A,C),C2: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F3,B2)),C2))
           => ( ! [X3: A,Y4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y4))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,X3)),aa(A,C,F3,Y4))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,A2)),C2)) ) ) ) ) ).

% order_less_le_subst2
tff(fact_698_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F3,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B2),C2))
           => ( ! [X3: B,Y4: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),X3),Y4))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,F3,Y4))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F3,C2))) ) ) ) ) ).

% order_less_le_subst1
tff(fact_699_order__le__less__subst2,axiom,
    ! [A: $tType,C: $tType] :
      ( ( order(C)
        & order(A) )
     => ! [A2: A,B2: A,F3: fun(A,C),C2: C] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,B2)),C2))
           => ( ! [X3: A,Y4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y4))
                 => pp(aa(C,bool,aa(C,fun(C,bool),ord_less_eq(C),aa(A,C,F3,X3)),aa(A,C,F3,Y4))) )
             => pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),aa(A,C,F3,A2)),C2)) ) ) ) ) ).

% order_le_less_subst2
tff(fact_700_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F3: fun(B,A),B2: B,C2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(B,A,F3,B2)))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),C2))
           => ( ! [X3: B,Y4: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),Y4))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X3)),aa(B,A,F3,Y4))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F3,C2))) ) ) ) ) ).

% order_le_less_subst1
tff(fact_701_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z)) ) ) ) ).

% order_less_le_trans
tff(fact_702_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z)) ) ) ) ).

% order_le_less_trans
tff(fact_703_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != B2 )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).

% order_neq_le_trans
tff(fact_704_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( ( A2 != B2 )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).

% order_le_neq_trans
tff(fact_705_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% order_less_imp_le
tff(fact_706_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% linorder_not_less
tff(fact_707_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% linorder_not_le
tff(fact_708_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & ( X != Y ) ) ) ) ).

% order_less_le
tff(fact_709_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
            | ( X = Y ) ) ) ) ).

% order_le_less
tff(fact_710_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% dual_order.strict_implies_order
tff(fact_711_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% order.strict_implies_order
tff(fact_712_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_713_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A2)) ) ) ) ).

% dual_order.strict_trans2
tff(fact_714_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A2)) ) ) ) ).

% dual_order.strict_trans1
tff(fact_715_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_716_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
            | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_717_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( ! [W2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),W2))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W2),Y))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),W2),Z)) ) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z)) ) ) ) ).

% dense_le_bounded
tff(fact_718_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),X))
         => ( ! [W2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),W2))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),W2),X))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),W2)) ) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z)) ) ) ) ).

% dense_ge_bounded
tff(fact_719_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% order.strict_iff_not
tff(fact_720_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2)) ) ) ) ).

% order.strict_trans2
tff(fact_721_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2)) ) ) ) ).

% order.strict_trans1
tff(fact_722_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
tff(fact_723_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
            | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
tff(fact_724_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% not_le_imp_less
tff(fact_725_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ) ).

% less_le_not_le
tff(fact_726_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y: A,Z: A] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z)) ) ) ).

% dense_le
tff(fact_727_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Y: A] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),X3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z)) ) ) ).

% dense_ge
tff(fact_728_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
          <=> ( X = Y ) ) ) ) ).

% antisym_conv2
tff(fact_729_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
          <=> ( X = Y ) ) ) ) ).

% antisym_conv1
tff(fact_730_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            | ( A2 = B2 ) ) ) ) ).

% nless_le
tff(fact_731_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% leI
tff(fact_732_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% leD
tff(fact_733_verit__comp__simplify1_I3_J,axiom,
    ! [B: $tType] :
      ( linorder(B)
     => ! [B4: B,A4: B] :
          ( ~ pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),B4),A4))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),A4),B4)) ) ) ).

% verit_comp_simplify1(3)
tff(fact_734_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),divide_divide(nat,aa(nat,nat,suc,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_735_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% div_2_gt_zero
tff(fact_736_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] : one2 != aa(num,num,bit0,X22) ).

% verit_eq_simplify(10)
tff(fact_737_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),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,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M) )
         => ( ( M = N )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
             => ( ! [I4: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)))
                   => ( ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),X_1))
                    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary),I4)) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
                       => ~ ? [X_13: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X3),X_13)) ) )
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi),Ma))
                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg)))
                     => ( ( ( Mi != Ma )
                         => ! [I4: nat] :
                              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)))
                             => ( ( ( vEBT_VEBT_high(Ma,N) = I4 )
                                 => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),vEBT_VEBT_low(Ma,N))) )
                                & ! [X3: nat] :
                                    ( ( ( vEBT_VEBT_high(X3,N) = I4 )
                                      & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),vEBT_VEBT_low(X3,N))) )
                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi),X3))
                                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),Ma)) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Mi),Ma)),Deg,TreeList,Summary),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_738_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) != aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = zero_zero(nat) ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_739_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),aa(num,num,bit0,one2))) != one_one(A) )
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_740_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),aa(num,num,bit0,one2))) != zero_zero(A) )
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_741_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
    <=> ( ( ? [A6: bool,B6: bool] : A1 = vEBT_Leaf(A6,B6)
          & ( A22 = aa(nat,nat,suc,zero_zero(nat)) ) )
        | ? [TreeList2: list(vEBT_VEBT),N5: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
               => vEBT_invar_vebt(X4,N5) )
            & vEBT_invar_vebt(Summary2,N5)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N5) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),N5) )
            & ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),X_1))
            & ! [X4: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
               => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_1)) ) )
        | ? [TreeList2: list(vEBT_VEBT),N5: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
               => vEBT_invar_vebt(X4,N5) )
            & vEBT_invar_vebt(Summary2,aa(nat,nat,suc,N5))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,N5)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),aa(nat,nat,suc,N5)) )
            & ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),X_1))
            & ! [X4: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
               => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_1)) ) )
        | ? [TreeList2: list(vEBT_VEBT),N5: nat,Summary2: vEBT_VEBT,Mi2: 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,Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
               => vEBT_invar_vebt(X4,N5) )
            & vEBT_invar_vebt(Summary2,N5)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N5) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),N5) )
            & ! [I3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N5)))
               => ( ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),X_1))
                <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),I3)) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X4: vEBT_VEBT] :
                  ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                 => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_1)) ) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi2),Ma2))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A22)))
            & ( ( Mi2 != Ma2 )
             => ! [I3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N5)))
                 => ( ( ( vEBT_VEBT_high(Ma2,N5) = I3 )
                     => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(Ma2,N5))) )
                    & ! [X4: nat] :
                        ( ( ( vEBT_VEBT_high(X4,N5) = I3 )
                          & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(X4,N5))) )
                       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi2),X4))
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),Ma2)) ) ) ) ) ) )
        | ? [TreeList2: list(vEBT_VEBT),N5: nat,Summary2: vEBT_VEBT,Mi2: 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,Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X4: vEBT_VEBT] :
                ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
               => vEBT_invar_vebt(X4,N5) )
            & vEBT_invar_vebt(Summary2,aa(nat,nat,suc,N5))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,N5)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N5),aa(nat,nat,suc,N5)) )
            & ! [I3: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,N5))))
               => ( ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),X_1))
                <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary2),I3)) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X4: vEBT_VEBT] :
                  ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2)))
                 => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_1)) ) )
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi2),Ma2))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma2),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),A22)))
            & ( ( Mi2 != Ma2 )
             => ! [I3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,N5))))
                 => ( ( ( vEBT_VEBT_high(Ma2,N5) = I3 )
                     => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(Ma2,N5))) )
                    & ! [X4: nat] :
                        ( ( ( vEBT_VEBT_high(X4,N5) = I3 )
                          & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(X4,N5))) )
                       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi2),X4))
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),Ma2)) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_742_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
     => ( ( ? [A5: bool,B5: bool] : A1 = vEBT_Leaf(A5,B5)
         => ( A22 != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ! [TreeList3: list(vEBT_VEBT),N2: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat] :
              ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList3,Summary3) )
             => ( ( A22 = Deg2 )
               => ( ! [X2: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                     => vEBT_invar_vebt(X2,N2) )
                 => ( vEBT_invar_vebt(Summary3,M5)
                   => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M5) )
                     => ( ( M5 = N2 )
                       => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M5) )
                         => ( ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3),X_12))
                           => ~ ! [X2: vEBT_VEBT] :
                                  ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                                 => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),X_12)) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList3: list(vEBT_VEBT),N2: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat] :
                ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList3,Summary3) )
               => ( ( A22 = Deg2 )
                 => ( ! [X2: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                       => vEBT_invar_vebt(X2,N2) )
                   => ( vEBT_invar_vebt(Summary3,M5)
                     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M5) )
                       => ( ( M5 = aa(nat,nat,suc,N2) )
                         => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M5) )
                           => ( ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3),X_12))
                             => ~ ! [X2: vEBT_VEBT] :
                                    ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                                   => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),X_12)) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList3: list(vEBT_VEBT),N2: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat,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)),Deg2,TreeList3,Summary3) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X2: vEBT_VEBT] :
                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                         => vEBT_invar_vebt(X2,N2) )
                     => ( vEBT_invar_vebt(Summary3,M5)
                       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M5) )
                         => ( ( M5 = N2 )
                           => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M5) )
                             => ( ! [I: nat] :
                                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M5)))
                                   => ( ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I)),X_1))
                                    <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3),I)) ) )
                               => ( ( ( Mi3 = Ma3 )
                                   => ! [X2: vEBT_VEBT] :
                                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                                       => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),X_12)) ) )
                                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi3),Ma3))
                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg2)))
                                     => ~ ( ( Mi3 != Ma3 )
                                         => ! [I: nat] :
                                              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M5)))
                                             => ( ( ( vEBT_VEBT_high(Ma3,N2) = I )
                                                 => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I)),vEBT_VEBT_low(Ma3,N2))) )
                                                & ! [X2: nat] :
                                                    ( ( ( vEBT_VEBT_high(X2,N2) = I )
                                                      & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I)),vEBT_VEBT_low(X2,N2))) )
                                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi3),X2))
                                                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X2),Ma3)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList3: list(vEBT_VEBT),N2: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat,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)),Deg2,TreeList3,Summary3) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X2: vEBT_VEBT] :
                            ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                           => vEBT_invar_vebt(X2,N2) )
                       => ( vEBT_invar_vebt(Summary3,M5)
                         => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M5) )
                           => ( ( M5 = aa(nat,nat,suc,N2) )
                             => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M5) )
                               => ( ! [I: nat] :
                                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M5)))
                                     => ( ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I)),X_1))
                                      <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3),I)) ) )
                                 => ( ( ( Mi3 = Ma3 )
                                     => ! [X2: vEBT_VEBT] :
                                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                                         => ~ ? [X_12: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X2),X_12)) ) )
                                   => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Mi3),Ma3))
                                     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Ma3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Deg2)))
                                       => ~ ( ( Mi3 != Ma3 )
                                           => ! [I: nat] :
                                                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M5)))
                                               => ( ( ( vEBT_VEBT_high(Ma3,N2) = I )
                                                   => pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I)),vEBT_VEBT_low(Ma3,N2))) )
                                                  & ! [X2: nat] :
                                                      ( ( ( vEBT_VEBT_high(X2,N2) = I )
                                                        & pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I)),vEBT_VEBT_low(X2,N2))) )
                                                     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Mi3),X2))
                                                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X2),Ma3)) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_743_nat__induct2,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( pp(aa(nat,bool,P,one_one(nat)))
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,P,N2))
             => pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_induct2
tff(fact_744_not__Some__eq,axiom,
    ! [A: $tType,X: option(A)] :
      ( ! [Y5: A] : X != aa(A,option(A),some(A),Y5)
    <=> ( X = none(A) ) ) ).

% not_Some_eq
tff(fact_745_not__None__eq,axiom,
    ! [A: $tType,X: option(A)] :
      ( ( X != none(A) )
    <=> ? [Y5: A] : X = aa(A,option(A),some(A),Y5) ) ).

% not_None_eq
tff(fact_746_pos2,axiom,
    pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ).

% pos2
tff(fact_747_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_748_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),L))
       => ( divide_divide(int,K,L) = zero_zero(int) ) ) ) ).

% div_pos_pos_trivial
tff(fact_749_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),K))
       => ( divide_divide(int,K,L) = zero_zero(int) ) ) ) ).

% div_neg_neg_trivial
tff(fact_750_VEBT_Oinject_I2_J,axiom,
    ! [X21: bool,X222: bool,Y21: bool,Y22: bool] :
      ( ( vEBT_Leaf(X21,X222) = vEBT_Leaf(Y21,Y22) )
    <=> ( ( pp(X21)
        <=> pp(Y21) )
        & ( pp(X222)
        <=> pp(Y22) ) ) ) ).

% VEBT.inject(2)
tff(fact_751_i0__less,axiom,
    ! [N: extended_enat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),zero_zero(extended_enat)),N))
    <=> ( N != zero_zero(extended_enat) ) ) ).

% i0_less
tff(fact_752_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% half_nonnegative_int_iff
tff(fact_753_half__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% half_negative_int_iff
tff(fact_754_less__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F3: fun(A,B),G: fun(A,B)] :
          ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less(fun(A,B)),F3),G))
        <=> ( pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),F3),G))
            & ~ pp(aa(fun(A,B),bool,aa(fun(A,B),fun(fun(A,B),bool),ord_less_eq(fun(A,B)),G),F3)) ) ) ) ).

% less_fun_def
tff(fact_755_VEBT_Osize_I4_J,axiom,
    ! [X21: bool,X222: bool] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Leaf(X21,X222)) = zero_zero(nat) ).

% VEBT.size(4)
tff(fact_756_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: bool,X223: bool] : Y != vEBT_Leaf(X212,X223) ) ).

% VEBT.exhaust
tff(fact_757_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT,X21: bool,X222: bool] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf(X21,X222) ).

% VEBT.distinct(1)
tff(fact_758_zdiv__mono__strict,axiom,
    ! [A3: int,B3: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),B3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
       => ( ( modulo_modulo(int,A3,N) = zero_zero(int) )
         => ( ( modulo_modulo(int,B3,N) = zero_zero(int) )
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),divide_divide(int,A3,N)),divide_divide(int,B3,N))) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_759_div__neg__pos__less0,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),divide_divide(int,A2,B2)),zero_zero(int))) ) ) ).

% div_neg_pos_less0
tff(fact_760_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),divide_divide(int,A2,B2)),zero_zero(int)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A2)) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_761_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),divide_divide(int,A2,B2)),zero_zero(int)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),zero_zero(int))) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_762_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),divide_divide(int,A2,B2)))
      <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),A2))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2)) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_763_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),divide_divide(int,A2,B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A2)) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_764_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),divide_divide(int,A2,B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),zero_zero(int))) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_765_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),divide_divide(int,I2,K)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I2)) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_766_div__nonpos__pos__le0,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),divide_divide(int,A2,B2)),zero_zero(int))) ) ) ).

% div_nonpos_pos_le0
tff(fact_767_div__nonneg__neg__le0,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),divide_divide(int,A2,B2)),zero_zero(int))) ) ) ).

% div_nonneg_neg_le0
tff(fact_768_verit__le__mono__div__int,axiom,
    ! [A3: int,B3: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A3),B3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),divide_divide(int,A3,N)),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),modulo_modulo(int,B3,N)),zero_zero(int)),one_one(int),zero_zero(int)))),divide_divide(int,B3,N))) ) ) ).

% verit_le_mono_div_int
tff(fact_769_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),K))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),divide_divide(int,X,K)),X)) ) ) ).

% int_div_less_self
tff(fact_770_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),L),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),divide_divide(int,K,L))) ) ) ).

% div_positive_int
tff(fact_771_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),divide_divide(int,K,L)))
    <=> ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ) ).

% div_int_pos_iff
tff(fact_772_zdiv__mono2__neg,axiom,
    ! [A2: int,B4: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B4),B2))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),divide_divide(int,A2,B4)),divide_divide(int,A2,B2))) ) ) ) ).

% zdiv_mono2_neg
tff(fact_773_zdiv__mono1__neg,axiom,
    ! [A2: int,A4: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),A4))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),divide_divide(int,A4,B2)),divide_divide(int,A2,B2))) ) ) ).

% zdiv_mono1_neg
tff(fact_774_zdiv__eq__0__iff,axiom,
    ! [I2: int,K: int] :
      ( ( divide_divide(int,I2,K) = zero_zero(int) )
    <=> ( ( K = zero_zero(int) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I2))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),K)) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),zero_zero(int)))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I2)) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_775_zdiv__mono2,axiom,
    ! [A2: int,B4: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B4),B2))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),divide_divide(int,A2,B2)),divide_divide(int,A2,B4))) ) ) ) ).

% zdiv_mono2
tff(fact_776_zdiv__mono1,axiom,
    ! [A2: int,A4: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),A4))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),divide_divide(int,A2,B2)),divide_divide(int,A4,B2))) ) ) ).

% zdiv_mono1
tff(fact_777_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,Uw: nat] : ~ vEBT_VEBT_membermima(vEBT_Leaf(Uu,Uv),Uw) ).

% VEBT_internal.membermima.simps(1)
tff(fact_778_VEBT_Osimps_I6_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),F2: fun(bool,fun(bool,A)),X21: bool,X222: bool] : vEBT_case_VEBT(A,F1,F2,vEBT_Leaf(X21,X222)) = aa(bool,A,aa(bool,fun(bool,A),F2,X21),X222) ).

% VEBT.simps(6)
tff(fact_779_not__iless0,axiom,
    ! [N: extended_enat] : ~ pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),N),zero_zero(extended_enat))) ).

% not_iless0
tff(fact_780_i0__lb,axiom,
    ! [N: extended_enat] : pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less_eq(extended_enat),zero_zero(extended_enat)),N)) ).

% i0_lb
tff(fact_781_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less_eq(extended_enat),N),zero_zero(extended_enat)))
    <=> ( N = zero_zero(extended_enat) ) ) ).

% ile0_eq
tff(fact_782_realpow__pos__nth2,axiom,
    ! [A2: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ? [R2: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
          & ( aa(nat,real,power_power(real,R2),aa(nat,nat,suc,N)) = A2 ) ) ) ).

% realpow_pos_nth2
tff(fact_783_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero(nat)) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(1)
tff(fact_784_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: bool,Uv: bool,D2: nat] :
      ( vEBT_VEBT_valid(vEBT_Leaf(Uu,Uv),D2)
    <=> ( D2 = one_one(nat) ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_785_realpow__pos__nth__unique,axiom,
    ! [N: nat,A2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
       => ? [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X3))
            & ( aa(nat,real,power_power(real,X3),N) = A2 )
            & ! [Y3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y3))
                  & ( aa(nat,real,power_power(real,Y3),N) = A2 ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_786_realpow__pos__nth,axiom,
    ! [N: nat,A2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
       => ? [R2: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
            & ( aa(nat,real,power_power(real,R2),N) = A2 ) ) ) ) ).

% realpow_pos_nth
tff(fact_787_invar__vebt_Ointros_I1_J,axiom,
    ! [A2: bool,B2: bool] : vEBT_invar_vebt(vEBT_Leaf(A2,B2),aa(nat,nat,suc,zero_zero(nat))) ).

% invar_vebt.intros(1)
tff(fact_788_not__exp__less__eq__0__int,axiom,
    ! [N: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),zero_zero(int))) ).

% not_exp_less_eq_0_int
tff(fact_789_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa(nat,nat,suc,zero_zero(nat))) = vEBT_Leaf(fFalse,fFalse) ).

% vebt_buildup.simps(2)
tff(fact_790_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A2: bool,B2: bool,X: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Leaf(A2,B2),X)
    <=> ( ( ( X = zero_zero(nat) )
         => pp(A2) )
        & ( ( X != zero_zero(nat) )
         => ( ( ( X = one_one(nat) )
             => pp(B2) )
            & ( X = one_one(nat) ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_791_real__arch__pow__inv,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => ? [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,power_power(real,X),N2)),Y)) ) ) ).

% real_arch_pow_inv
tff(fact_792_option_Odistinct_I1_J,axiom,
    ! [A: $tType,X22: A] : none(A) != aa(A,option(A),some(A),X22) ).

% option.distinct(1)
tff(fact_793_option_OdiscI,axiom,
    ! [A: $tType,Option: option(A),X22: A] :
      ( ( Option = aa(A,option(A),some(A),X22) )
     => ( Option != none(A) ) ) ).

% option.discI
tff(fact_794_option_Oexhaust,axiom,
    ! [A: $tType,Y: option(A)] :
      ( ( Y != none(A) )
     => ~ ! [X23: A] : Y != aa(A,option(A),some(A),X23) ) ).

% option.exhaust
tff(fact_795_split__option__ex,axiom,
    ! [A: $tType,P: fun(option(A),bool)] :
      ( ? [X_1: option(A)] : pp(aa(option(A),bool,P,X_1))
    <=> ( pp(aa(option(A),bool,P,none(A)))
        | ? [X4: A] : pp(aa(option(A),bool,P,aa(A,option(A),some(A),X4))) ) ) ).

% split_option_ex
tff(fact_796_split__option__all,axiom,
    ! [A: $tType,P: fun(option(A),bool)] :
      ( ! [X_1: option(A)] : pp(aa(option(A),bool,P,X_1))
    <=> ( pp(aa(option(A),bool,P,none(A)))
        & ! [X4: A] : pp(aa(option(A),bool,P,aa(A,option(A),some(A),X4))) ) ) ).

% split_option_all
tff(fact_797_combine__options__cases,axiom,
    ! [A: $tType,B: $tType,X: option(A),P: fun(option(A),fun(option(B),bool)),Y: option(B)] :
      ( ( ( X = none(A) )
       => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y)) )
     => ( ( ( Y = none(B) )
         => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y)) )
       => ( ! [A5: A,B5: B] :
              ( ( X = aa(A,option(A),some(A),A5) )
             => ( ( Y = aa(B,option(B),some(B),B5) )
               => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y)) ) )
         => pp(aa(option(B),bool,aa(option(A),fun(option(B),bool),P,X),Y)) ) ) ) ).

% combine_options_cases
tff(fact_798_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_799_option_Osize_I4_J,axiom,
    ! [A: $tType,X22: A] : aa(option(A),nat,size_size(option(A)),aa(A,option(A),some(A),X22)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(4)
tff(fact_800_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q3: A,R3: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),product_Pair(A,A,Q3),R3))
        <=> ( R3 = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_801_gcd__nat__induct,axiom,
    ! [P: fun(nat,fun(nat,bool)),M: nat,N: nat] :
      ( ! [M5: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M5),zero_zero(nat)))
     => ( ! [M5: nat,N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N2),modulo_modulo(nat,M5,N2)))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M5),N2)) ) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,M),N)) ) ) ).

% gcd_nat_induct
tff(fact_802_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X22: A] : size_option(A,X,aa(A,option(A),some(A),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_803_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),modulo_modulo(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_804_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = divide_divide(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_805_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),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),aa(num,num,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),aa(num,num,bit0,one2))),divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)))),one_one(A)) = divide_divide(A,A2,B2) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_806_num_Osize__gen_I2_J,axiom,
    ! [X22: num] : size_num(aa(num,num,bit0,X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(2)
tff(fact_807_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),W))),zero_zero(A)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(num,nat,numeral_numeral(nat),W)))
            & ( ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(num,nat,numeral_numeral(nat),W)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) )
              | ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(num,nat,numeral_numeral(nat),W)))
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_808_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : modulo_modulo(A,one_one(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% one_mod_2_pow_eq
tff(fact_809_arith__geo__mean,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,X: A,Y: A] :
          ( ( aa(nat,A,power_power(A,U),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ) ) ) ).

% arith_geo_mean
tff(fact_810_semiring__norm_I13_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit0,M)),aa(num,num,bit0,N)) = aa(num,num,bit0,aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ).

% semiring_norm(13)
tff(fact_811_semiring__norm_I11_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),times_times(num),M),one2) = M ).

% semiring_norm(11)
tff(fact_812_semiring__norm_I12_J,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),one2),N) = N ).

% semiring_norm(12)
tff(fact_813_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        | ( N = zero_zero(nat) ) ) ) ).

% mult_is_0
tff(fact_814_mult__0__right,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),zero_zero(nat)) = zero_zero(nat) ).

% mult_0_right
tff(fact_815_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
    <=> ( ( M = N )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_816_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K) )
    <=> ( ( M = N )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_817_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = one_one(nat) )
    <=> ( ( M = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_818_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
    <=> ( ( M = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_819_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_820_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_821_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_822_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_823_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_824_numeral__times__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ) ).

% numeral_times_numeral
tff(fact_825_mult__numeral__left__semiring__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [V: num,W: num,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W))),Z) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_826_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_827_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_828_num__double,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit0,one2)),N) = aa(num,num,bit0,N) ).

% num_double
tff(fact_829_times__divide__eq__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),divide_divide(A,B2,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ).

% times_divide_eq_right
tff(fact_830_divide__divide__eq__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,C2: A] : divide_divide(A,A2,divide_divide(A,B2,C2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),B2) ) ).

% divide_divide_eq_right
tff(fact_831_divide__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,C2: A] : divide_divide(A,divide_divide(A,A2,B2),C2) = divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% divide_divide_eq_left
tff(fact_832_times__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,C2: A,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,B2,C2)),A2) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2),C2) ) ).

% times_divide_eq_left
tff(fact_833_dvd__0__right,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : pp(dvd_dvd(A,A2,zero_zero(A))) ) ).

% dvd_0_right
tff(fact_834_dvd__0__left__iff,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] :
          ( pp(dvd_dvd(A,zero_zero(A),A2))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% dvd_0_left_iff
tff(fact_835_dvd__add__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A] :
          ( pp(dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)))
        <=> pp(dvd_dvd(A,A2,B2)) ) ) ).

% dvd_add_triv_right_iff
tff(fact_836_dvd__add__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A] :
          ( pp(dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)))
        <=> pp(dvd_dvd(A,A2,B2)) ) ) ).

% dvd_add_triv_left_iff
tff(fact_837_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% mult_eq_1_iff
tff(fact_838_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% one_eq_mult_iff
tff(fact_839_div__dvd__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(dvd_dvd(A,A2,B2))
         => ( pp(dvd_dvd(A,A2,C2))
           => ( pp(dvd_dvd(A,divide_divide(A,B2,A2),divide_divide(A,C2,A2)))
            <=> pp(dvd_dvd(A,B2,C2)) ) ) ) ) ).

% div_dvd_div
tff(fact_840_power__mult__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: num,N: num] : aa(nat,A,power_power(A,aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),M))),aa(num,nat,numeral_numeral(nat),N)) = aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ) ).

% power_mult_numeral
tff(fact_841_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_842_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% nat_0_less_mult_iff
tff(fact_843_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% mult_less_cancel2
tff(fact_844_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),times_times(real),X),X)))
    <=> ( X = zero_zero(real) ) ) ).

% not_real_square_gt_zero
tff(fact_845_mult__Suc__right,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) ).

% mult_Suc_right
tff(fact_846_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
    <=> ( ( K = zero_zero(nat) )
        | pp(dvd_dvd(nat,M,N)) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_847_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: bool,Q: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)))
        <=> ( pp(P)
           => pp(Q) ) ) ) ).

% of_bool_less_eq_iff
tff(fact_848_of__bool__eq_I1_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa(bool,A,zero_neq_one_of_bool(A),fFalse) = zero_zero(A) ) ) ).

% of_bool_eq(1)
tff(fact_849_of__bool__eq__0__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: bool] :
          ( ( aa(bool,A,zero_neq_one_of_bool(A),P) = zero_zero(A) )
        <=> ~ pp(P) ) ) ).

% of_bool_eq_0_iff
tff(fact_850_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( ( K = zero_zero(nat) )
       => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = zero_zero(nat) ) )
      & ( ( K != zero_zero(nat) )
       => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = divide_divide(nat,M,N) ) ) ) ).

% nat_mult_div_cancel_disj
tff(fact_851_of__bool__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: bool,Q: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)))
        <=> ( ~ pp(P)
            & pp(Q) ) ) ) ).

% of_bool_less_iff
tff(fact_852_of__bool__eq__1__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: bool] :
          ( ( aa(bool,A,zero_neq_one_of_bool(A),P) = one_one(A) )
        <=> pp(P) ) ) ).

% of_bool_eq_1_iff
tff(fact_853_of__bool__eq_I2_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa(bool,A,zero_neq_one_of_bool(A),fTrue) = one_one(A) ) ) ).

% of_bool_eq(2)
tff(fact_854_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(dvd_dvd(nat,M,one_one(nat)))
    <=> ( M = one_one(nat) ) ) ).

% nat_dvd_1_iff_1
tff(fact_855_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_856_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_857_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_858_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_859_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_860_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_861_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_862_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_863_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_864_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_865_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_866_mult__divide__mult__cancel__left__if,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)) = zero_zero(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) ) ) ) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_867_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_868_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_869_div__mult__mult1__if,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)) = zero_zero(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_if
tff(fact_870_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [A2: A,B2: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V))) ) ).

% distrib_right_numeral
tff(fact_871_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).

% distrib_left_numeral
tff(fact_872_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(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) )
            | pp(dvd_dvd(A,A2,B2)) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_873_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(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) )
            | pp(dvd_dvd(A,A2,B2)) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_874_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( pp(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)))
          <=> pp(dvd_dvd(A,B2,C2)) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_875_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( pp(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)))
          <=> pp(dvd_dvd(A,B2,C2)) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_876_unit__prod,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(dvd_dvd(A,A2,one_one(A)))
         => ( pp(dvd_dvd(A,B2,one_one(A)))
           => pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),one_one(A))) ) ) ) ).

% unit_prod
tff(fact_877_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(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))))
        <=> pp(dvd_dvd(A,A2,B2)) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_878_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(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)))
        <=> pp(dvd_dvd(A,A2,B2)) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_879_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_880_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_881_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(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_882_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(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_883_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( pp(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_884_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( pp(dvd_dvd(A,A2,one_one(A)))
         => pp(dvd_dvd(A,divide_divide(A,one_one(A),A2),one_one(A))) ) ) ).

% unit_div_1_unit
tff(fact_885_unit__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(dvd_dvd(A,A2,one_one(A)))
         => ( pp(dvd_dvd(A,B2,one_one(A)))
           => pp(dvd_dvd(A,divide_divide(A,A2,B2),one_one(A))) ) ) ) ).

% unit_div
tff(fact_886_div__add,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(dvd_dvd(A,C2,A2))
         => ( pp(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_887_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_888_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_889_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_890_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_891_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( pp(dvd_dvd(A,A2,B2))
         => ( modulo_modulo(A,B2,A2) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_892_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),M))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),N)) ) ) ).

% one_le_mult_iff
tff(fact_893_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% mult_le_cancel2
tff(fact_894_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_895_zero__less__of__bool__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(bool,A,zero_neq_one_of_bool(A),P)))
        <=> pp(P) ) ) ).

% zero_less_of_bool_iff
tff(fact_896_of__bool__less__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(bool,A,zero_neq_one_of_bool(A),P)),one_one(A)))
        <=> ~ pp(P) ) ) ).

% of_bool_less_one_iff
tff(fact_897_dvd__1__left,axiom,
    ! [K: nat] : pp(dvd_dvd(nat,aa(nat,nat,suc,zero_zero(nat)),K)) ).

% dvd_1_left
tff(fact_898_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( pp(dvd_dvd(nat,M,aa(nat,nat,suc,zero_zero(nat))))
    <=> ( M = aa(nat,nat,suc,zero_zero(nat)) ) ) ).

% dvd_1_iff_1
tff(fact_899_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M),N) = M ) ) ).

% div_mult_self1_is_m
tff(fact_900_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N),N) = M ) ) ).

% div_mult_self_is_m
tff(fact_901_Suc__0__mod__eq,axiom,
    ! [N: nat] : modulo_modulo(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),aa(nat,nat,suc,zero_zero(nat))))) ).

% Suc_0_mod_eq
tff(fact_902_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N))),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% Suc_mod_mult_self1
tff(fact_903_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: nat,K: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% Suc_mod_mult_self2
tff(fact_904_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: nat,M: 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),N)),M)),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% Suc_mod_mult_self3
tff(fact_905_Suc__mod__mult__self4,axiom,
    ! [N: nat,K: nat,M: 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),N),K)),M)),N) = modulo_modulo(nat,aa(nat,nat,suc,M),N) ).

% Suc_mod_mult_self4
tff(fact_906_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_907_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W))),A2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_908_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)) )
        <=> ( ( ( 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 ) )
            & ( ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_909_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 )
        <=> ( ( ( 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)) ) )
            & ( ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_910_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_911_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,aa(num,A,numeral_numeral(A),W))),A2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_912_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_913_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_914_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_915_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_916_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_917_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_918_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(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_919_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(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_920_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [N: nat,A2: A,B2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( pp(dvd_dvd(A,aa(nat,A,power_power(A,A2),N),aa(nat,A,power_power(A,B2),N)))
          <=> pp(dvd_dvd(A,A2,B2)) ) ) ) ).

% pow_divides_pow_iff
tff(fact_921_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,V)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),modulo_modulo(int,aa(num,int,numeral_numeral(int),V),aa(num,int,numeral_numeral(int),W))) ).

% zmod_numeral_Bit0
tff(fact_922_power__add__numeral2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: num,N: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),M))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),N))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)))),B2) ) ).

% power_add_numeral2
tff(fact_923_power__add__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),M))),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),N))) = aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N))) ) ).

% power_add_numeral
tff(fact_924_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: 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)),N)),aa(num,nat,numeral_numeral(nat),K)) = one_one(nat) ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_925_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W),Z)) ) ).

% zle_add1_eq_le
tff(fact_926_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),K))
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_927_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),L))
       => ( modulo_modulo(int,K,L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_928_even__mult__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
        <=> ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
            | pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2)) ) ) ) ).

% even_mult_iff
tff(fact_929_even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)))
        <=> ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
          <=> pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2)) ) ) ) ).

% even_add
tff(fact_930_odd__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)))
        <=> ~ ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
            <=> ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2)) ) ) ) ).

% odd_add
tff(fact_931_even__mod__2__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))
        <=> pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)) ) ) ).

% even_mod_2_iff
tff(fact_932_even__Suc__Suc__iff,axiom,
    ! [N: nat] :
      ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(nat,nat,suc,aa(nat,nat,suc,N))))
    <=> pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N)) ) ).

% even_Suc_Suc_iff
tff(fact_933_even__Suc,axiom,
    ! [N: nat] :
      ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(nat,nat,suc,N)))
    <=> ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N)) ) ).

% even_Suc
tff(fact_934_odd__of__bool__self,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [P2: bool] :
          ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(bool,A,zero_neq_one_of_bool(A),P2)))
        <=> pp(P2) ) ) ).

% odd_of_bool_self
tff(fact_935_even__plus__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))))
        <=> ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)) ) ) ).

% even_plus_one_iff
tff(fact_936_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [B2: bool] : divide_divide(A,aa(bool,A,zero_neq_one_of_bool(A),B2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ).

% of_bool_half_eq_0
tff(fact_937_even__Suc__div__two,axiom,
    ! [N: nat] :
      ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
     => ( divide_divide(nat,aa(nat,nat,suc,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ) ).

% even_Suc_div_two
tff(fact_938_odd__Suc__div__two,axiom,
    ! [N: nat] :
      ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
     => ( divide_divide(nat,aa(nat,nat,suc,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,nat,suc,divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% odd_Suc_div_two
tff(fact_939_even__succ__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))) = divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ).

% even_succ_div_2
tff(fact_940_even__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))) = divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ).

% even_succ_div_two
tff(fact_941_odd__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ) ) ).

% odd_succ_div_two
tff(fact_942_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),W))))
        <=> ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(num,nat,numeral_numeral(nat),W)))
            | ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(num,nat,numeral_numeral(nat),W)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_943_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,N: nat] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(nat,A,power_power(A,A2),N)))
        <=> ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% even_power
tff(fact_944_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),W))),zero_zero(A)))
        <=> ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(num,nat,numeral_numeral(nat),W)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_945_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A2),N)),zero_zero(A)))
        <=> ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% power_less_zero_eq
tff(fact_946_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),one_one(A)) = A2 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_947_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),W))))
        <=> ( ( aa(num,nat,numeral_numeral(nat),W) = zero_zero(nat) )
            | ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(num,nat,numeral_numeral(nat),W)))
              & ( A2 != zero_zero(A) ) )
            | ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(num,nat,numeral_numeral(nat),W)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_948_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : divide_divide(A,one_one(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).

% one_div_2_pow_eq
tff(fact_949_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] : divide_divide(A,one_one(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).

% bits_1_div_exp
tff(fact_950_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),one_one(int))),Z))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z)) ) ).

% add1_zle_eq
tff(fact_951_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( pp(dvd_dvd(int,Z,N))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),N)) ) ) ).

% zdvd_imp_le
tff(fact_952_int__gr__induct,axiom,
    ! [K: int,I2: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I2))
     => ( pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))))
       => ( ! [I4: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I4))
             => ( pp(aa(int,bool,P,I4))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I4),one_one(int)))) ) )
         => pp(aa(int,bool,P,I2)) ) ) ) ).

% int_gr_induct
tff(fact_953_le__imp__0__less,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => pp(aa(int,bool,aa(int,fun(int,bool),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_954_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int))))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z))
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_955_split__zmod,axiom,
    ! [P: fun(int,bool),N: int,K: int] :
      ( pp(aa(int,bool,P,modulo_modulo(int,N,K)))
    <=> ( ( ( K = zero_zero(int) )
         => pp(aa(int,bool,P,N)) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
         => ! [I3: int,J3: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J3))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J3),K))
                & ( N = 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) ) )
             => pp(aa(int,bool,P,J3)) ) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
         => ! [I3: int,J3: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),J3))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J3),zero_zero(int)))
                & ( N = 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) ) )
             => pp(aa(int,bool,P,J3)) ) ) ) ) ).

% split_zmod
tff(fact_956_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),zero_zero(int))) ) ).

% odd_less_0_iff
tff(fact_957_q__pos__lemma,axiom,
    ! [B4: int,Q4: int,R4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q4)),R4)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R4),B4))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Q4)) ) ) ) ).

% q_pos_lemma
tff(fact_958_neg__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),modulo_modulo(int,A2,B2)),zero_zero(int)))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),modulo_modulo(int,A2,B2))) ) ) ).

% neg_mod_conj
tff(fact_959_pos__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,A2,B2)))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),modulo_modulo(int,A2,B2)),B2)) ) ) ).

% pos_mod_conj
tff(fact_960_int__mod__neg__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R3: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R3) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R3),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R3))
         => ( modulo_modulo(int,A2,B2) = R3 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_961_int__mod__pos__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R3: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R3) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R3))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R3),B2))
         => ( modulo_modulo(int,A2,B2) = R3 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_962_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
     => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N) = one_one(int) )
      <=> ( ( M = one_one(int) )
          & ( N = one_one(int) ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_963_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z))
     => pp(aa(int,bool,aa(int,fun(int,bool),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_964_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K,L)))
    <=> ( pp(dvd_dvd(int,L,K))
        | ( ( L = zero_zero(int) )
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) )
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L)) ) ) ).

% mod_int_pos_iff
tff(fact_965_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q3: int,R3: int,B4: int,Q4: int,R4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q4)),R4) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q4)),R4)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R4),B4))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R3))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B4),B2))
               => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q3),Q4)) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_966_zmod__trivial__iff,axiom,
    ! [I2: int,K: int] :
      ( ( modulo_modulo(int,I2,K) = I2 )
    <=> ( ( K = zero_zero(int) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I2))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),K)) )
        | ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),zero_zero(int)))
          & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),I2)) ) ) ) ).

% zmod_trivial_iff
tff(fact_967_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q3: int,R3: int,B4: int,Q4: int,R4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q4)),R4) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q4)),R4)),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R3),B2))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R4))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B4))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B4),B2))
               => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q4),Q3)) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_968_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)) ) ).

% int_one_le_iff_zero_less
tff(fact_969_unique__quotient__lemma,axiom,
    ! [B2: int,Q4: int,R4: int,Q3: int,R3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R3)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R4))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R4),B2))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R3),B2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q4),Q3)) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_970_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),modulo_modulo(int,K,L)),zero_zero(int))) ) ).

% neg_mod_sign
tff(fact_971_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),modulo_modulo(int,K,L))) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_972_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q4: int,R4: int,Q3: int,R3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q4)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R3)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R3),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R3))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R4))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Q3),Q4)) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_973_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),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_974_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),modulo_modulo(int,M,K)),M)) ) ).

% zmod_le_nonneg_dividend
tff(fact_975_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),M))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
       => ( pp(dvd_dvd(int,M,N))
         => ( pp(dvd_dvd(int,N,M))
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_976_verit__la__generic,axiom,
    ! [A2: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),X))
      | ( A2 = X )
      | pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),A2)) ) ).

% verit_la_generic
tff(fact_977_less__eq__int__code_I1_J,axiom,
    pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),zero_zero(int))) ).

% less_eq_int_code(1)
tff(fact_978_zmod__eq__0D,axiom,
    ! [M: int,D2: int] :
      ( ( modulo_modulo(int,M,D2) = zero_zero(int) )
     => ? [Q2: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q2) ) ).

% zmod_eq_0D
tff(fact_979_zmod__eq__0__iff,axiom,
    ! [M: int,D2: int] :
      ( ( modulo_modulo(int,M,D2) = zero_zero(int) )
    <=> ? [Q5: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q5) ) ).

% zmod_eq_0_iff
tff(fact_980_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),modulo_modulo(int,K,L)),L)) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_981_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),modulo_modulo(int,K,L))) ) ).

% neg_mod_bound
tff(fact_982_zmult__zless__mono2,axiom,
    ! [I2: int,J: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),J))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I2)),aa(int,int,aa(int,fun(int,int),times_times(int),K),J))) ) ) ).

% zmult_zless_mono2
tff(fact_983_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),M))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),M),N))
       => ~ pp(dvd_dvd(int,N,M)) ) ) ).

% zdvd_not_zless
tff(fact_984_less__int__code_I1_J,axiom,
    ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),zero_zero(int))) ).

% less_int_code(1)
tff(fact_985_iadd__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),M),N) = zero_zero(extended_enat) )
    <=> ( ( M = zero_zero(extended_enat) )
        & ( N = zero_zero(extended_enat) ) ) ) ).

% iadd_is_0
tff(fact_986_imult__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M),N) = zero_zero(extended_enat) )
    <=> ( ( M = zero_zero(extended_enat) )
        | ( N = zero_zero(extended_enat) ) ) ) ).

% imult_is_0
tff(fact_987_zero__one__enat__neq_I1_J,axiom,
    zero_zero(extended_enat) != one_one(extended_enat) ).

% zero_one_enat_neq(1)
tff(fact_988_int__ge__induct,axiom,
    ! [K: int,I2: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I2))
     => ( pp(aa(int,bool,P,K))
       => ( ! [I4: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I4))
             => ( pp(aa(int,bool,P,I4))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I4),one_one(int)))) ) )
         => pp(aa(int,bool,P,I2)) ) ) ) ).

% int_ge_induct
tff(fact_989_is__unit__mult__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),one_one(A)))
        <=> ( pp(dvd_dvd(A,A2,one_one(A)))
            & pp(dvd_dvd(A,B2,one_one(A))) ) ) ) ).

% is_unit_mult_iff
tff(fact_990_dvd__mult__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(dvd_dvd(A,B2,one_one(A)))
         => ( pp(dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
          <=> pp(dvd_dvd(A,A2,C2)) ) ) ) ).

% dvd_mult_unit_iff
tff(fact_991_mult__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(dvd_dvd(A,B2,one_one(A)))
         => ( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2))
          <=> pp(dvd_dvd(A,A2,C2)) ) ) ) ).

% mult_unit_dvd_iff
tff(fact_992_dvd__mult__unit__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(dvd_dvd(A,B2,one_one(A)))
         => ( pp(dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
          <=> pp(dvd_dvd(A,A2,C2)) ) ) ) ).

% dvd_mult_unit_iff'
tff(fact_993_mult__unit__dvd__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(dvd_dvd(A,A2,one_one(A)))
         => ( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2))
          <=> pp(dvd_dvd(A,B2,C2)) ) ) ) ).

% mult_unit_dvd_iff'
tff(fact_994_unit__mult__left__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(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_995_unit__mult__right__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(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_996_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(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_997_div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(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_998_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(dvd_dvd(A,C2,B2))
         => ( pp(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_999_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(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_1000_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),B2))
         => pp(dvd_dvd(A,A2,divide_divide(A,B2,C2))) ) ) ).

% dvd_mult_imp_div
tff(fact_1001_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,D2: A,C2: A] :
          ( pp(dvd_dvd(A,B2,A2))
         => ( pp(dvd_dvd(A,D2,C2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,A2,B2)),divide_divide(A,C2,D2)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_1002_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( pp(dvd_dvd(nat,M,N))
     => ( pp(dvd_dvd(nat,N,M))
       => ( M = N ) ) ) ).

% dvd_antisym
tff(fact_1003_bezout__lemma__nat,axiom,
    ! [D2: nat,A2: nat,B2: nat,X: nat,Y: nat] :
      ( pp(dvd_dvd(nat,D2,A2))
     => ( pp(dvd_dvd(nat,D2,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),Y)),D2) )
            | ( 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),Y)),D2) ) )
         => ? [X3: nat,Y4: nat] :
              ( pp(dvd_dvd(nat,D2,A2))
              & pp(dvd_dvd(nat,D2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)))
              & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3) = 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)),Y4)),D2) )
                | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y4)),D2) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_1004_bezout__add__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D3: nat,X3: nat,Y4: nat] :
      ( pp(dvd_dvd(nat,D3,A2))
      & pp(dvd_dvd(nat,D3,B2))
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)),D3) )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y4)),D3) ) ) ) ).

% bezout_add_nat
tff(fact_1005_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_1006_dvdE,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( pp(dvd_dvd(A,B2,A2))
         => ~ ! [K2: A] : A2 != aa(A,A,aa(A,fun(A,A),times_times(A),B2),K2) ) ) ).

% dvdE
tff(fact_1007_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) )
         => pp(dvd_dvd(A,B2,A2)) ) ) ).

% dvdI
tff(fact_1008_dvd__def,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( pp(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_1009_of__bool__conj,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P: bool,Q: bool] : aa(bool,A,zero_neq_one_of_bool(A),fconj(P,Q)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)) ) ).

% of_bool_conj
tff(fact_1010_dvd__mult,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(dvd_dvd(A,A2,C2))
         => pp(dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ).

% dvd_mult
tff(fact_1011_dvd__refl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : pp(dvd_dvd(A,A2,A2)) ) ).

% dvd_refl
tff(fact_1012_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_1013_dvd__mult2,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(dvd_dvd(A,A2,B2))
         => pp(dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))) ) ) ).

% dvd_mult2
tff(fact_1014_dvd__trans,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(dvd_dvd(A,A2,B2))
         => ( pp(dvd_dvd(A,B2,C2))
           => pp(dvd_dvd(A,A2,C2)) ) ) ) ).

% dvd_trans
tff(fact_1015_of__bool__eq__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: bool,Q3: bool] :
          ( ( aa(bool,A,zero_neq_one_of_bool(A),P2) = aa(bool,A,zero_neq_one_of_bool(A),Q3) )
        <=> ( pp(P2)
          <=> pp(Q3) ) ) ) ).

% of_bool_eq_iff
tff(fact_1016_dvd__mult__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2))
         => pp(dvd_dvd(A,A2,C2)) ) ) ).

% dvd_mult_left
tff(fact_1017_dvd__triv__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] : pp(dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ).

% dvd_triv_left
tff(fact_1018_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(dvd_dvd(A,A2,B2))
         => ( pp(dvd_dvd(A,C2,D2))
           => pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ).

% mult_dvd_mono
tff(fact_1019_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_1020_dvd__mult__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2))
         => pp(dvd_dvd(A,B2,C2)) ) ) ).

% dvd_mult_right
tff(fact_1021_dvd__triv__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] : pp(dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2))) ) ).

% dvd_triv_right
tff(fact_1022_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_1023_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [D3: nat,X3: nat,Y4: nat] :
          ( pp(dvd_dvd(nat,D3,A2))
          & pp(dvd_dvd(nat,D3,B2))
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)),D3) ) ) ) ).

% bezout_add_strong_nat
tff(fact_1024_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(dvd_dvd(A,A2,one_one(A)))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [C4: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A2),C4) ) ) ) ).

% unit_dvdE
tff(fact_1025_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( pp(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_1026_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( pp(dvd_dvd(A,B2,A2))
           => ( pp(dvd_dvd(A,divide_divide(A,A2,B2),C2))
            <=> pp(dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_1027_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( pp(dvd_dvd(A,C2,B2))
           => ( pp(dvd_dvd(A,A2,divide_divide(A,B2,C2)))
            <=> pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),B2)) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_1028_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( C2 != zero_zero(A) )
           => ( pp(dvd_dvd(A,A2,B2))
             => ( pp(dvd_dvd(A,C2,D2))
               => ( ( divide_divide(A,B2,A2) = divide_divide(A,D2,C2) )
                <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),D2) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_1029_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(dvd_dvd(A,B2,one_one(A)))
         => ( pp(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_1030_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(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_1031_unit__div__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(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_1032_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(dvd_dvd(A,C2,one_one(A)))
         => ( pp(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_1033_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(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_1034_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(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_1035_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( pp(dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
      <=> pp(dvd_dvd(nat,M,N)) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_1036_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(dvd_dvd(nat,M,N)) ) ) ).

% dvd_mult_cancel
tff(fact_1037_dvd__field__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( pp(dvd_dvd(A,A2,B2))
        <=> ( ( A2 = zero_zero(A) )
           => ( B2 = zero_zero(A) ) ) ) ) ).

% dvd_field_iff
tff(fact_1038_dvd__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] :
          ( pp(dvd_dvd(A,zero_zero(A),A2))
         => ( A2 = zero_zero(A) ) ) ) ).

% dvd_0_left
tff(fact_1039_dvd__unit__imp__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( pp(dvd_dvd(A,A2,B2))
         => ( pp(dvd_dvd(A,B2,one_one(A)))
           => pp(dvd_dvd(A,A2,one_one(A))) ) ) ) ).

% dvd_unit_imp_unit
tff(fact_1040_unit__imp__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( pp(dvd_dvd(A,B2,one_one(A)))
         => pp(dvd_dvd(A,B2,A2)) ) ) ).

% unit_imp_dvd
tff(fact_1041_one__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : pp(dvd_dvd(A,one_one(A),A2)) ) ).

% one_dvd
tff(fact_1042_dvd__add__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(dvd_dvd(A,A2,B2))
         => ( pp(dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
          <=> pp(dvd_dvd(A,A2,C2)) ) ) ) ).

% dvd_add_right_iff
tff(fact_1043_dvd__add__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(dvd_dvd(A,A2,C2))
         => ( pp(dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)))
          <=> pp(dvd_dvd(A,A2,B2)) ) ) ) ).

% dvd_add_left_iff
tff(fact_1044_dvd__add,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(dvd_dvd(A,A2,B2))
         => ( pp(dvd_dvd(A,A2,C2))
           => pp(dvd_dvd(A,A2,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))) ) ) ) ).

% dvd_add
tff(fact_1045_div__div__div__same,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [D2: A,B2: A,A2: A] :
          ( pp(dvd_dvd(A,D2,B2))
         => ( pp(dvd_dvd(A,B2,A2))
           => ( divide_divide(A,divide_divide(A,A2,D2),divide_divide(A,B2,D2)) = divide_divide(A,A2,B2) ) ) ) ) ).

% div_div_div_same
tff(fact_1046_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) )
         => ( pp(dvd_dvd(A,C2,A2))
           => ( pp(dvd_dvd(A,C2,B2))
             => ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_1047_dvd__div__eq__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(dvd_dvd(A,C2,A2))
         => ( pp(dvd_dvd(A,C2,B2))
           => ( ( divide_divide(A,A2,C2) = divide_divide(A,B2,C2) )
            <=> ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_1048_dvd__power__same,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y: A,N: nat] :
          ( pp(dvd_dvd(A,X,Y))
         => pp(dvd_dvd(A,aa(nat,A,power_power(A,X),N),aa(nat,A,power_power(A,Y),N))) ) ) ).

% dvd_power_same
tff(fact_1049_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_1050_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_1051_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_1052_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_1053_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_1054_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_1055_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_1056_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_1057_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_1058_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_1059_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_1060_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_1061_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_1062_dvd__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [K: A,M: A,N: A] :
          ( pp(dvd_dvd(A,K,M))
         => ( pp(dvd_dvd(A,K,N))
           => pp(dvd_dvd(A,K,modulo_modulo(A,M,N))) ) ) ) ).

% dvd_mod
tff(fact_1063_mod__mod__cancel,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(dvd_dvd(A,C2,B2))
         => ( modulo_modulo(A,modulo_modulo(A,A2,B2),C2) = modulo_modulo(A,A2,C2) ) ) ) ).

% mod_mod_cancel
tff(fact_1064_dvd__mod__iff,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(dvd_dvd(A,C2,B2))
         => ( pp(dvd_dvd(A,C2,modulo_modulo(A,A2,B2)))
          <=> pp(dvd_dvd(A,C2,A2)) ) ) ) ).

% dvd_mod_iff
tff(fact_1065_dvd__mod__imp__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(dvd_dvd(A,C2,modulo_modulo(A,A2,B2)))
         => ( pp(dvd_dvd(A,C2,B2))
           => pp(dvd_dvd(A,C2,A2)) ) ) ) ).

% dvd_mod_imp_dvd
tff(fact_1066_divide__divide__eq__left_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : divide_divide(A,divide_divide(A,A2,B2),C2) = divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% divide_divide_eq_left'
tff(fact_1067_divide__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y: A,Z: A,W: A] : divide_divide(A,divide_divide(A,X,Y),divide_divide(A,Z,W)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),W),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ).

% divide_divide_times_eq
tff(fact_1068_times__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,Y: A,Z: A,W: A] : aa(A,A,aa(A,fun(A,A),times_times(A),divide_divide(A,X,Y)),divide_divide(A,Z,W)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z),aa(A,A,aa(A,fun(A,A),times_times(A),Y),W)) ) ).

% times_divide_times_eq
tff(fact_1069_power__commuting__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Y: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),X) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),N)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,power_power(A,X),N)) ) ) ) ).

% power_commuting_commutes
tff(fact_1070_power__mult__distrib,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,N: nat] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),N)),aa(nat,A,power_power(A,B2),N)) ) ).

% power_mult_distrib
tff(fact_1071_power__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),N)),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,A2),N)) ) ).

% power_commutes
tff(fact_1072_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N) )
    <=> ( M = N ) ) ).

% Suc_mult_cancel1
tff(fact_1073_power__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: nat,N: nat] : aa(nat,A,power_power(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) = aa(nat,A,power_power(A,aa(nat,A,power_power(A,A2),M)),N) ) ).

% power_mult
tff(fact_1074_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))] :
      ~ ! [F4: fun(nat,fun(A,A)),A5: nat,B5: nat,Acc: A] : X != 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)),F4),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),A5),aa(A,product_prod(nat,A),product_Pair(nat,A,B5),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_1075_mult__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),N) = zero_zero(nat) ).

% mult_0
tff(fact_1076_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
    <=> ( ( K = zero_zero(nat) )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_1077_mult__le__mono2,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J))) ) ).

% mult_le_mono2
tff(fact_1078_mult__le__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K))) ) ).

% mult_le_mono1
tff(fact_1079_mult__le__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),L))) ) ) ).

% mult_le_mono
tff(fact_1080_le__square,axiom,
    ! [M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M))) ).

% le_square
tff(fact_1081_le__cube,axiom,
    ! [M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M)))) ).

% le_cube
tff(fact_1082_mod__mult__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ).

% mod_mult_eq
tff(fact_1083_mod__mult__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,A4: A,B2: A,B4: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A4,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B4,C2) )
           => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A4),B4),C2) ) ) ) ) ).

% mod_mult_cong
tff(fact_1084_mod__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,B2)),C2) ) ).

% mod_mult_mult2
tff(fact_1085_mult__mod__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),modulo_modulo(A,A2,B2)) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% mult_mod_right
tff(fact_1086_mod__mult__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,C2)),B2),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ).

% mod_mult_left_eq
tff(fact_1087_mod__mult__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),C2) ) ).

% mod_mult_right_eq
tff(fact_1088_left__add__mult__distrib,axiom,
    ! [I2: 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),I2),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),I2),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_1089_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) ).

% add_mult_distrib2
tff(fact_1090_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)) ).

% add_mult_distrib
tff(fact_1091_nat__mult__1__right,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),one_one(nat)) = N ).

% nat_mult_1_right
tff(fact_1092_nat__mult__1,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),one_one(nat)),N) = N ).

% nat_mult_1
tff(fact_1093_div__mult2__eq,axiom,
    ! [M: nat,N: nat,Q3: nat] : divide_divide(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q3)) = divide_divide(nat,divide_divide(nat,M,N),Q3) ).

% div_mult2_eq
tff(fact_1094_div__mod__decomp__int,axiom,
    ! [A3: int,N: 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,N)),N)),modulo_modulo(int,A3,N)) ).

% div_mod_decomp_int
tff(fact_1095_split__zdiv,axiom,
    ! [P: fun(int,bool),N: int,K: int] :
      ( pp(aa(int,bool,P,divide_divide(int,N,K)))
    <=> ( ( ( K = zero_zero(int) )
         => pp(aa(int,bool,P,zero_zero(int))) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
         => ! [I3: int,J3: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J3))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J3),K))
                & ( N = 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) ) )
             => pp(aa(int,bool,P,I3)) ) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
         => ! [I3: int,J3: int] :
              ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),J3))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J3),zero_zero(int)))
                & ( N = 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) ) )
             => pp(aa(int,bool,P,I3)) ) ) ) ) ).

% split_zdiv
tff(fact_1096_int__div__neg__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R3: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R3) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R3),zero_zero(int)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),R3))
         => ( divide_divide(int,A2,B2) = Q3 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1097_int__div__pos__eq,axiom,
    ! [A2: int,B2: int,Q3: int,R3: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q3)),R3) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R3))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R3),B2))
         => ( divide_divide(int,A2,B2) = Q3 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1098_split__neg__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,bool)),N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),P,divide_divide(int,N,K)),modulo_modulo(int,N,K)))
      <=> ! [I3: int,J3: int] :
            ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),J3))
              & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J3),zero_zero(int)))
              & ( N = 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) ) )
           => pp(aa(int,bool,aa(int,fun(int,bool),P,I3),J3)) ) ) ) ).

% split_neg_lemma
tff(fact_1099_split__pos__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,bool)),N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),P,divide_divide(int,N,K)),modulo_modulo(int,N,K)))
      <=> ! [I3: int,J3: int] :
            ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),J3))
              & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J3),K))
              & ( N = 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) ) )
           => pp(aa(int,bool,aa(int,fun(int,bool),P,I3),J3)) ) ) ) ).

% split_pos_lemma
tff(fact_1100_zdiv__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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_1101_zmod__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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_1102_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(dvd_dvd(nat,M,N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M)) ) ) ).

% dvd_pos_nat
tff(fact_1103_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),zero_zero(extended_enat)),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M),N)))
    <=> ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),zero_zero(extended_enat)),M))
        & pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less(extended_enat),zero_zero(extended_enat)),N)) ) ) ).

% enat_0_less_mult_iff
tff(fact_1104_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( pp(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_1105_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( pp(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_1106_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A] :
          ( pp(dvd_dvd(A,A2,one_one(A)))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [B5: A] :
                  ( ( B5 != zero_zero(A) )
                 => ( pp(dvd_dvd(A,B5,one_one(A)))
                   => ( ( divide_divide(A,one_one(A),A2) = B5 )
                     => ( ( divide_divide(A,one_one(A),B5) = A2 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B5) = one_one(A) )
                         => ( divide_divide(A,C2,A2) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B5) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_1107_evenE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
         => ~ ! [B5: A] : A2 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B5) ) ) ).

% evenE
tff(fact_1108_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M),M))
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel2
tff(fact_1109_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N),M))
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel1
tff(fact_1110_of__bool__odd__eq__mod__2,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] : aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% of_bool_odd_eq_mod_2
tff(fact_1111_even__two__times__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = A2 ) ) ) ).

% even_two_times_div_two
tff(fact_1112_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(bool,A,zero_neq_one_of_bool(A),P))) ) ).

% zero_less_eq_of_bool
tff(fact_1113_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,zero_neq_one_of_bool(A),P)),one_one(A))) ) ).

% of_bool_less_eq_one
tff(fact_1114_split__of__bool__asm,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,bool),P2: bool] :
          ( pp(aa(A,bool,P,aa(bool,A,zero_neq_one_of_bool(A),P2)))
        <=> ~ ( ( pp(P2)
                & ~ pp(aa(A,bool,P,one_one(A))) )
              | ( ~ pp(P2)
                & ~ pp(aa(A,bool,P,zero_zero(A))) ) ) ) ) ).

% split_of_bool_asm
tff(fact_1115_split__of__bool,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,bool),P2: bool] :
          ( pp(aa(A,bool,P,aa(bool,A,zero_neq_one_of_bool(A),P2)))
        <=> ( ( pp(P2)
             => pp(aa(A,bool,P,one_one(A))) )
            & ( ~ pp(P2)
             => pp(aa(A,bool,P,zero_zero(A))) ) ) ) ) ).

% split_of_bool
tff(fact_1116_of__bool__def,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: bool] :
          ( ( pp(P2)
           => ( aa(bool,A,zero_neq_one_of_bool(A),P2) = one_one(A) ) )
          & ( ~ pp(P2)
           => ( aa(bool,A,zero_neq_one_of_bool(A),P2) = zero_zero(A) ) ) ) ) ).

% of_bool_def
tff(fact_1117_not__is__unit__0,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ~ pp(dvd_dvd(A,zero_zero(A),one_one(A))) ) ).

% not_is_unit_0
tff(fact_1118_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( pp(dvd_dvd(A,B2,A2))
         => ( ( divide_divide(A,A2,B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_1119_dvd__div__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(dvd_dvd(A,B2,one_one(A)))
         => ( pp(dvd_dvd(A,A2,divide_divide(A,C2,B2)))
          <=> pp(dvd_dvd(A,A2,C2)) ) ) ) ).

% dvd_div_unit_iff
tff(fact_1120_div__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(dvd_dvd(A,B2,one_one(A)))
         => ( pp(dvd_dvd(A,divide_divide(A,A2,B2),C2))
          <=> pp(dvd_dvd(A,A2,C2)) ) ) ) ).

% div_unit_dvd_iff
tff(fact_1121_unit__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(dvd_dvd(A,A2,one_one(A)))
         => ( ( divide_divide(A,B2,A2) = divide_divide(A,C2,A2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_div_cancel
tff(fact_1122_div__plus__div__distrib__dvd__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(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_1123_div__plus__div__distrib__dvd__left,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(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_1124_div__power,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,N: nat] :
          ( pp(dvd_dvd(A,B2,A2))
         => ( aa(nat,A,power_power(A,divide_divide(A,A2,B2)),N) = divide_divide(A,aa(nat,A,power_power(A,A2),N),aa(nat,A,power_power(A,B2),N)) ) ) ) ).

% div_power
tff(fact_1125_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = zero_zero(A) )
         => pp(dvd_dvd(A,B2,A2)) ) ) ).

% mod_0_imp_dvd
tff(fact_1126_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( pp(dvd_dvd(A,A2,B2))
        <=> ( modulo_modulo(A,B2,A2) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_1127_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( modulo_modulo(A,A2,B2) = zero_zero(A) )
        <=> pp(dvd_dvd(A,B2,A2)) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_1128_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1129_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_1130_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1131_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1132_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1133_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1134_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),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_1135_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) ) ) ) ) ).

% mult_le_0_iff
tff(fact_1136_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1137_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1138_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1139_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1140_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1141_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),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_1142_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),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_1143_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).

% mult_mono'
tff(fact_1144_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).

% mult_mono
tff(fact_1145_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ) ) ).

% mult_neg_neg
tff(fact_1146_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),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_1147_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ) ).

% mult_less_0_iff
tff(fact_1148_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))) ) ) ) ).

% mult_neg_pos
tff(fact_1149_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))) ) ) ) ).

% mult_pos_neg
tff(fact_1150_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ) ) ).

% mult_pos_pos
tff(fact_1151_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),zero_zero(A))) ) ) ) ).

% mult_pos_neg2
tff(fact_1152_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A))) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_1153_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ).

% zero_less_mult_pos
tff(fact_1154_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2)) ) ) ) ).

% zero_less_mult_pos2
tff(fact_1155_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1156_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1157_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1158_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1159_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1160_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1161_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1162_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) )
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1163_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_1164_dvd__power__le,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y: A,N: nat,M: nat] :
          ( pp(dvd_dvd(A,X,Y))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => pp(dvd_dvd(A,aa(nat,A,power_power(A,X),N),aa(nat,A,power_power(A,Y),M))) ) ) ) ).

% dvd_power_le
tff(fact_1165_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat,B2: A,M: nat] :
          ( pp(dvd_dvd(A,aa(nat,A,power_power(A,A2),N),B2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => pp(dvd_dvd(A,aa(nat,A,power_power(A,A2),M),B2)) ) ) ) ).

% power_le_dvd
tff(fact_1166_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,N: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => pp(dvd_dvd(A,aa(nat,A,power_power(A,A2),M),aa(nat,A,power_power(A,A2),N))) ) ) ).

% le_imp_power_dvd
tff(fact_1167_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),M),N))) ) ) ) ).

% less_1_mult
tff(fact_1168_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_1169_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_1170_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_1171_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_1172_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = divide_divide(A,B2,C2) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2 ) )
            & ( ( C2 = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq
tff(fact_1173_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( divide_divide(A,B2,C2) = A2 )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq
tff(fact_1174_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,X: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( ( divide_divide(A,X,Y) = divide_divide(A,W,Z) )
            <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Z) = aa(A,A,aa(A,fun(A,A),times_times(A),W),Y) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_1175_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_1176_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_1177_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ~ pp(dvd_dvd(nat,N,M)) ) ) ).

% nat_dvd_not_less
tff(fact_1178_div__mult2__numeral__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,K: num,L: num] : divide_divide(A,divide_divide(A,A2,aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),L)) = divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),K),L))) ) ).

% div_mult2_numeral_eq
tff(fact_1179_left__right__inverse__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A,Y: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = one_one(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),N)),aa(nat,A,power_power(A,Y),N)) = one_one(A) ) ) ) ).

% left_right_inverse_power
tff(fact_1180_power__Suc,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A2: A,N: nat] : aa(nat,A,power_power(A,A2),aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,A2),N)) ) ).

% power_Suc
tff(fact_1181_power__Suc2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,N: nat] : aa(nat,A,power_power(A,A2),aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),N)),A2) ) ).

% power_Suc2
tff(fact_1182_mod__eqE,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,B2,C2) )
         => ~ ! [D3: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D3)) ) ) ).

% mod_eqE
tff(fact_1183_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% Suc_mult_less_cancel1
tff(fact_1184_power__add,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: nat,N: nat] : aa(nat,A,power_power(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),M)),aa(nat,A,power_power(A,A2),N)) ) ).

% power_add
tff(fact_1185_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% nat_mult_less_cancel1
tff(fact_1186_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
      <=> ( M = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_1187_mult__less__mono2,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J))) ) ) ).

% mult_less_mono2
tff(fact_1188_mult__less__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K))) ) ) ).

% mult_less_mono1
tff(fact_1189_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% Suc_mult_le_cancel1
tff(fact_1190_mult__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) ).

% mult_Suc
tff(fact_1191_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
     => ( ( N = one_one(nat) )
        | ( M = zero_zero(nat) ) ) ) ).

% mult_eq_self_implies_10
tff(fact_1192_less__mult__imp__div__less,axiom,
    ! [M: nat,I2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),divide_divide(nat,M,N)),I2)) ) ).

% less_mult_imp_div_less
tff(fact_1193_div__times__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,M,N)),N)),M)) ).

% div_times_less_eq_dividend
tff(fact_1194_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),divide_divide(nat,M,N))),M)) ).

% times_div_less_eq_dividend
tff(fact_1195_mod__eq__0D,axiom,
    ! [M: nat,D2: nat] :
      ( ( modulo_modulo(nat,M,D2) = zero_zero(nat) )
     => ? [Q2: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D2),Q2) ) ).

% mod_eq_0D
tff(fact_1196_nat__mod__eq__iff,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( modulo_modulo(nat,X,N) = modulo_modulo(nat,Y,N) )
    <=> ? [Q1: nat,Q22: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_1197_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P: fun(A,bool),A2: A] :
          ( ! [A5: A] :
              ( ( divide_divide(A,A5,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A5 )
             => pp(aa(A,bool,P,A5)) )
         => ( ! [A5: A,B5: bool] :
                ( pp(aa(A,bool,P,A5))
               => ( ( divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),B5)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A5)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A5 )
                 => pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(bool,A,zero_neq_one_of_bool(A),B5)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),A5)))) ) )
           => pp(aa(A,bool,P,A2)) ) ) ) ).

% bits_induct
tff(fact_1198_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
         => ~ ! [B5: 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),aa(num,num,bit0,one2))),B5)),one_one(A)) ) ) ).

% oddE
tff(fact_1199_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: bool,Uv2: bool,D3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(Uu2,Uv2)),D3)
     => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList3: list(vEBT_VEBT),Summary3: vEBT_VEBT,Deg3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Mima,Deg2,TreeList3,Summary3)),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_1200_power__odd__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,N: nat] : aa(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),aa(num,num,bit0,one2))),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,aa(nat,A,power_power(A,A2),N)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% power_odd_eq
tff(fact_1201_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] : modulo_modulo(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) ) ).

% exp_mod_exp
tff(fact_1202_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( pp(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_1203_even__numeral,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: num] : pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)))) ) ).

% even_numeral
tff(fact_1204_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A2: A] :
          ( pp(dvd_dvd(A,B2,one_one(A)))
         => ( modulo_modulo(A,A2,B2) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_1205_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(dvd_dvd(A,aa(nat,A,power_power(A,A2),N),one_one(A)))
        <=> ( pp(dvd_dvd(A,A2,one_one(A)))
            | ( N = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_1206_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_1207_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_1208_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).

% mult_left_less_imp_less
tff(fact_1209_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_1210_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_1211_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ).

% mult_right_less_imp_less
tff(fact_1212_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1213_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_1214_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1215_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1216_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% mult_left_le_imp_le
tff(fact_1217_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% mult_right_le_imp_le
tff(fact_1218_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1219_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2))) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1220_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)),X)) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_1221_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)),X)) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_1222_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),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_1223_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),one_one(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),A2)) ) ) ) ).

% mult_left_le
tff(fact_1224_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))) ) ).

% sum_squares_ge_zero
tff(fact_1225_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)))
        <=> ( ( X = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_1226_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))) ) ).

% not_sum_squares_lt_zero
tff(fact_1227_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))))
        <=> ( ( X != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1228_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_1229_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,C2,A2)),divide_divide(A,C2,B2))) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_1230_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,C2,A2)),divide_divide(A,C2,B2))) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_1231_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),divide_divide(A,X,Y))) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_1232_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,X,Y)),Z)) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_1233_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),divide_divide(A,B2,C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) ) ) ) ).

% pos_less_divide_eq
tff(fact_1234_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C2)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% pos_divide_less_eq
tff(fact_1235_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),divide_divide(A,B2,C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% neg_less_divide_eq
tff(fact_1236_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C2)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) ) ) ) ).

% neg_divide_less_eq
tff(fact_1237_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),divide_divide(A,B2,C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq
tff(fact_1238_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C2)),A2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ) ) ) ) ).

% divide_less_eq
tff(fact_1239_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) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2) = B2 ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_1240_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) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(num,A,numeral_numeral(A),W) = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_1241_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,X,Z)),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)),Z) ) ) ) ).

% divide_add_eq_iff
tff(fact_1242_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),X),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),X),Z)),Y),Z) ) ) ) ).

% add_divide_eq_iff
tff(fact_1243_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,X: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),divide_divide(A,X,Y)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Y) ) ) ) ).

% add_num_frac
tff(fact_1244_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,X: A,Z: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,X,Y)),Z) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Y) ) ) ) ).

% add_frac_num
tff(fact_1245_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,X: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,X,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),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ) ) ) ).

% add_frac_eq
tff(fact_1246_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),divide_divide(A,B2,Z)) = A2 ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),divide_divide(A,B2,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),A2),Z)),B2),Z) ) ) ) ) ).

% add_divide_eq_if_simps(1)
tff(fact_1247_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,Z)),B2) = B2 ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,Z)),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_1248_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,A2),N)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,A2),N)))) ) ) ).

% power_less_power_Suc
tff(fact_1249_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,A2),N)))) ) ) ).

% power_gt1_lemma
tff(fact_1250_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( pp(dvd_dvd(nat,K,N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ) ).

% dvd_imp_le
tff(fact_1251_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_1252_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_1253_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_1254_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_1255_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_1256_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_1257_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_1258_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_1259_pos__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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),aa(num,num,bit0,one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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),aa(num,num,bit0,one2))),modulo_modulo(int,B2,A2))) ) ) ).

% pos_zmod_mult_2
tff(fact_1260_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N))) ) ) ).

% one_less_mult
tff(fact_1261_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N))) ) ) ).

% n_less_m_mult_n
tff(fact_1262_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M))) ) ) ).

% n_less_n_mult_m
tff(fact_1263_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),modulo_modulo(nat,M,N)))
    <=> ~ pp(dvd_dvd(nat,N,M)) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_1264_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% nat_mult_le_cancel1
tff(fact_1265_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Q3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),divide_divide(nat,M,Q3)),N))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q3))) ) ) ).

% div_less_iff_less_mult
tff(fact_1266_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = divide_divide(nat,M,N) ) ) ).

% nat_mult_div_cancel1
tff(fact_1267_mod__eq__nat1E,axiom,
    ! [M: nat,Q3: nat,N: nat] :
      ( ( modulo_modulo(nat,M,Q3) = modulo_modulo(nat,N,Q3) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
       => ~ ! [S3: nat] : M != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q3),S3)) ) ) ).

% mod_eq_nat1E
tff(fact_1268_mod__eq__nat2E,axiom,
    ! [M: nat,Q3: nat,N: nat] :
      ( ( modulo_modulo(nat,M,Q3) = modulo_modulo(nat,N,Q3) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ~ ! [S3: nat] : N != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q3),S3)) ) ) ).

% mod_eq_nat2E
tff(fact_1269_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( modulo_modulo(nat,X,N) = modulo_modulo(nat,Y,N) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),X))
       => ? [Q2: nat] : X = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) ) ) ).

% nat_mod_eq_lemma
tff(fact_1270_div__mod__decomp,axiom,
    ! [A3: nat,N: 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,N)),N)),modulo_modulo(nat,A3,N)) ).

% div_mod_decomp
tff(fact_1271_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q3: nat] : modulo_modulo(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),modulo_modulo(nat,divide_divide(nat,M,N),Q3))),modulo_modulo(nat,M,N)) ).

% mod_mult2_eq
tff(fact_1272_even__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),zero_zero(A))) ) ).

% even_zero
tff(fact_1273_odd__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),one_one(A))) ) ).

% odd_one
tff(fact_1274_odd__even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
         => ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2))
           => pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ) ) ).

% odd_even_add
tff(fact_1275_bit__eq__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
            <=> pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2)) )
            & ( divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = divide_divide(A,B2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ) ) ).

% bit_eq_rec
tff(fact_1276_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [X: A,M: nat,N: nat] :
          ( ( X != zero_zero(A) )
         => ( pp(dvd_dvd(A,aa(nat,A,power_power(A,X),M),aa(nat,A,power_power(A,X),N)))
          <=> ( pp(dvd_dvd(A,X,one_one(A)))
              | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ) ).

% dvd_power_iff
tff(fact_1277_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat,X: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
            | ( X = one_one(A) ) )
         => pp(dvd_dvd(A,X,aa(nat,A,power_power(A,X),N))) ) ) ).

% dvd_power
tff(fact_1278_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] :
          ( ! [Z2: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),one_one(A)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z2),X)),Y)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% field_le_mult_one_interval
tff(fact_1279_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),C2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2)) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_1280_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A))) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_1281_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),C2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),A2)) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_1282_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),one_one(A))) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_1283_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),C2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A2)) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_1284_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A))) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1285_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),C2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),one_one(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),A2)) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_1286_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),B2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),one_one(A))) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1287_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,C2,A2)),divide_divide(A,C2,B2))) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_1288_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),divide_divide(A,X,Y))) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_1289_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Y)),Z)) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_1290_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),divide_divide(A,B2,C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) ) ) ) ).

% pos_le_divide_eq
tff(fact_1291_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C2)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% pos_divide_le_eq
tff(fact_1292_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),divide_divide(A,B2,C2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) ) ) ) ).

% neg_le_divide_eq
tff(fact_1293_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C2)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) ) ) ) ).

% neg_divide_le_eq
tff(fact_1294_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,C2,A2)),divide_divide(A,C2,B2))) ) ) ) ) ).

% divide_left_mono
tff(fact_1295_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),divide_divide(A,B2,C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq
tff(fact_1296_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C2)),A2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ) ) ) ).

% divide_le_eq
tff(fact_1297_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [X: A,A2: A,Y: A,U: A,V: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),U))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),V))
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A2)) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1298_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),W)),divide_divide(A,B2,C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_1299_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C2)),aa(num,A,numeral_numeral(A),W)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_1300_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,A2),N))),aa(nat,A,power_power(A,A2),N))) ) ) ) ).

% power_Suc_less
tff(fact_1301_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),aa(num,num,bit0,one2))),A2)),B2) ) ).

% left_add_twice
tff(fact_1302_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),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Z) ) ).

% mult_2_right
tff(fact_1303_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),aa(num,num,bit0,one2))),Z) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Z) ) ).

% mult_2
tff(fact_1304_power2__eq__square,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2) ) ).

% power2_eq_square
tff(fact_1305_power4__eq__xxxx,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [X: A] : aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),X)),X) ) ).

% power4_eq_xxxx
tff(fact_1306_double__not__eq__Suc__double,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M) != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ).

% double_not_eq_Suc_double
tff(fact_1307_Suc__double__not__eq__double,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)) != aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) ).

% Suc_double_not_eq_double
tff(fact_1308_power__even__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,N: nat] : aa(nat,A,power_power(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) = aa(nat,A,power_power(A,aa(nat,A,power_power(A,A2),N)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% power_even_eq
tff(fact_1309_power__dvd__imp__le,axiom,
    ! [I2: nat,M: nat,N: nat] :
      ( pp(dvd_dvd(nat,aa(nat,nat,power_power(nat,I2),M),aa(nat,nat,power_power(nat,I2),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),I2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% power_dvd_imp_le
tff(fact_1310_neg__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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),aa(num,num,bit0,one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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_1311_pos__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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),aa(num,num,bit0,one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A2)) = divide_divide(int,B2,A2) ) ) ).

% pos_zdiv_mult_2
tff(fact_1312_div__nat__eqI,axiom,
    ! [N: nat,Q3: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q3)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q3))))
       => ( divide_divide(nat,M,N) = Q3 ) ) ) ).

% div_nat_eqI
tff(fact_1313_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Q3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),divide_divide(nat,N,Q3)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)),N)) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_1314_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),divide_divide(nat,M,N))))) ) ).

% dividend_less_times_div
tff(fact_1315_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,M,N)),N)))) ) ).

% dividend_less_div_times
tff(fact_1316_split__div,axiom,
    ! [P: fun(nat,bool),M: nat,N: nat] :
      ( pp(aa(nat,bool,P,divide_divide(nat,M,N)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(nat,bool,P,zero_zero(nat))) )
        & ( ( N != zero_zero(nat) )
         => ! [I3: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N))
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I3)),J3) )
               => pp(aa(nat,bool,P,I3)) ) ) ) ) ) ).

% split_div
tff(fact_1317_split__mod,axiom,
    ! [P: fun(nat,bool),M: nat,N: nat] :
      ( pp(aa(nat,bool,P,modulo_modulo(nat,M,N)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(nat,bool,P,M)) )
        & ( ( N != zero_zero(nat) )
         => ! [I3: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),N))
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I3)),J3) )
               => pp(aa(nat,bool,P,J3)) ) ) ) ) ) ).

% split_mod
tff(fact_1318_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [A5: bool,B5: bool,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(A5,B5)),X3)
     => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Ux2: nat] : X != 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)),V3: nat,TreeList3: list(vEBT_VEBT),S3: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList3,S3)),X3) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_1319_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_1320_odd__iff__mod__2__eq__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
        <=> ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_1321_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: A,B2: A] :
          ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A2),N)),aa(nat,A,power_power(A,B2),N))) ) ) ) ).

% power_mono_odd
tff(fact_1322_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [X: A,A2: A,Y: A,U: A,V: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),A2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),U))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),V))
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),X)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A2)) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1323_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),W)),divide_divide(A,B2,C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_1324_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C2)),aa(num,A,numeral_numeral(A),W)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_1325_odd__pos,axiom,
    ! [N: nat] :
      ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% odd_pos
tff(fact_1326_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_1327_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
     => ( pp(dvd_dvd(nat,aa(nat,nat,power_power(nat,K),M),aa(nat,nat,power_power(nat,K),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% dvd_power_iff_le
tff(fact_1328_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_1329_four__x__squared,axiom,
    ! [X: real] : aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% four_x_squared
tff(fact_1330_split__div_H,axiom,
    ! [P: fun(nat,bool),M: nat,N: nat] :
      ( pp(aa(nat,bool,P,divide_divide(nat,M,N)))
    <=> ( ( ( N = zero_zero(nat) )
          & pp(aa(nat,bool,P,zero_zero(nat))) )
        | ? [Q5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q5)),M))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q5))))
            & pp(aa(nat,bool,P,Q5)) ) ) ) ).

% split_div'
tff(fact_1331_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),M))
     => ( modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),M) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_1332_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
           => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != zero_zero(A) ) )
         => ~ ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
             => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) != one_one(A) ) ) ) ) ).

% parity_cases
tff(fact_1333_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
           => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = zero_zero(A) ) )
          & ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
           => ( modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = one_one(A) ) ) ) ) ).

% mod2_eq_if
tff(fact_1334_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: A] :
          ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),N))) ) ) ).

% zero_le_even_power
tff(fact_1335_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: A] :
          ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),N)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ).

% zero_le_odd_power
tff(fact_1336_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),N)))
        <=> ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
            | ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ) ).

% zero_le_power_eq
tff(fact_1337_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))),B2)) = modulo_modulo(A,A2,B2) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_1338_power2__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [X: A,Y: A] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,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),aa(num,num,bit0,one2))),X)),Y)) ) ).

% power2_sum
tff(fact_1339_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) ) ).

% zero_le_even_power'
tff(fact_1340_nat__bit__induct,axiom,
    ! [P: fun(nat,bool),N: nat] :
      ( pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,P,N2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
             => pp(aa(nat,bool,P,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2))) ) )
       => ( ! [N2: nat] :
              ( pp(aa(nat,bool,P,N2))
             => pp(aa(nat,bool,P,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N2)))) )
         => pp(aa(nat,bool,P,N)) ) ) ) ).

% nat_bit_induct
tff(fact_1341_L2__set__mult__ineq__lemma,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,aa(real,fun(real,real),times_times(real),A2),C2))),aa(real,real,aa(real,fun(real,real),times_times(real),B2),D2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,D2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,B2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,C2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ).

% L2_set_mult_ineq_lemma
tff(fact_1342_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,A2),N)))
        <=> ( ( N = zero_zero(nat) )
            | ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
              & ( A2 != zero_zero(A) ) )
            | ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ) ) ).

% zero_less_power_eq
tff(fact_1343_Euclid__induct,axiom,
    ! [P: fun(nat,fun(nat,bool)),A2: nat,B2: nat] :
      ( ! [A5: nat,B5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A5),B5))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,B5),A5)) )
     => ( ! [A5: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A5),zero_zero(nat)))
       => ( ! [A5: nat,B5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A5),B5))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A5),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A5),B5))) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),P,A2),B2)) ) ) ) ).

% Euclid_induct
tff(fact_1344_sum__squares__bound,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),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),aa(num,num,bit0,one2))),X)),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% sum_squares_bound
tff(fact_1345_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))),B2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2))) = divide_divide(A,A2,B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_1346_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(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),aa(num,num,bit0,one2))),N)))))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_1347_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(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),aa(num,num,bit0,one2))),N)))),zero_zero(A))) ) ) ).

% odd_power_less_zero
tff(fact_1348_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: bool,Uv2: bool,Uw2: nat] : X != 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] : X != 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)
       => ( ! [Mi3: nat,Ma3: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT,X3: nat] : X != 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,Mi3),Ma3)),zero_zero(nat),Va3,Vb2)),X3)
         => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList3: list(vEBT_VEBT),Vc: vEBT_VEBT,X3: nat] : X != 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,Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList3,Vc)),X3)
           => ~ ! [V3: nat,TreeList3: list(vEBT_VEBT),Vd: vEBT_VEBT,X3: nat] : X != aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList3,Vd)),X3) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_1349_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A2),N)),zero_zero(A)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
            & ( ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) )
              | ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_1350_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => ( ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) = modulo_modulo(A,X,M) )
              | ( modulo_modulo(A,X,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,X,M)),M) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_1351_option_Osize__gen_I1_J,axiom,
    ! [A: $tType,X: fun(A,nat)] : size_option(A,X,none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size_gen(1)
tff(fact_1352_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(bool,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% flip_bit_0
tff(fact_1353_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),zero_zero(nat)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% set_bit_0
tff(fact_1354_even__even__mod__4__iff,axiom,
    ! [N: nat] :
      ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
    <=> pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))))) ) ).

% even_even_mod_4_iff
tff(fact_1355_div2__even__ext__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( divide_divide(nat,X,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = divide_divide(nat,Y,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
     => ( ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),X))
        <=> pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Y)) )
       => ( X = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_1356_unset__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),zero_zero(nat)),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% unset_bit_0
tff(fact_1357_incr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,bool),K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => ( ! [X3: int] :
            ( pp(aa(int,bool,P,X3))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D2))) )
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
         => ! [X2: int] :
              ( pp(aa(int,bool,P,X2))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2)))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1358_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),aa(nat,nat,suc,N)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),N),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% unset_bit_Suc
tff(fact_1359_flip__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : bit_se8732182000553998342ip_bit(A,aa(nat,nat,suc,N),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),bit_se8732182000553998342ip_bit(A,N,divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% flip_bit_Suc
tff(fact_1360_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),aa(nat,nat,suc,N)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),N),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% set_bit_Suc
tff(fact_1361_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P: fun(A,bool),L: A] :
          ( ? [X4: A] : pp(aa(A,bool,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X4)))
        <=> ? [X4: A] :
              ( pp(dvd_dvd(A,L,aa(A,A,aa(A,fun(A,A),plus_plus(A),X4),zero_zero(A))))
              & pp(aa(A,bool,P,X4)) ) ) ) ).

% unity_coeff_ex
tff(fact_1362_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_1363_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% set_bit_nonnegative_int_iff
tff(fact_1364_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se8732182000553998342ip_bit(int,N,K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_1365_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% unset_bit_negative_int_iff
tff(fact_1366_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% set_bit_negative_int_iff
tff(fact_1367_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se8732182000553998342ip_bit(int,N,K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% flip_bit_negative_int_iff
tff(fact_1368_unset__bit__less__eq,axiom,
    ! [N: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K)),K)) ).

% unset_bit_less_eq
tff(fact_1369_set__bit__greater__eq,axiom,
    ! [K: int,N: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K))) ).

% set_bit_greater_eq
tff(fact_1370_minf_I11_J,axiom,
    ! [C: $tType,D: $tType] :
      ( ord(C)
     => ! [F5: D] :
        ? [Z2: C] :
        ! [X2: C] :
          ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),X2),Z2))
         => ( F5 = F5 ) ) ) ).

% minf(11)
tff(fact_1371_minf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T2),X2)) ) ) ).

% minf(7)
tff(fact_1372_minf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),T2)) ) ) ).

% minf(5)
tff(fact_1373_minf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z2))
         => ( X2 != T2 ) ) ) ).

% minf(4)
tff(fact_1374_minf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z2))
         => ( X2 != T2 ) ) ) ).

% minf(3)
tff(fact_1375_minf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool),P3: fun(A,bool),Q: fun(A,bool),Q6: fun(A,bool)] :
          ( ? [Z3: A] :
            ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z3))
             => ( pp(aa(A,bool,P,X3))
              <=> pp(aa(A,bool,P3,X3)) ) )
         => ( ? [Z3: A] :
              ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z3))
               => ( pp(aa(A,bool,Q,X3))
                <=> pp(aa(A,bool,Q6,X3)) ) )
           => ? [Z2: A] :
              ! [X2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z2))
               => ( ( pp(aa(A,bool,P,X2))
                    | pp(aa(A,bool,Q,X2)) )
                <=> ( pp(aa(A,bool,P3,X2))
                    | pp(aa(A,bool,Q6,X2)) ) ) ) ) ) ) ).

% minf(2)
tff(fact_1376_minf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool),P3: fun(A,bool),Q: fun(A,bool),Q6: fun(A,bool)] :
          ( ? [Z3: A] :
            ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z3))
             => ( pp(aa(A,bool,P,X3))
              <=> pp(aa(A,bool,P3,X3)) ) )
         => ( ? [Z3: A] :
              ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z3))
               => ( pp(aa(A,bool,Q,X3))
                <=> pp(aa(A,bool,Q6,X3)) ) )
           => ? [Z2: A] :
              ! [X2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z2))
               => ( ( pp(aa(A,bool,P,X2))
                    & pp(aa(A,bool,Q,X2)) )
                <=> ( pp(aa(A,bool,P3,X2))
                    & pp(aa(A,bool,Q6,X2)) ) ) ) ) ) ) ).

% minf(1)
tff(fact_1377_pinf_I11_J,axiom,
    ! [C: $tType,D: $tType] :
      ( ord(C)
     => ! [F5: D] :
        ? [Z2: C] :
        ! [X2: C] :
          ( pp(aa(C,bool,aa(C,fun(C,bool),ord_less(C),Z2),X2))
         => ( F5 = F5 ) ) ) ).

% pinf(11)
tff(fact_1378_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T2),X2)) ) ) ).

% pinf(7)
tff(fact_1379_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),T2)) ) ) ).

% pinf(5)
tff(fact_1380_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X2))
         => ( X2 != T2 ) ) ) ).

% pinf(4)
tff(fact_1381_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X2))
         => ( X2 != T2 ) ) ) ).

% pinf(3)
tff(fact_1382_pinf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool),P3: fun(A,bool),Q: fun(A,bool),Q6: fun(A,bool)] :
          ( ? [Z3: A] :
            ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X3))
             => ( pp(aa(A,bool,P,X3))
              <=> pp(aa(A,bool,P3,X3)) ) )
         => ( ? [Z3: A] :
              ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X3))
               => ( pp(aa(A,bool,Q,X3))
                <=> pp(aa(A,bool,Q6,X3)) ) )
           => ? [Z2: A] :
              ! [X2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X2))
               => ( ( pp(aa(A,bool,P,X2))
                    | pp(aa(A,bool,Q,X2)) )
                <=> ( pp(aa(A,bool,P3,X2))
                    | pp(aa(A,bool,Q6,X2)) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_1383_pinf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool),P3: fun(A,bool),Q: fun(A,bool),Q6: fun(A,bool)] :
          ( ? [Z3: A] :
            ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X3))
             => ( pp(aa(A,bool,P,X3))
              <=> pp(aa(A,bool,P3,X3)) ) )
         => ( ? [Z3: A] :
              ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z3),X3))
               => ( pp(aa(A,bool,Q,X3))
                <=> pp(aa(A,bool,Q6,X3)) ) )
           => ? [Z2: A] :
              ! [X2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X2))
               => ( ( pp(aa(A,bool,P,X2))
                    & pp(aa(A,bool,Q,X2)) )
                <=> ( pp(aa(A,bool,P3,X2))
                    & pp(aa(A,bool,Q6,X2)) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_1384_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T2),X2)) ) ) ).

% minf(8)
tff(fact_1385_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),Z2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),T2)) ) ) ).

% minf(6)
tff(fact_1386_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T2),X2)) ) ) ).

% pinf(8)
tff(fact_1387_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [T2: A] :
        ? [Z2: A] :
        ! [X2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z2),X2))
         => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),T2)) ) ) ).

% pinf(6)
tff(fact_1388_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero(nat) )
     => ~ ! [N2: nat] : X != aa(nat,nat,suc,N2) ) ).

% list_decode.cases
tff(fact_1389_imp__le__cong,axiom,
    ! [X: int,X5: int,P: bool,P3: bool] :
      ( ( X = X5 )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X5))
         => ( pp(P)
          <=> pp(P3) ) )
       => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
           => pp(P) )
        <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X5))
           => pp(P3) ) ) ) ) ).

% imp_le_cong
tff(fact_1390_conj__le__cong,axiom,
    ! [X: int,X5: int,P: bool,P3: bool] :
      ( ( X = X5 )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X5))
         => ( pp(P)
          <=> pp(P3) ) )
       => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
            & pp(P) )
        <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X5))
            & pp(P3) ) ) ) ) ).

% conj_le_cong
tff(fact_1391_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A2)))
        <=> ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
            & ( M != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_1392_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),bit_se8732182000553998342ip_bit(A,M,A2)))
        <=> ~ ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
            <=> ( M = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_1393_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2)))
        <=> ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
            | ( M = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_1394_minf_I10_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D2: B,S: B] :
        ? [Z2: B] :
        ! [X2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),Z2))
         => ( ~ pp(dvd_dvd(B,D2,aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S)))
          <=> ~ pp(dvd_dvd(B,D2,aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S))) ) ) ) ).

% minf(10)
tff(fact_1395_minf_I9_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D2: B,S: B] :
        ? [Z2: B] :
        ! [X2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X2),Z2))
         => ( pp(dvd_dvd(B,D2,aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S)))
          <=> pp(dvd_dvd(B,D2,aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S))) ) ) ) ).

% minf(9)
tff(fact_1396_pinf_I10_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D2: B,S: B] :
        ? [Z2: B] :
        ! [X2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z2),X2))
         => ( ~ pp(dvd_dvd(B,D2,aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S)))
          <=> ~ pp(dvd_dvd(B,D2,aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S))) ) ) ) ).

% pinf(10)
tff(fact_1397_pinf_I9_J,axiom,
    ! [B: $tType] :
      ( ( plus(B)
        & linorder(B)
        & dvd(B) )
     => ! [D2: B,S: B] :
        ? [Z2: B] :
        ! [X2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z2),X2))
         => ( pp(dvd_dvd(B,D2,aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S)))
          <=> pp(dvd_dvd(B,D2,aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),S))) ) ) ) ).

% pinf(9)
tff(fact_1398_length__mul__elem,axiom,
    ! [A: $tType,Xs: list(list(A)),N: nat] :
      ( ! [X3: list(A)] :
          ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),X3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
         => ( aa(list(A),nat,size_size(list(A)),X3) = N ) )
     => ( 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)),N) ) ) ).

% length_mul_elem
tff(fact_1399_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_1400_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_1401_product__nth,axiom,
    ! [A: $tType,B: $tType,N: nat,Xs: list(A),Ys: list(B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys))))
     => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xs,Ys)),N) = aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Xs),divide_divide(nat,N,aa(list(B),nat,size_size(list(B)),Ys)))),aa(nat,B,nth(B,Ys),modulo_modulo(nat,N,aa(list(B),nat,size_size(list(B)),Ys)))) ) ) ).

% product_nth
tff(fact_1402_triangle__def,axiom,
    ! [N: nat] : nat_triangle(N) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% triangle_def
tff(fact_1403_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q3: int,R3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B2))
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q3),R3))
       => 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),aa(num,num,bit0,one2))),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2),aa(int,product_prod(int,int),product_Pair(int,int,Q3),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),R3)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_1404_length__product,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),product(A,B,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) ).

% length_product
tff(fact_1405_triangle__Suc,axiom,
    ! [N: nat] : nat_triangle(aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(N)),aa(nat,nat,suc,N)) ).

% triangle_Suc
tff(fact_1406_unique__remainder,axiom,
    ! [A2: int,B2: int,Q3: int,R3: int,Q4: int,R4: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q3),R3))
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q4),R4))
       => ( R3 = R4 ) ) ) ).

% unique_remainder
tff(fact_1407_unique__quotient,axiom,
    ! [A2: int,B2: int,Q3: int,R3: int,Q4: int,R4: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q3),R3))
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q4),R4))
       => ( Q3 = Q4 ) ) ) ).

% unique_quotient
tff(fact_1408_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_1409_div__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R3: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q3),R3))
     => ( divide_divide(int,K,L) = Q3 ) ) ).

% div_int_unique
tff(fact_1410_mod__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R3: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q3),R3))
     => ( modulo_modulo(int,K,L) = R3 ) ) ).

% mod_int_unique
tff(fact_1411_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q3: int] :
      ( ( L != zero_zero(int) )
     => ( ( K = aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L) )
       => eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q3),zero_zero(int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_1412_eucl__rel__int,axiom,
    ! [K: int,L: int] : eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,divide_divide(int,K,L)),modulo_modulo(int,K,L))) ).

% eucl_rel_int
tff(fact_1413_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R3: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q3),R3))
    <=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q3)),R3) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),R3))
            & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),R3),L)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
         => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),R3))
                & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),R3),zero_zero(int))) ) )
            & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
             => ( Q3 = zero_zero(int) ) ) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_1414_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Z))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).

% mult_less_iff1
tff(fact_1415_Divides_Oadjust__div__eq,axiom,
    ! [Q3: int,R3: int] : adjust_div(aa(int,product_prod(int,int),product_Pair(int,int,Q3),R3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q3),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),fequal(int),R3),zero_zero(int))))) ).

% Divides.adjust_div_eq
tff(fact_1416_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_concat_bit(aa(nat,nat,suc,N),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,bit_concat_bit(N,divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),L))) ).

% concat_bit_Suc
tff(fact_1417_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q3: int,R3: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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,Q3),R3))
       => 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),aa(num,num,bit0,one2))),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),B2),aa(int,product_prod(int,int),product_Pair(int,int,Q3),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),R3)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_1418_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,N),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_1419_set__decode__Suc,axiom,
    ! [N: nat,X: nat] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,suc,N)),nat_set_decode(X)))
    <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N),nat_set_decode(divide_divide(nat,X,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% set_decode_Suc
tff(fact_1420_set__decode__0,axiom,
    ! [X: nat] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),nat_set_decode(X)))
    <=> ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),X)) ) ).

% set_decode_0
tff(fact_1421_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R3: A,A2: A,B2: A,C2: A,D2: A] :
          ( ( R3 != zero_zero(A) )
         => ( ( ( A2 = B2 )
              & ( C2 != D2 ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),R3),C2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R3),D2)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_1422_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,M: nat,N: nat] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
            | ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) = zero_zero(A) )
            | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
              & pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),divide_divide(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,minus_minus(nat,N),M))))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_1423_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,A2),A2) = zero_zero(A) ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1424_diff__zero,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,A2),zero_zero(A)) = A2 ) ).

% diff_zero
tff(fact_1425_zero__diff,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,zero_zero(A)),A2) = zero_zero(A) ) ).

% zero_diff
tff(fact_1426_diff__0__right,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,A2),zero_zero(A)) = A2 ) ).

% diff_0_right
tff(fact_1427_diff__self,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,A2),A2) = zero_zero(A) ) ).

% diff_self
tff(fact_1428_add__diff__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(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_1429_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,minus_minus(A,A2),B2)),B2) = A2 ) ).

% diff_add_cancel
tff(fact_1430_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C2: A,A2: A,B2: A] : aa(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,minus_minus(A,A2),B2) ) ).

% add_diff_cancel_left
tff(fact_1431_add__diff__cancel__left_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A] : aa(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_1432_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C2: A,B2: A] : aa(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,minus_minus(A,A2),B2) ) ).

% add_diff_cancel_right
tff(fact_1433_add__diff__cancel__right_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A] : aa(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_1434_minus__mod__self2,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : modulo_modulo(A,aa(A,A,minus_minus(A,A2),B2),B2) = modulo_modulo(A,A2,B2) ) ).

% minus_mod_self2
tff(fact_1435_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,minus_minus(nat,M),N) ).

% diff_Suc_Suc
tff(fact_1436_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),N)),aa(nat,nat,suc,K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),N)),K) ).

% Suc_diff_diff
tff(fact_1437_diff__self__eq__0,axiom,
    ! [M: nat] : aa(nat,nat,minus_minus(nat,M),M) = zero_zero(nat) ).

% diff_self_eq_0
tff(fact_1438_diff__0__eq__0,axiom,
    ! [N: nat] : aa(nat,nat,minus_minus(nat,zero_zero(nat)),N) = zero_zero(nat) ).

% diff_0_eq_0
tff(fact_1439_diff__diff__cancel,axiom,
    ! [I2: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),N))
     => ( aa(nat,nat,minus_minus(nat,N),aa(nat,nat,minus_minus(nat,N),I2)) = I2 ) ) ).

% diff_diff_cancel
tff(fact_1440_diff__diff__left,axiom,
    ! [I2: nat,J: nat,K: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,I2),J)),K) = aa(nat,nat,minus_minus(nat,I2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ).

% diff_diff_left
tff(fact_1441_signed__take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).

% signed_take_bit_of_0
tff(fact_1442_concat__bit__0,axiom,
    ! [K: int,L: int] : aa(int,int,bit_concat_bit(zero_zero(nat),K),L) = L ).

% concat_bit_0
tff(fact_1443_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,minus_minus(A,A2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% diff_ge_0_iff_ge
tff(fact_1444_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,minus_minus(A,A2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ).

% diff_gt_0_iff_gt
tff(fact_1445_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,A2),B2)),B2) = A2 ) ) ) ).

% le_add_diff_inverse2
tff(fact_1446_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,minus_minus(A,A2),B2)) = A2 ) ) ) ).

% le_add_diff_inverse
tff(fact_1447_diff__numeral__special_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,one_one(A)),one_one(A)) = zero_zero(A) ) ) ).

% diff_numeral_special(9)
tff(fact_1448_diff__add__zero,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A2: A,B2: A] : aa(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_1449_right__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [V: num,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,minus_minus(A,B2),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).

% right_diff_distrib_numeral
tff(fact_1450_left__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [A2: A,B2: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),B2)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V))) ) ).

% left_diff_distrib_numeral
tff(fact_1451_div__diff,axiom,
    ! [A: $tType] :
      ( idom_modulo(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(dvd_dvd(A,C2,A2))
         => ( pp(dvd_dvd(A,C2,B2))
           => ( divide_divide(A,aa(A,A,minus_minus(A,A2),B2),C2) = aa(A,A,minus_minus(A,divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ) ) ) ).

% div_diff
tff(fact_1452_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,minus_minus(nat,N),M)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% zero_less_diff
tff(fact_1453_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(nat,nat,minus_minus(nat,M),N) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_1454_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,minus_minus(nat,M),N) = zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% diff_is_0_eq
tff(fact_1455_of__bool__not__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [P: bool] : aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,P)) = aa(A,A,minus_minus(A,one_one(A)),aa(bool,A,zero_neq_one_of_bool(A),P)) ) ).

% of_bool_not_iff
tff(fact_1456_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,minus_minus(nat,I2),aa(nat,nat,minus_minus(nat,J),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_1457_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,J),K)),I2) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I2)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_1458_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),aa(nat,nat,minus_minus(nat,J),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_1459_diff__Suc__1,axiom,
    ! [N: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,N)),one_one(nat)) = N ).

% diff_Suc_1
tff(fact_1460_signed__take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N)),one_one(A)) = one_one(A) ) ).

% signed_take_bit_Suc_1
tff(fact_1461_signed__take__bit__numeral__of__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(num,nat,numeral_numeral(nat),K)),one_one(A)) = one_one(A) ) ).

% signed_take_bit_numeral_of_1
tff(fact_1462_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_concat_bit(N,K),L)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) ) ).

% concat_bit_nonnegative_iff
tff(fact_1463_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_concat_bit(N,K),L)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ).

% concat_bit_negative_iff
tff(fact_1464_Suc__pred,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))) = N ) ) ).

% Suc_pred
tff(fact_1465_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,J),K))),I2) = aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,J)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I2)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_1466_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,minus_minus(nat,I2),aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,J),K))) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),aa(nat,nat,suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_1467_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W),aa(int,int,minus_minus(int,Z),one_one(int))))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z)) ) ).

% zle_diff1_eq
tff(fact_1468_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% signed_take_bit_Suc_bit0
tff(fact_1469_Suc__diff__1,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,N),one_one(nat))) = N ) ) ).

% Suc_diff_1
tff(fact_1470_even__diff,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A2: A,B2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,minus_minus(A,A2),B2)))
        <=> pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ) ).

% even_diff
tff(fact_1471_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))) = N ) ) ).

% odd_Suc_minus_one
tff(fact_1472_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(nat,nat,minus_minus(nat,M),N)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
        | pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ) ).

% even_diff_nat
tff(fact_1473_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A))))
        <=> ( N = zero_zero(nat) ) ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_1474_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(nat,nat,minus_minus(nat,N),one_one(nat)) ) ) ).

% odd_two_times_div_two_nat
tff(fact_1475_diff__right__commute,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A2),C2)),B2) = aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A2),B2)),C2) ) ).

% diff_right_commute
tff(fact_1476_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,minus_minus(A,A2),B2) = aa(A,A,minus_minus(A,C2),D2) )
         => ( ( A2 = B2 )
          <=> ( C2 = D2 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_1477_signed__take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,minus_minus(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,minus_minus(int,K),L)) ).

% signed_take_bit_diff
tff(fact_1478_diff__commute,axiom,
    ! [I2: nat,J: nat,K: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,I2),J)),K) = aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,I2),K)),J) ).

% diff_commute
tff(fact_1479_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,minus_minus(A,A2),C2)),aa(A,A,minus_minus(A,B2),D2))) ) ) ) ).

% diff_mono
tff(fact_1480_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,minus_minus(A,C2),A2)),aa(A,A,minus_minus(A,C2),B2))) ) ) ).

% diff_left_mono
tff(fact_1481_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,minus_minus(A,A2),C2)),aa(A,A,minus_minus(A,B2),C2))) ) ) ).

% diff_right_mono
tff(fact_1482_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,minus_minus(A,A2),B2) = aa(A,A,minus_minus(A,C2),D2) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2)) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1483_eq__iff__diff__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,A,minus_minus(A,A2),B2) = zero_zero(A) ) ) ) ).

% eq_iff_diff_eq_0
tff(fact_1484_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),D2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,minus_minus(A,A2),C2)),aa(A,A,minus_minus(A,B2),D2))) ) ) ) ).

% diff_strict_mono
tff(fact_1485_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,minus_minus(A,A2),B2) = aa(A,A,minus_minus(A,C2),D2) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2)) ) ) ) ).

% diff_eq_diff_less
tff(fact_1486_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,minus_minus(A,C2),A2)),aa(A,A,minus_minus(A,C2),B2))) ) ) ).

% diff_strict_left_mono
tff(fact_1487_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,minus_minus(A,A2),C2)),aa(A,A,minus_minus(A,B2),C2))) ) ) ).

% diff_strict_right_mono
tff(fact_1488_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,minus_minus(A,B2),C2)) = aa(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_1489_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,minus_minus(A,B2),C2)),A2) = aa(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_1490_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,minus_minus(A,B2),C2)) = aa(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_1491_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,minus_minus(A,A2),B2)),C2) = aa(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_1492_add__diff__add,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,C2: A,B2: A,D2: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,A2),B2)),aa(A,A,minus_minus(A,C2),D2)) ) ).

% add_diff_add
tff(fact_1493_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,minus_minus(A,A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,minus_minus(A,A2),B2)) ) ) ) ).

% group_cancel.sub1
tff(fact_1494_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(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_1495_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( A2 = aa(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_1496_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,minus_minus(A,B2),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ).

% add_diff_eq
tff(fact_1497_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,minus_minus(A,B2),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ).

% diff_diff_eq2
tff(fact_1498_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,minus_minus(A,A2),B2)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ).

% diff_add_eq
tff(fact_1499_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A2),C2)),B2) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_1500_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,minus_minus(A,A2),B2) ) ) ) ).

% add_implies_diff
tff(fact_1501_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A2),B2)),C2) = aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% diff_diff_eq
tff(fact_1502_diff__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : divide_divide(A,aa(A,A,minus_minus(A,A2),B2),C2) = aa(A,A,minus_minus(A,divide_divide(A,A2,C2)),divide_divide(A,B2,C2)) ) ).

% diff_divide_distrib
tff(fact_1503_dvd__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(dvd_dvd(A,X,Y))
         => ( pp(dvd_dvd(A,X,Z))
           => pp(dvd_dvd(A,X,aa(A,A,minus_minus(A,Y),Z))) ) ) ) ).

% dvd_diff
tff(fact_1504_dvd__diff__commute,axiom,
    ! [A: $tType] :
      ( euclid5891614535332579305n_ring(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(dvd_dvd(A,A2,aa(A,A,minus_minus(A,C2),B2)))
        <=> pp(dvd_dvd(A,A2,aa(A,A,minus_minus(A,B2),C2))) ) ) ).

% dvd_diff_commute
tff(fact_1505_zero__induct__lemma,axiom,
    ! [P: fun(nat,bool),K: nat,I2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,P,aa(nat,nat,suc,N2)))
           => pp(aa(nat,bool,P,N2)) )
       => pp(aa(nat,bool,P,aa(nat,nat,minus_minus(nat,K),I2))) ) ) ).

% zero_induct_lemma
tff(fact_1506_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,minus_minus(nat,M),N) = zero_zero(nat) )
     => ( ( aa(nat,nat,minus_minus(nat,N),M) = zero_zero(nat) )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
tff(fact_1507_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa(nat,nat,minus_minus(nat,M),zero_zero(nat)) = M ).

% minus_nat.diff_0
tff(fact_1508_mod__diff__right__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,C2: A] : modulo_modulo(A,aa(A,A,minus_minus(A,A2),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,minus_minus(A,A2),B2),C2) ) ).

% mod_diff_right_eq
tff(fact_1509_mod__diff__left__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,minus_minus(A,modulo_modulo(A,A2,C2)),B2),C2) = modulo_modulo(A,aa(A,A,minus_minus(A,A2),B2),C2) ) ).

% mod_diff_left_eq
tff(fact_1510_mod__diff__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,A4: A,B2: A,B4: A] :
          ( ( modulo_modulo(A,A2,C2) = modulo_modulo(A,A4,C2) )
         => ( ( modulo_modulo(A,B2,C2) = modulo_modulo(A,B4,C2) )
           => ( modulo_modulo(A,aa(A,A,minus_minus(A,A2),B2),C2) = modulo_modulo(A,aa(A,A,minus_minus(A,A4),B4),C2) ) ) ) ) ).

% mod_diff_cong
tff(fact_1511_mod__diff__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] : modulo_modulo(A,aa(A,A,minus_minus(A,modulo_modulo(A,A2,C2)),modulo_modulo(A,B2,C2)),C2) = modulo_modulo(A,aa(A,A,minus_minus(A,A2),B2),C2) ) ).

% mod_diff_eq
tff(fact_1512_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),L))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,minus_minus(nat,L),N)),aa(nat,nat,minus_minus(nat,L),M))) ) ) ).

% diff_less_mono2
tff(fact_1513_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),K))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,minus_minus(nat,J),N)),K)) ) ).

% less_imp_diff_less
tff(fact_1514_signed__take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).

% signed_take_bit_mult
tff(fact_1515_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
       => ( ( aa(nat,nat,minus_minus(nat,M),K) = aa(nat,nat,minus_minus(nat,N),K) )
        <=> ( M = N ) ) ) ) ).

% eq_diff_iff
tff(fact_1516_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,N),K)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).

% le_diff_iff
tff(fact_1517_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
       => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,N),K)) = aa(nat,nat,minus_minus(nat,M),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_1518_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),L)),aa(nat,nat,minus_minus(nat,N),L))) ) ).

% diff_le_mono
tff(fact_1519_diff__le__self,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),N)),M)) ).

% diff_le_self
tff(fact_1520_le__diff__iff_H,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),C2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),C2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,C2),A2)),aa(nat,nat,minus_minus(nat,C2),B2)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),B2),A2)) ) ) ) ).

% le_diff_iff'
tff(fact_1521_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,L),N)),aa(nat,nat,minus_minus(nat,L),M))) ) ).

% diff_le_mono2
tff(fact_1522_signed__take__bit__add,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ).

% signed_take_bit_add
tff(fact_1523_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)) = aa(nat,nat,minus_minus(nat,M),N) ).

% Nat.diff_cancel
tff(fact_1524_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)) = aa(nat,nat,minus_minus(nat,M),N) ).

% diff_cancel2
tff(fact_1525_diff__add__inverse,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),N) = M ).

% diff_add_inverse
tff(fact_1526_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),N) = M ).

% diff_add_inverse2
tff(fact_1527_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,M),N)),K) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)) ).

% diff_mult_distrib
tff(fact_1528_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,minus_minus(nat,M),N)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) ).

% diff_mult_distrib2
tff(fact_1529_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(dvd_dvd(nat,K,M))
     => ( pp(dvd_dvd(nat,K,N))
       => pp(dvd_dvd(nat,K,aa(nat,nat,minus_minus(nat,M),N))) ) ) ).

% dvd_diff_nat
tff(fact_1530_concat__bit__assoc,axiom,
    ! [N: nat,K: int,M: nat,L: int,R3: int] : aa(int,int,bit_concat_bit(N,K),aa(int,int,bit_concat_bit(M,L),R3)) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),aa(int,int,bit_concat_bit(N,K),L)),R3) ).

% concat_bit_assoc
tff(fact_1531_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),nat_set_decode(M)),nat_set_decode(N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% subset_decode_imp_le
tff(fact_1532_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,minus_minus(A,A2),B2)),zero_zero(A))) ) ) ).

% le_iff_diff_le_0
tff(fact_1533_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,minus_minus(A,A2),B2)),zero_zero(A))) ) ) ).

% less_iff_diff_less_0
tff(fact_1534_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( ( aa(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_1535_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,minus_minus(A,B2),A2)) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1536_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,minus_minus(A,C2),aa(A,A,minus_minus(A,B2),A2)) = aa(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_1537_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(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,minus_minus(A,B2),A2)),C2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1538_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B2),A2)),C2) = aa(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_1539_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(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,minus_minus(A,B2),A2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1540_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,minus_minus(A,B2),A2)) = aa(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_1541_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,minus_minus(A,B2),A2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),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_1542_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2))) ) ) ).

% le_add_diff
tff(fact_1543_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: A,K: A,N: A,J: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),N))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,minus_minus(A,N),K)),J)) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_1544_diff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B2),A2)),A2) = B2 ) ) ) ).

% diff_add
tff(fact_1545_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: A,K: A,N: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K)),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),aa(A,A,minus_minus(A,N),K))) ) ) ).

% add_le_imp_le_diff
tff(fact_1546_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,minus_minus(A,C2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2)) ) ) ).

% le_diff_eq
tff(fact_1547_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,minus_minus(A,A2),B2)),C2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).

% diff_le_eq
tff(fact_1548_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,minus_minus(A,A2),B2)),C2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))) ) ) ).

% diff_less_eq
tff(fact_1549_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,minus_minus(A,C2),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2)) ) ) ).

% less_diff_eq
tff(fact_1550_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,minus_minus(A,A2),B2)) = A2 ) ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_1551_square__diff__square__factored,axiom,
    ! [A: $tType] :
      ( comm_ring(A)
     => ! [X: A,Y: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,minus_minus(A,X),Y)) ) ).

% square_diff_square_factored
tff(fact_1552_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),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)),D2) )
        <=> ( C2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B2),A2)),E2)),D2) ) ) ) ).

% eq_add_iff2
tff(fact_1553_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),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)),D2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),B2)),E2)),C2) = D2 ) ) ) ).

% eq_add_iff1
tff(fact_1554_mult__diff__mult,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [X: A,Y: A,A2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,minus_minus(A,Y),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),A2)),B2)) ) ).

% mult_diff_mult
tff(fact_1555_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) )
        <=> pp(dvd_dvd(A,C2,aa(A,A,minus_minus(A,A2),B2))) ) ) ).

% mod_eq_dvd_iff
tff(fact_1556_dvd__minus__mod,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A2: A] : pp(dvd_dvd(A,B2,aa(A,A,minus_minus(A,A2),modulo_modulo(A,A2,B2)))) ) ).

% dvd_minus_mod
tff(fact_1557_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,minus_minus(nat,M),N)),aa(nat,nat,suc,M))) ).

% diff_less_Suc
tff(fact_1558_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,N))) = aa(nat,nat,minus_minus(nat,M),N) ) ) ).

% Suc_diff_Suc
tff(fact_1559_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,minus_minus(nat,M),N)),M)) ) ) ).

% diff_less
tff(fact_1560_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),N) = aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,M),N)) ) ) ).

% Suc_diff_le
tff(fact_1561_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A2),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),C2),A2))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,minus_minus(nat,A2),C2)),aa(nat,nat,minus_minus(nat,B2),C2))) ) ) ).

% diff_less_mono
tff(fact_1562_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,N),K)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ) ).

% less_diff_iff
tff(fact_1563_diff__add__0,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,minus_minus(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) = zero_zero(nat) ).

% diff_add_0
tff(fact_1564_less__diff__conv,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,minus_minus(nat,J),K)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),J)) ) ).

% less_diff_conv
tff(fact_1565_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,minus_minus(nat,M),N)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_1566_Nat_Ole__imp__diff__is__add,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( ( aa(nat,nat,minus_minus(nat,J),I2) = K )
      <=> ( J = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I2) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_1567_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I2)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,J),K)),I2) ) ) ).

% Nat.diff_add_assoc2
tff(fact_1568_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),aa(nat,nat,minus_minus(nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_1569_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(nat,nat,minus_minus(nat,J),K)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),J)) ) ) ).

% Nat.le_diff_conv2
tff(fact_1570_le__diff__conv,axiom,
    ! [J: nat,K: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,J),K)),I2))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K))) ) ).

% le_diff_conv
tff(fact_1571_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,N)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),one_one(nat))),N) ).

% diff_Suc_eq_diff_pred
tff(fact_1572_int__le__induct,axiom,
    ! [I2: int,K: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),K))
     => ( pp(aa(int,bool,P,K))
       => ( ! [I4: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I4),K))
             => ( pp(aa(int,bool,P,I4))
               => pp(aa(int,bool,P,aa(int,int,minus_minus(int,I4),one_one(int)))) ) )
         => pp(aa(int,bool,P,I2)) ) ) ) ).

% int_le_induct
tff(fact_1573_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(dvd_dvd(nat,M,aa(nat,nat,minus_minus(nat,N),M)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
        | pp(dvd_dvd(nat,M,N)) ) ) ).

% dvd_minus_self
tff(fact_1574_int__less__induct,axiom,
    ! [I2: int,K: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),K))
     => ( pp(aa(int,bool,P,aa(int,int,minus_minus(int,K),one_one(int))))
       => ( ! [I4: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I4),K))
             => ( pp(aa(int,bool,P,I4))
               => pp(aa(int,bool,P,aa(int,int,minus_minus(int,I4),one_one(int)))) ) )
         => pp(aa(int,bool,P,I2)) ) ) ) ).

% int_less_induct
tff(fact_1575_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(dvd_dvd(nat,K,aa(nat,nat,minus_minus(nat,M),N)))
     => ( pp(dvd_dvd(nat,K,N))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => pp(dvd_dvd(nat,K,M)) ) ) ) ).

% dvd_diffD
tff(fact_1576_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(dvd_dvd(nat,K,aa(nat,nat,minus_minus(nat,M),N)))
     => ( pp(dvd_dvd(nat,K,M))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => pp(dvd_dvd(nat,K,N)) ) ) ) ).

% dvd_diffD1
tff(fact_1577_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( pp(dvd_dvd(nat,M,N))
      <=> pp(dvd_dvd(nat,M,aa(nat,nat,minus_minus(nat,N),M))) ) ) ).

% less_eq_dvd_minus
tff(fact_1578_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
     => ( modulo_modulo(nat,M,N) = modulo_modulo(nat,aa(nat,nat,minus_minus(nat,M),N),N) ) ) ).

% mod_geq
tff(fact_1579_mod__if,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ( modulo_modulo(nat,M,N) = M ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ( modulo_modulo(nat,M,N) = modulo_modulo(nat,aa(nat,nat,minus_minus(nat,M),N),N) ) ) ) ).

% mod_if
tff(fact_1580_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( modulo_modulo(nat,M,N) = modulo_modulo(nat,aa(nat,nat,minus_minus(nat,M),N),N) ) ) ).

% le_mod_geq
tff(fact_1581_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),K))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(int,int,minus_minus(int,K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(nat,nat,suc,N))))) ) ).

% signed_take_bit_int_less_eq
tff(fact_1582_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)),D2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B2),A2)),E2)),D2))) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1583_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)),D2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),B2)),E2)),C2)),D2)) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1584_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)),D2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),B2)),E2)),C2)),D2)) ) ) ).

% less_add_iff1
tff(fact_1585_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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)),D2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B2),A2)),E2)),D2))) ) ) ).

% less_add_iff2
tff(fact_1586_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,divide_divide(A,X,Z)),Y) = divide_divide(A,aa(A,A,minus_minus(A,X),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)),Z) ) ) ) ).

% divide_diff_eq_iff
tff(fact_1587_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,X),divide_divide(A,Y,Z)) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),Y),Z) ) ) ) ).

% diff_divide_eq_iff
tff(fact_1588_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,X: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,divide_divide(A,X,Y)),divide_divide(A,W,Z)) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ) ) ) ).

% diff_frac_eq
tff(fact_1589_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,minus_minus(A,A2),divide_divide(A,B2,Z)) = A2 ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,A2),divide_divide(A,B2,Z)) = divide_divide(A,aa(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_1590_square__diff__one__factored,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))),aa(A,A,minus_minus(A,X),one_one(A))) ) ).

% square_diff_one_factored
tff(fact_1591_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D2: A,D4: A,T2: A] :
          ( pp(dvd_dvd(A,D2,D4))
         => ! [X2: A,K4: A] :
              ( ~ pp(dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),T2)))
            <=> ~ pp(dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,X2),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4))),T2))) ) ) ) ).

% inf_period(4)
tff(fact_1592_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D2: A,D4: A,T2: A] :
          ( pp(dvd_dvd(A,D2,D4))
         => ! [X2: A,K4: A] :
              ( pp(dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),X2),T2)))
            <=> pp(dvd_dvd(A,D2,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,X2),aa(A,A,aa(A,fun(A,A),times_times(A),K4),D4))),T2))) ) ) ) ).

% inf_period(3)
tff(fact_1593_minus__mult__div__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(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_1594_minus__mod__eq__mult__div,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(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_1595_minus__mod__eq__div__mult,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(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_1596_minus__div__mult__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(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_1597_diff__Suc__less,axiom,
    ! [N: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,I2))),N)) ) ).

% diff_Suc_less
tff(fact_1598_nat__diff__split,axiom,
    ! [P: fun(nat,bool),A2: nat,B2: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,minus_minus(nat,A2),B2)))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A2),B2))
         => pp(aa(nat,bool,P,zero_zero(nat))) )
        & ! [D5: nat] :
            ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D5) )
           => pp(aa(nat,bool,P,D5)) ) ) ) ).

% nat_diff_split
tff(fact_1599_nat__diff__split__asm,axiom,
    ! [P: fun(nat,bool),A2: nat,B2: nat] :
      ( pp(aa(nat,bool,P,aa(nat,nat,minus_minus(nat,A2),B2)))
    <=> ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A2),B2))
            & ~ pp(aa(nat,bool,P,zero_zero(nat))) )
          | ? [D5: nat] :
              ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D5) )
              & ~ pp(aa(nat,bool,P,D5)) ) ) ) ).

% nat_diff_split_asm
tff(fact_1600_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,minus_minus(nat,J),K)),I2))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K))) ) ) ).

% less_diff_conv2
tff(fact_1601_nat__diff__add__eq2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)) = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I2)),U)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_1602_nat__diff__add__eq1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I2))
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I2),J)),U)),M)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_1603_nat__le__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I2)),U)),N))) ) ) ).

% nat_le_add_iff2
tff(fact_1604_nat__le__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I2),J)),U)),M)),N)) ) ) ).

% nat_le_add_iff1
tff(fact_1605_nat__eq__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N) )
      <=> ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I2)),U)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_1606_nat__eq__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I2))
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I2),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N) )
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I2),J)),U)),M) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_1607_plusinfinity,axiom,
    ! [D2: int,P3: fun(int,bool),P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => ( ! [X3: int,K2: int] :
            ( pp(aa(int,bool,P3,X3))
          <=> pp(aa(int,bool,P3,aa(int,int,minus_minus(int,X3),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2)))) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z3),X3))
             => ( pp(aa(int,bool,P,X3))
              <=> pp(aa(int,bool,P3,X3)) ) )
         => ( ? [X_12: int] : pp(aa(int,bool,P3,X_12))
           => ? [X_13: int] : pp(aa(int,bool,P,X_13)) ) ) ) ) ).

% plusinfinity
tff(fact_1608_minusinfinity,axiom,
    ! [D2: int,P1: fun(int,bool),P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => ( ! [X3: int,K2: int] :
            ( pp(aa(int,bool,P1,X3))
          <=> pp(aa(int,bool,P1,aa(int,int,minus_minus(int,X3),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2)))) )
       => ( ? [Z3: int] :
            ! [X3: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X3),Z3))
             => ( pp(aa(int,bool,P,X3))
              <=> pp(aa(int,bool,P1,X3)) ) )
         => ( ? [X_12: int] : pp(aa(int,bool,P1,X_12))
           => ? [X_13: int] : pp(aa(int,bool,P,X_13)) ) ) ) ) ).

% minusinfinity
tff(fact_1609_int__induct,axiom,
    ! [P: fun(int,bool),K: int,I2: int] :
      ( pp(aa(int,bool,P,K))
     => ( ! [I4: int] :
            ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),I4))
           => ( pp(aa(int,bool,P,I4))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I4),one_one(int)))) ) )
       => ( ! [I4: int] :
              ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I4),K))
             => ( pp(aa(int,bool,P,I4))
               => pp(aa(int,bool,P,aa(int,int,minus_minus(int,I4),one_one(int)))) ) )
         => pp(aa(int,bool,P,I2)) ) ) ) ).

% int_induct
tff(fact_1610_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( ( modulo_modulo(nat,M,Q3) = modulo_modulo(nat,N,Q3) )
      <=> pp(dvd_dvd(nat,Q3,aa(nat,nat,minus_minus(nat,M),N))) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_1611_modulo__nat__def,axiom,
    ! [M: nat,N: nat] : modulo_modulo(nat,M,N) = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),divide_divide(nat,M,N)),N)) ).

% modulo_nat_def
tff(fact_1612_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,X: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,X,Y)),divide_divide(A,W,Z)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),zero_zero(A))) ) ) ) ) ).

% frac_le_eq
tff(fact_1613_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,X: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,X,Y)),divide_divide(A,W,Z)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),zero_zero(A))) ) ) ) ) ).

% frac_less_eq
tff(fact_1614_power2__commute,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] : aa(nat,A,power_power(A,aa(A,A,minus_minus(A,X),Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,power_power(A,aa(A,A,minus_minus(A,Y),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% power2_commute
tff(fact_1615_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,N: nat,M: nat] :
          ( ( A2 != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(nat,A,power_power(A,A2),aa(nat,nat,minus_minus(nat,M),N)) = divide_divide(A,aa(nat,A,power_power(A,A2),M),aa(nat,A,power_power(A,A2),N)) ) ) ) ) ).

% power_diff
tff(fact_1616_Suc__pred_H,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( N = aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,N),one_one(nat))) ) ) ).

% Suc_pred'
tff(fact_1617_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),N) = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,minus_minus(nat,N),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_1618_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => ( divide_divide(nat,M,N) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,minus_minus(nat,M),N),N)) ) ) ) ).

% div_geq
tff(fact_1619_div__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
          | ( N = zero_zero(nat) ) )
       => ( divide_divide(nat,M,N) = zero_zero(nat) ) )
      & ( ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
            | ( N = zero_zero(nat) ) )
       => ( divide_divide(nat,M,N) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,minus_minus(nat,M),N),N)) ) ) ) ).

% div_if
tff(fact_1620_add__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = N ) )
      & ( ( M != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,M),one_one(nat))),N)) ) ) ) ).

% add_eq_if
tff(fact_1621_nat__less__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I2),J)),U)),M)),N)) ) ) ).

% nat_less_add_iff1
tff(fact_1622_nat__less__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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),I2),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)))
      <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I2)),U)),N))) ) ) ).

% nat_less_add_iff2
tff(fact_1623_mult__eq__if,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = zero_zero(nat) ) )
      & ( ( M != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,M),one_one(nat))),N)) ) ) ) ).

% mult_eq_if
tff(fact_1624_decr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,bool),K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => ( ! [X3: int] :
            ( pp(aa(int,bool,P,X3))
           => pp(aa(int,bool,P,aa(int,int,minus_minus(int,X3),D2))) )
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
         => ! [X2: int] :
              ( pp(aa(int,bool,P,X2))
             => pp(aa(int,bool,P,aa(int,int,minus_minus(int,X2),aa(int,int,aa(int,fun(int,int),times_times(int),K),D2)))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1625_dvd__minus__add,axiom,
    ! [Q3: nat,N: nat,R3: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Q3),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Q3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R3),M)))
       => ( pp(dvd_dvd(nat,M,aa(nat,nat,minus_minus(nat,N),Q3)))
        <=> pp(dvd_dvd(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R3),M)),Q3)))) ) ) ) ).

% dvd_minus_add
tff(fact_1626_mod__nat__eqI,axiom,
    ! [R3: nat,N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),R3),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R3),M))
       => ( pp(dvd_dvd(nat,N,aa(nat,nat,minus_minus(nat,M),R3)))
         => ( modulo_modulo(nat,M,N) = R3 ) ) ) ) ).

% mod_nat_eqI
tff(fact_1627_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),L),K))
       => ( modulo_modulo(int,K,L) = modulo_modulo(int,aa(int,int,minus_minus(int,K),L),L) ) ) ) ).

% mod_pos_geq
tff(fact_1628_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V: A,R3: A,S: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),R3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R3),S))
             => pp(aa(A,bool,aa(A,fun(A,bool),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),R3),aa(A,A,minus_minus(A,V),U)),S))),V)) ) ) ) ) ).

% scaling_mono
tff(fact_1629_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat,M: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) != zero_zero(A) )
         => ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,minus_minus(nat,N),M)) != zero_zero(A) ) ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_1630_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,N: nat,M: nat] :
          ( ( A2 != zero_zero(A) )
         => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
             => ( divide_divide(A,aa(nat,A,power_power(A,A2),M),aa(nat,A,power_power(A,A2),N)) = aa(nat,A,power_power(A,A2),aa(nat,nat,minus_minus(nat,M),N)) ) )
            & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
             => ( divide_divide(A,aa(nat,A,power_power(A,A2),M),aa(nat,A,power_power(A,A2),N)) = divide_divide(A,one_one(A),aa(nat,A,power_power(A,A2),aa(nat,nat,minus_minus(nat,N),M))) ) ) ) ) ) ).

% power_diff_power_eq
tff(fact_1631_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ).

% signed_take_bit_int_less_exp
tff(fact_1632_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [M: nat,P2: A] :
          ( ( ( M = zero_zero(nat) )
           => ( aa(nat,A,power_power(A,P2),M) = one_one(A) ) )
          & ( ( M != zero_zero(nat) )
           => ( aa(nat,A,power_power(A,P2),M) = aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(nat,A,power_power(A,P2),aa(nat,nat,minus_minus(nat,M),one_one(nat)))) ) ) ) ) ).

% power_eq_if
tff(fact_1633_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [N: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),aa(nat,nat,minus_minus(nat,N),one_one(nat)))),A2) = aa(nat,A,power_power(A,A2),N) ) ) ) ).

% power_minus_mult
tff(fact_1634_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),N)),aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,K),M)),aa(nat,nat,power_power(nat,K),N)))) ) ).

% diff_le_diff_pow
tff(fact_1635_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
       => ( divide_divide(nat,M,N) = aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,minus_minus(nat,M),N),N)) ) ) ) ).

% le_div_geq
tff(fact_1636_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),aa(int,int,minus_minus(int,K),L)))
    <=> pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L))) ) ).

% even_diff_iff
tff(fact_1637_even__signed__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,A2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,bit_ri4674362597316999326ke_bit(A,M),A2)))
        <=> pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)) ) ) ).

% even_signed_take_bit_iff
tff(fact_1638_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_1639_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),K))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),K)) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_1640_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),L))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),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,minus_minus(int,K),L),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_1641_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_1642_crossproduct__eq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [W: A,Y: A,X: A,Z: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),X),Y)) )
        <=> ( ( W = X )
            | ( Y = Z ) ) ) ) ).

% crossproduct_eq
tff(fact_1643_crossproduct__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ( A2 != B2 )
            & ( C2 != D2 ) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),D2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% crossproduct_noteq
tff(fact_1644_power2__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] : aa(nat,A,power_power(A,aa(A,A,minus_minus(A,X),Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,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),aa(num,num,bit0,one2))),X)),Y)) ) ).

% power2_diff
tff(fact_1645_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( modulo_modulo(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),modulo_modulo(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,minus_minus(nat,N),M)))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) ) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_1646_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
       => ( divide_divide(int,aa(nat,int,power_power(int,K),M),K) = aa(nat,int,power_power(int,K),aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% int_power_div_base
tff(fact_1647_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),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),aa(num,num,bit0,one2))),B2))))
             => ( aa(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),aa(num,num,bit0,one2))),B2))),B2) = modulo_modulo(A,A2,B2) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_1648_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),divide_divide(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),one_one(A)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% even_mask_div_iff'
tff(fact_1649_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(nat,nat,suc,zero_zero(nat)) )
     => pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),divide_divide(nat,aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% even_mod_4_div_2
tff(fact_1650_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),divide_divide(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),one_one(A)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))))
        <=> ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) = zero_zero(A) )
            | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ) ).

% even_mask_div_iff
tff(fact_1651_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] : divide_divide(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),zero_zero(A))),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M)))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,minus_minus(nat,M),N))) ) ).

% exp_div_exp_eq
tff(fact_1652_neg__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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),aa(num,num,bit0,one2))),B2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),A2)) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,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_1653_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,R3: A,Q3: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R3))
           => ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),product_Pair(A,A,Q3),R3)) = 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),aa(num,num,bit0,one2))),Q3)),one_one(A))),aa(A,A,minus_minus(A,R3),aa(num,A,numeral_numeral(A),L))) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R3))
           => ( unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),product_Pair(A,A,Q3),R3)) = 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),aa(num,num,bit0,one2))),Q3)),R3) ) ) ) ) ).

% divmod_step_eq
tff(fact_1654_Suc__if__eq,axiom,
    ! [A: $tType,F3: fun(nat,A),H: fun(nat,A),G: A,N: nat] :
      ( ! [N2: nat] : aa(nat,A,F3,aa(nat,nat,suc,N2)) = aa(nat,A,H,N2)
     => ( ( aa(nat,A,F3,zero_zero(nat)) = G )
       => ( ( ( N = zero_zero(nat) )
           => ( aa(nat,A,F3,N) = G ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,A,F3,N) = aa(nat,A,H,aa(nat,nat,minus_minus(nat,N),one_one(nat))) ) ) ) ) ) ).

% Suc_if_eq
tff(fact_1655_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,minus_minus(nat,N),one_one(nat))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ) ) ) ).

% signed_take_bit_rec
tff(fact_1656_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = zero_zero(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,minus_minus(nat,N),one_one(nat))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ) ) ) ).

% take_bit_rec
tff(fact_1657_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
     => ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),divide_divide(nat,aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% odd_mod_4_div_2
tff(fact_1658_Bernoulli__inequality__even,axiom,
    ! [N: nat,X: real] :
      ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
     => pp(aa(real,bool,aa(real,fun(real,bool),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),N)),X))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),N))) ) ).

% Bernoulli_inequality_even
tff(fact_1659_Suc__0__xor__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N)))) ).

% Suc_0_xor_eq
tff(fact_1660_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_1661_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_1662_Compl__anti__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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_1663_Compl__subset__Compl__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3)) ) ).

% Compl_subset_Compl_iff
tff(fact_1664_of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( M = N ) ) ) ).

% of_nat_eq_iff
tff(fact_1665_semiring__norm_I90_J,axiom,
    ! [M: num,N: num] :
      ( ( aa(num,num,bit1,M) = aa(num,num,bit1,N) )
    <=> ( M = N ) ) ).

% semiring_norm(90)
tff(fact_1666_bit_Oxor__left__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y)) = Y ) ).

% bit.xor_left_self
tff(fact_1667_idiff__0,axiom,
    ! [N: extended_enat] : aa(extended_enat,extended_enat,minus_minus(extended_enat,zero_zero(extended_enat)),N) = zero_zero(extended_enat) ).

% idiff_0
tff(fact_1668_idiff__0__right,axiom,
    ! [N: extended_enat] : aa(extended_enat,extended_enat,minus_minus(extended_enat,N),zero_zero(extended_enat)) = N ).

% idiff_0_right
tff(fact_1669_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% neg_le_iff_le
tff(fact_1670_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_1671_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_1672_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_1673_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_1674_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_1675_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% neg_less_iff_less
tff(fact_1676_neg__numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,N: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) )
        <=> ( M = N ) ) ) ).

% neg_numeral_eq_iff
tff(fact_1677_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_1678_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_1679_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_1680_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_1681_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_1682_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_1683_minus__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,minus_minus(A,A2),B2)) = aa(A,A,minus_minus(A,B2),A2) ) ).

% minus_diff_eq
tff(fact_1684_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_1685_dvd__minus__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] :
          ( pp(dvd_dvd(A,X,aa(A,A,uminus_uminus(A),Y)))
        <=> pp(dvd_dvd(A,X,Y)) ) ) ).

% dvd_minus_iff
tff(fact_1686_minus__dvd__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [X: A,Y: A] :
          ( pp(dvd_dvd(A,aa(A,A,uminus_uminus(A),X),Y))
        <=> pp(dvd_dvd(A,X,Y)) ) ) ).

% minus_dvd_iff
tff(fact_1687_semiring__norm_I88_J,axiom,
    ! [M: num,N: num] : aa(num,num,bit0,M) != aa(num,num,bit1,N) ).

% semiring_norm(88)
tff(fact_1688_semiring__norm_I89_J,axiom,
    ! [M: num,N: num] : aa(num,num,bit1,M) != aa(num,num,bit0,N) ).

% semiring_norm(89)
tff(fact_1689_semiring__norm_I84_J,axiom,
    ! [N: num] : one2 != aa(num,num,bit1,N) ).

% semiring_norm(84)
tff(fact_1690_semiring__norm_I86_J,axiom,
    ! [M: num] : aa(num,num,bit1,M) != one2 ).

% semiring_norm(86)
tff(fact_1691_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_1692_take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).

% take_bit_of_0
tff(fact_1693_real__add__minus__iff,axiom,
    ! [X: real,A2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,uminus_uminus(real),A2)) = zero_zero(real) )
    <=> ( X = A2 ) ) ).

% real_add_minus_iff
tff(fact_1694_bit_Oxor__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),X) = zero_zero(A) ) ).

% bit.xor_self
tff(fact_1695_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_1696_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_1697_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_1698_take__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2)) ) ).

% take_bit_xor
tff(fact_1699_concat__bit__of__zero__2,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_concat_bit(N,K),zero_zero(int)) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) ).

% concat_bit_of_zero_2
tff(fact_1700_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ).

% semiring_norm(80)
tff(fact_1701_semiring__norm_I73_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ).

% semiring_norm(73)
tff(fact_1702_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),A2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% neg_less_eq_nonneg
tff(fact_1703_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% less_eq_neg_nonpos
tff(fact_1704_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% neg_le_0_iff_le
tff(fact_1705_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% neg_0_le_iff_le
tff(fact_1706_less__neg__neg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% less_neg_neg
tff(fact_1707_neg__less__pos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A2)),A2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% neg_less_pos
tff(fact_1708_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% neg_0_less_iff_less
tff(fact_1709_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% neg_less_0_iff_less
tff(fact_1710_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_1711_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_1712_diff__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,zero_zero(A)),A2) = aa(A,A,uminus_uminus(A),A2) ) ).

% diff_0
tff(fact_1713_add__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N))) ) ).

% add_neg_numeral_simps(3)
tff(fact_1714_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_1715_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_1716_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,minus_minus(A,B2),A2) ) ).

% uminus_add_conv_diff
tff(fact_1717_diff__minus__eq__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(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_1718_divide__minus1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A] : divide_divide(A,X,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),X) ) ).

% divide_minus1
tff(fact_1719_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_1720_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_1721_of__nat__0__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( zero_zero(A) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( zero_zero(nat) = N ) ) ) ).

% of_nat_0_eq_iff
tff(fact_1722_of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M) = zero_zero(A) )
        <=> ( M = zero_zero(nat) ) ) ) ).

% of_nat_eq_0_iff
tff(fact_1723_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% of_nat_less_iff
tff(fact_1724_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ) ).

% of_nat_le_iff
tff(fact_1725_of__nat__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: num] : aa(nat,A,semiring_1_of_nat(A),aa(num,nat,numeral_numeral(nat),N)) = aa(num,A,numeral_numeral(A),N) ) ).

% of_nat_numeral
tff(fact_1726_minus__mod__self1,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [B2: A,A2: A] : modulo_modulo(A,aa(A,A,minus_minus(A,B2),A2),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2) ) ).

% minus_mod_self1
tff(fact_1727_of__nat__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_add
tff(fact_1728_of__nat__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_mult
tff(fact_1729_take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,zero_zero(nat)),A2) = zero_zero(A) ) ).

% take_bit_0
tff(fact_1730_of__nat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),N) = one_one(A) )
        <=> ( N = one_one(nat) ) ) ) ).

% of_nat_eq_1_iff
tff(fact_1731_of__nat__1__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( one_one(A) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( N = one_one(nat) ) ) ) ).

% of_nat_1_eq_iff
tff(fact_1732_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_1733_take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),one_one(A)) = one_one(A) ) ).

% take_bit_Suc_1
tff(fact_1734_take__bit__numeral__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),one_one(A)) = one_one(A) ) ).

% take_bit_numeral_1
tff(fact_1735_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [X: nat,B2: nat,W: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),X) = aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B2)),W) )
        <=> ( X = aa(nat,nat,power_power(nat,B2),W) ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_1736_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [B2: nat,W: nat,X: nat] :
          ( ( aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B2)),W) = aa(nat,A,semiring_1_of_nat(A),X) )
        <=> ( aa(nat,nat,power_power(nat,B2),W) = X ) ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_1737_of__nat__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,power_power(nat,M),N)) = aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),M)),N) ) ).

% of_nat_power
tff(fact_1738_signed__take__bit__of__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),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_1739_semiring__norm_I9_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),aa(num,num,bit0,N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).

% semiring_norm(9)
tff(fact_1740_semiring__norm_I7_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit0,M)),aa(num,num,bit1,N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).

% semiring_norm(7)
tff(fact_1741_semiring__norm_I15_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),aa(num,num,bit0,N)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),N)) ).

% semiring_norm(15)
tff(fact_1742_semiring__norm_I14_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit0,M)),aa(num,num,bit1,N)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),times_times(num),M),aa(num,num,bit1,N))) ).

% semiring_norm(14)
tff(fact_1743_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit1,M)),aa(num,num,bit0,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ).

% semiring_norm(81)
tff(fact_1744_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit0,M)),aa(num,num,bit1,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ).

% semiring_norm(72)
tff(fact_1745_semiring__norm_I77_J,axiom,
    ! [N: num] : pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),one2),aa(num,num,bit1,N))) ).

% semiring_norm(77)
tff(fact_1746_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit1,M)),one2)) ).

% semiring_norm(70)
tff(fact_1747_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_1748_of__nat__of__bool,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P: bool] : aa(nat,A,semiring_1_of_nat(A),aa(bool,nat,zero_neq_one_of_bool(nat),P)) = aa(bool,A,zero_neq_one_of_bool(A),P) ) ).

% of_nat_of_bool
tff(fact_1749_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_1750_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_1751_diff__numeral__special_I12_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).

% diff_numeral_special(12)
tff(fact_1752_numeral__eq__neg__one__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( N = one2 ) ) ) ).

% numeral_eq_neg_one_iff
tff(fact_1753_neg__one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] :
          ( ( aa(A,A,uminus_uminus(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) )
        <=> ( N = one2 ) ) ) ).

% neg_one_eq_numeral_iff
tff(fact_1754_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A)))
        <=> ( M = zero_zero(nat) ) ) ) ).

% of_nat_le_0_iff
tff(fact_1755_left__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),N)),A2)) = A2 ) ).

% left_minus_one_mult_self
tff(fact_1756_minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),N)) = one_one(A) ) ).

% minus_one_mult_self
tff(fact_1757_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_1758_of__nat__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),M)) ) ).

% of_nat_Suc
tff(fact_1759_take__bit__of__1__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),one_one(A)) = zero_zero(A) )
        <=> ( N = zero_zero(nat) ) ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_1760_semiring__norm_I168_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),W)))),Y) ) ).

% semiring_norm(168)
tff(fact_1761_diff__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N))) ) ).

% diff_numeral_simps(3)
tff(fact_1762_diff__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ) ).

% diff_numeral_simps(2)
tff(fact_1763_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : divide_divide(int,aa(num,int,numeral_numeral(int),aa(num,num,bit1,V)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W))) = divide_divide(int,aa(num,int,numeral_numeral(int),V),aa(num,int,numeral_numeral(int),W)) ).

% zdiv_numeral_Bit1
tff(fact_1764_semiring__norm_I3_J,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),aa(num,num,bit0,N)) = aa(num,num,bit1,N) ).

% semiring_norm(3)
tff(fact_1765_semiring__norm_I4_J,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),aa(num,num,bit1,N)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ).

% semiring_norm(4)
tff(fact_1766_semiring__norm_I5_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit0,M)),one2) = aa(num,num,bit1,M) ).

% semiring_norm(5)
tff(fact_1767_semiring__norm_I8_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),one2) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one2)) ).

% semiring_norm(8)
tff(fact_1768_semiring__norm_I10_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)) = aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)),one2)) ).

% semiring_norm(10)
tff(fact_1769_semiring__norm_I172_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W))),Y) ) ).

% semiring_norm(172)
tff(fact_1770_semiring__norm_I171_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W)))),Y) ) ).

% semiring_norm(171)
tff(fact_1771_semiring__norm_I170_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W)))),Y) ) ).

% semiring_norm(170)
tff(fact_1772_mult__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ) ).

% mult_neg_numeral_simps(3)
tff(fact_1773_mult__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ) ).

% mult_neg_numeral_simps(2)
tff(fact_1774_mult__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ) ).

% mult_neg_numeral_simps(1)
tff(fact_1775_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),M)) ) ) ).

% neg_numeral_le_iff
tff(fact_1776_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),N),M)) ) ) ).

% neg_numeral_less_iff
tff(fact_1777_take__bit__of__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(nat,nat,suc,zero_zero(nat))) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ).

% take_bit_of_Suc_0
tff(fact_1778_semiring__norm_I16_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)),aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),times_times(num),M),N)))) ).

% semiring_norm(16)
tff(fact_1779_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),aa(num,num,bit0,M)),aa(num,num,bit1,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ).

% semiring_norm(79)
tff(fact_1780_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),aa(num,num,bit1,M)),aa(num,num,bit0,N)))
    <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ).

% semiring_norm(74)
tff(fact_1781_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))))
        <=> ( M != one2 ) ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_1782_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_1783_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_1784_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))) )
        <=> ( ( ( 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 ) )
            & ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_1785_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 )
        <=> ( ( ( 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))) ) )
            & ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_1786_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))))
        <=> ( M != one2 ) ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_1787_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_1788_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),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_1789_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ).

% of_nat_0_less_iff
tff(fact_1790_power2__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% power2_minus
tff(fact_1791_xor__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% xor_numerals(3)
tff(fact_1792_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,B2: nat,W: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B2)),W)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,power_power(nat,B2),W))) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_1793_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B2)),W)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,power_power(nat,B2),W)),X)) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_1794_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),aa(num,num,bit0,Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).

% xor_numerals(1)
tff(fact_1795_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),aa(num,num,bit0,Y)) ) ).

% xor_numerals(2)
tff(fact_1796_xor__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).

% xor_numerals(5)
tff(fact_1797_xor__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,X)) ) ).

% xor_numerals(8)
tff(fact_1798_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,B2: nat,W: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B2)),W)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(nat,nat,power_power(nat,B2),W))) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_1799_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),B2)),W)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,power_power(nat,B2),W)),X)) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_1800_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Y: nat,X: num,N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Y) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),N) )
        <=> ( Y = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),N) ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_1801_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [X: num,N: nat,Y: nat] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),N) = aa(nat,A,semiring_1_of_nat(A),Y) )
        <=> ( aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),N) = Y ) ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_1802_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W: num] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(num,real,numeral_numeral(real),W)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(num,nat,numeral_numeral(nat),W))) ) ).

% real_of_nat_less_numeral_iff
tff(fact_1803_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(num,real,numeral_numeral(real),W)),aa(nat,real,semiring_1_of_nat(real),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),W)),N)) ) ).

% numeral_less_real_of_nat_iff
tff(fact_1804_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(num,real,numeral_numeral(real),N)),aa(nat,real,semiring_1_of_nat(real),M)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),N)),M)) ) ).

% numeral_le_real_of_nat_iff
tff(fact_1805_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),one_one(A)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% take_bit_of_1
tff(fact_1806_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),aa(num,num,bit0,one2))) ) ) ).

% add_neg_numeral_special(9)
tff(fact_1807_diff__numeral__special_I10_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ) ).

% diff_numeral_special(10)
tff(fact_1808_diff__numeral__special_I11_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)) ) ) ).

% diff_numeral_special(11)
tff(fact_1809_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),aa(num,num,bit0,one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% minus_1_div_2_eq
tff(fact_1810_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),aa(num,num,bit0,one2))) = one_one(A) ) ) ).

% bits_minus_1_mod_2_eq
tff(fact_1811_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),aa(num,num,bit0,one2))) = one_one(A) ) ) ).

% minus_1_mod_2_eq
tff(fact_1812_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,aa(nat,A,semiring_1_of_nat(A),X)),N)))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),X))
            | ( N = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1813_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,N: nat] : aa(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),aa(num,num,bit0,one2))),N)) = aa(nat,A,power_power(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ) ).

% Power.ring_1_class.power_minus_even
tff(fact_1814_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat,A2: A] :
          ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A2)),N) = aa(nat,A,power_power(A,A2),N) ) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_1815_power__minus__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat,A2: A] :
          ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A2)),N) = aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,A2),N)) ) ) ) ).

% power_minus_odd
tff(fact_1816_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)))
        <=> ( ( N = zero_zero(nat) )
            | pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)) ) ) ) ).

% even_take_bit_eq
tff(fact_1817_xor__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% xor_numerals(7)
tff(fact_1818_diff__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,minus_minus(A,one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N)) ) ).

% diff_numeral_special(3)
tff(fact_1819_diff__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one2))) ) ).

% diff_numeral_special(4)
tff(fact_1820_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M))),aa(num,nat,numeral_numeral(nat),V)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M),aa(num,nat,numeral_numeral(nat),V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_1821_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: nat] : divide_divide(nat,M,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N)))) = divide_divide(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N)) ).

% div_Suc_eq_div_add3
tff(fact_1822_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M))),aa(num,nat,numeral_numeral(nat),V)) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M),aa(num,nat,numeral_numeral(nat),V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_1823_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: nat] : modulo_modulo(nat,M,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N)))) = modulo_modulo(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N)) ).

% mod_Suc_eq_mod_add3
tff(fact_1824_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_1825_xor__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,X)) ).

% xor_nat_numerals(4)
tff(fact_1826_xor__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% xor_nat_numerals(3)
tff(fact_1827_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),aa(num,num,bit0,Y)) ).

% xor_nat_numerals(2)
tff(fact_1828_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),aa(num,num,bit0,Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_1829_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),aa(num,num,bit0,one2))) ) ) ).

% dbl_simps(4)
tff(fact_1830_power__minus1__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) = one_one(A) ) ).

% power_minus1_even
tff(fact_1831_neg__one__even__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] :
          ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),N) = one_one(A) ) ) ) ).

% neg_one_even_power
tff(fact_1832_neg__one__odd__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] :
          ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),N) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% neg_one_odd_power
tff(fact_1833_even__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N)) ) ) ).

% even_of_nat
tff(fact_1834_take__bit__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,zero_zero(nat))),A2) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_Suc_0
tff(fact_1835_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,I2: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I2)),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I2)),N))) ) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_1836_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: num,N: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I2)),N)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I2)),N)),X)) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_1837_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: nat,I2: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I2)),N)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I2)),N))) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_1838_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I2: num,N: nat,X: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),I2)),N)),aa(nat,A,semiring_1_of_nat(A),X)))
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),I2)),N)),X)) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_1839_signed__take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,zero_zero(nat)),A2) = aa(A,A,uminus_uminus(A),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% signed_take_bit_0
tff(fact_1840_xor__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X))),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),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(4)
tff(fact_1841_xor__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(6)
tff(fact_1842_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) ) ).

% take_bit_of_exp
tff(fact_1843_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_of_2
tff(fact_1844_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),aa(num,num,bit1,V)),aa(num,int,numeral_numeral(int),aa(num,num,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),aa(num,num,bit0,one2))),modulo_modulo(int,aa(num,int,numeral_numeral(int),V),aa(num,int,numeral_numeral(int),W)))),one_one(int)) ).

% zmod_numeral_Bit1
tff(fact_1845_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_Suc_bit1
tff(fact_1846_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_1847_signed__take__bit__minus,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,uminus_uminus(int),K)) ).

% signed_take_bit_minus
tff(fact_1848_minus__real__def,axiom,
    ! [X: real,Y: real] : aa(real,real,minus_minus(real,X),Y) = aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,uminus_uminus(real),Y)) ).

% minus_real_def
tff(fact_1849_xor_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),C2)) ) ).

% xor.assoc
tff(fact_1850_xor_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),A2) ) ).

% xor.commute
tff(fact_1851_of__nat__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_xor_eq
tff(fact_1852_take__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)) ) ).

% take_bit_of_nat
tff(fact_1853_xor_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B2),C2)) ) ).

% xor.left_commute
tff(fact_1854_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_1855_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_1856_power__minus__Bit1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,K: num] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K))) = aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K)))) ) ).

% power_minus_Bit1
tff(fact_1857_take__bit__add,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2))) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ).

% take_bit_add
tff(fact_1858_take__bit__tightened,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A,M: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),B2) ) ) ) ) ).

% take_bit_tightened
tff(fact_1859_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,M),Q3)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),Q3))) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_1860_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)),M)) ).

% take_bit_nat_less_eq_self
tff(fact_1861_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [X: nat,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),X)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,semiring_1_of_nat(A),X)) ) ).

% mult_of_nat_commute
tff(fact_1862_take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),L))) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).

% take_bit_mult
tff(fact_1863_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2))) ) ) ).

% le_imp_neg_le
tff(fact_1864_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),A2)) ) ) ).

% minus_le_iff
tff(fact_1865_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,uminus_uminus(A),A2))) ) ) ).

% le_minus_iff
tff(fact_1866_verit__negate__coefficient_I2_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2))) ) ) ).

% verit_negate_coefficient(2)
tff(fact_1867_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A2)),B2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),A2)) ) ) ).

% minus_less_iff
tff(fact_1868_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,uminus_uminus(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,uminus_uminus(A),A2))) ) ) ).

% less_minus_iff
tff(fact_1869_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] : aa(num,num,bit0,X22) != aa(num,num,bit1,X32) ).

% verit_eq_simplify(14)
tff(fact_1870_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] : one2 != aa(num,num,bit1,X32) ).

% verit_eq_simplify(12)
tff(fact_1871_numeral__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,N: num] : aa(num,A,numeral_numeral(A),M) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).

% numeral_neq_neg_numeral
tff(fact_1872_neg__numeral__neq__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,N: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) != aa(num,A,numeral_numeral(A),N) ) ).

% neg_numeral_neq_numeral
tff(fact_1873_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_1874_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_1875_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_1876_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_1877_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_1878_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_1879_take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),L))) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,minus_minus(int,K),L)) ).

% take_bit_diff
tff(fact_1880_minus__diff__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(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,minus_minus(A,A2),B2)) ) ).

% minus_diff_minus
tff(fact_1881_minus__diff__commute,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B2: A,A2: A] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),B2)),A2) = aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% minus_diff_commute
tff(fact_1882_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_1883_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_1884_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_1885_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_1886_Diff__mono,axiom,
    ! [A: $tType,A3: set(A),C5: set(A),D4: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),D4),B3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),B3)),aa(set(A),set(A),minus_minus(set(A),C5),D4))) ) ) ).

% Diff_mono
tff(fact_1887_Diff__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),B3)),A3)) ).

% Diff_subset
tff(fact_1888_double__diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C5))
       => ( aa(set(A),set(A),minus_minus(set(A),B3),aa(set(A),set(A),minus_minus(set(A),C5),A3)) = A3 ) ) ) ).

% double_diff
tff(fact_1889_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_1890_mod__minus__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,A4: A] :
          ( ( modulo_modulo(A,A2,B2) = modulo_modulo(A,A4,B2) )
         => ( modulo_modulo(A,aa(A,A,uminus_uminus(A),A2),B2) = modulo_modulo(A,aa(A,A,uminus_uminus(A),A4),B2) ) ) ) ).

% mod_minus_cong
tff(fact_1891_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_1892_psubset__imp__ex__mem,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
     => ? [B5: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B5),aa(set(A),set(A),minus_minus(set(A),B3),A3))) ) ).

% psubset_imp_ex_mem
tff(fact_1893_concat__bit__eq__iff,axiom,
    ! [N: nat,K: int,L: int,R3: int,S: int] :
      ( ( aa(int,int,bit_concat_bit(N,K),L) = aa(int,int,bit_concat_bit(N,R3),S) )
    <=> ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),R3) )
        & ( L = S ) ) ) ).

% concat_bit_eq_iff
tff(fact_1894_concat__bit__take__bit__eq,axiom,
    ! [N: nat,B2: int] : bit_concat_bit(N,aa(int,int,bit_se2584673776208193580ke_bit(int,N),B2)) = bit_concat_bit(N,B2) ).

% concat_bit_take_bit_eq
tff(fact_1895_real__of__nat__div2,axiom,
    ! [N: nat,X: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,minus_minus(real,divide_divide(real,aa(nat,real,semiring_1_of_nat(real),N),aa(nat,real,semiring_1_of_nat(real),X))),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,N,X))))) ).

% real_of_nat_div2
tff(fact_1896_real__of__nat__div3,axiom,
    ! [N: nat,X: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,minus_minus(real,divide_divide(real,aa(nat,real,semiring_1_of_nat(real),N),aa(nat,real,semiring_1_of_nat(real),X))),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,N,X)))),one_one(real))) ).

% real_of_nat_div3
tff(fact_1897_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N))) ) ).

% of_nat_0_le_iff
tff(fact_1898_of__nat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A))) ) ).

% of_nat_less_0_iff
tff(fact_1899_of__nat__neq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)) != zero_zero(A) ) ).

% of_nat_neq_0
tff(fact_1900_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,M: nat,N: nat] : divide_divide(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) = divide_divide(A,divide_divide(A,A2,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% div_mult2_eq'
tff(fact_1901_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ) ).

% of_nat_less_imp_less
tff(fact_1902_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) ) ) ).

% less_imp_of_nat_less
tff(fact_1903_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K))) ) ).

% take_bit_tightened_less_eq_int
tff(fact_1904_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I2: nat,J: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),I2)),aa(nat,A,semiring_1_of_nat(A),J))) ) ) ).

% of_nat_mono
tff(fact_1905_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),K))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_1906_take__bit__nonnegative,axiom,
    ! [N: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K))) ).

% take_bit_nonnegative
tff(fact_1907_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% take_bit_int_greater_self_iff
tff(fact_1908_not__take__bit__negative,axiom,
    ! [N: nat,K: int] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),zero_zero(int))) ).

% not_take_bit_negative
tff(fact_1909_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A,B2: A] :
          ( ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = aa(A,A,bit_ri4674362597316999326ke_bit(A,N),B2) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),B2) ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_1910_signed__take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) = aa(A,A,if(fun(A,A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M),bit_se2584673776208193580ke_bit(A,N),bit_ri4674362597316999326ke_bit(A,M)),A2) ) ).

% signed_take_bit_take_bit
tff(fact_1911_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N))) ) ).

% neg_numeral_le_numeral
tff(fact_1912_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_numeral_le_neg_numeral
tff(fact_1913_zero__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] : zero_zero(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).

% zero_neq_neg_numeral
tff(fact_1914_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_numeral_less_neg_numeral
tff(fact_1915_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N))) ) ).

% neg_numeral_less_numeral
tff(fact_1916_le__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% le_minus_one_simps(4)
tff(fact_1917_le__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A))) ) ).

% le_minus_one_simps(2)
tff(fact_1918_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_1919_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_1920_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_1921_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_1922_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_1923_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_1924_xor__num_Ocases,axiom,
    ! [X: product_prod(num,num)] :
      ( ( X != aa(num,product_prod(num,num),product_Pair(num,num,one2),one2) )
     => ( ! [N2: num] : X != aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit0,N2))
       => ( ! [N2: num] : X != aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit1,N2))
         => ( ! [M5: num] : X != aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M5)),one2)
           => ( ! [M5: num,N2: num] : X != aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M5)),aa(num,num,bit0,N2))
             => ( ! [M5: num,N2: num] : X != aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M5)),aa(num,num,bit1,N2))
               => ( ! [M5: num] : X != aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M5)),one2)
                 => ( ! [M5: num,N2: num] : X != aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M5)),aa(num,num,bit0,N2))
                   => ~ ! [M5: num,N2: num] : X != aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M5)),aa(num,num,bit1,N2)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_1925_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one2 )
     => ( ! [X23: num] : Y != aa(num,num,bit0,X23)
       => ~ ! [X33: num] : Y != aa(num,num,bit1,X33) ) ) ).

% num.exhaust
tff(fact_1926_less__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A))) ) ).

% less_minus_one_simps(2)
tff(fact_1927_less__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% less_minus_one_simps(4)
tff(fact_1928_numeral__times__minus__swap,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [W: num,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),aa(A,A,uminus_uminus(A),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) ) ).

% numeral_times_minus_swap
tff(fact_1929_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_1930_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_1931_one__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] : one_one(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).

% one_neq_neg_numeral
tff(fact_1932_numeral__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),N) != aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% numeral_neq_neg_one
tff(fact_1933_square__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [X: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),X) = one_one(A) )
        <=> ( ( X = one_one(A) )
            | ( X = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% square_eq_1_iff
tff(fact_1934_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,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,minus_minus(A,A2),B2)) ) ) ) ).

% group_cancel.sub2
tff(fact_1935_diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(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_1936_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(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_1937_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),divide_divide(nat,M,N)) = divide_divide(A,aa(nat,A,semiring_1_of_nat(A),M),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_1938_of__nat__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] :
          ( pp(dvd_dvd(A,aa(nat,A,semiring_1_of_nat(A),M),aa(nat,A,semiring_1_of_nat(A),N)))
        <=> pp(dvd_dvd(nat,M,N)) ) ) ).

% of_nat_dvd_iff
tff(fact_1939_take__bit__unset__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat,A2: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2)) = aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) ) ) ) ) ).

% take_bit_unset_bit_eq
tff(fact_1940_take__bit__set__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat,A2: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A2)) = aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) ) ) ) ) ).

% take_bit_set_bit_eq
tff(fact_1941_take__bit__flip__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat,A2: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se8732182000553998342ip_bit(A,M,A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se8732182000553998342ip_bit(A,M,A2)) = bit_se8732182000553998342ip_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) ) ) ) ) ).

% take_bit_flip_bit_eq
tff(fact_1942_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( pp(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_1943_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( pp(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_1944_real__of__nat__div4,axiom,
    ! [N: nat,X: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,N,X))),divide_divide(real,aa(nat,real,semiring_1_of_nat(real),N),aa(nat,real,semiring_1_of_nat(real),X)))) ).

% real_of_nat_div4
tff(fact_1945_of__nat__mod,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),modulo_modulo(nat,M,N)) = modulo_modulo(A,aa(nat,A,semiring_1_of_nat(A),M),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_mod
tff(fact_1946_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),U),U))),aa(real,real,aa(real,fun(real,real),times_times(real),X),X))) ).

% real_minus_mult_self_le
tff(fact_1947_real__of__nat__div,axiom,
    ! [D2: nat,N: nat] :
      ( pp(dvd_dvd(nat,D2,N))
     => ( aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,N,D2)) = divide_divide(real,aa(nat,real,semiring_1_of_nat(real),N),aa(nat,real,semiring_1_of_nat(real),D2)) ) ) ).

% real_of_nat_div
tff(fact_1948_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_1949_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_1950_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% take_bit_Suc_minus_bit0
tff(fact_1951_Bernoulli__inequality,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),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),N)),X))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),N))) ) ).

% Bernoulli_inequality
tff(fact_1952_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,X: extended_enat] :
      ( pp(aa(extended_enat,bool,aa(extended_enat,fun(extended_enat,bool),ord_less_eq(extended_enat),Z),Y))
     => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),X),aa(extended_enat,extended_enat,minus_minus(extended_enat,Y),Z)) = aa(extended_enat,extended_enat,minus_minus(extended_enat,aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),X),Y)),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_1953_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),A2) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_1954_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),zero_zero(A))) ) ).

% neg_numeral_le_zero
tff(fact_1955_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_zero_le_neg_numeral
tff(fact_1956_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))) ) ).

% not_zero_less_neg_numeral
tff(fact_1957_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),zero_zero(A))) ) ).

% neg_numeral_less_zero
tff(fact_1958_le__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% le_minus_one_simps(3)
tff(fact_1959_le__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A))) ) ).

% le_minus_one_simps(1)
tff(fact_1960_numeral__Bit1,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)) = 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),N)),aa(num,A,numeral_numeral(A),N))),one_one(A)) ) ).

% numeral_Bit1
tff(fact_1961_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) != zero_zero(int) )
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,minus_minus(int,K),one_one(int))) = aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),one_one(int)) ) ) ).

% take_bit_decr_eq
tff(fact_1962_less__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A))) ) ).

% less_minus_one_simps(1)
tff(fact_1963_less__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% less_minus_one_simps(3)
tff(fact_1964_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,M),N)) = aa(A,A,minus_minus(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ) ) ).

% of_nat_diff
tff(fact_1965_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))) ) ).

% not_one_le_neg_numeral
tff(fact_1966_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% not_numeral_le_neg_one
tff(fact_1967_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% neg_numeral_le_neg_one
tff(fact_1968_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M))) ) ).

% neg_one_le_numeral
tff(fact_1969_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A))) ) ).

% neg_numeral_le_one
tff(fact_1970_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))) ) ).

% not_neg_one_less_neg_numeral
tff(fact_1971_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))) ) ).

% not_one_less_neg_numeral
tff(fact_1972_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A)))) ) ).

% not_numeral_less_neg_one
tff(fact_1973_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M))) ) ).

% neg_one_less_numeral
tff(fact_1974_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A))) ) ).

% neg_numeral_less_one
tff(fact_1975_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)) )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = aa(A,A,uminus_uminus(A),B2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% eq_minus_divide_eq
tff(fact_1976_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 )
        <=> ( ( ( C2 != zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) ) )
            & ( ( C2 = zero_zero(A) )
             => ( A2 = zero_zero(A) ) ) ) ) ) ).

% minus_divide_eq_eq
tff(fact_1977_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_1978_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_1979_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_1980_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_1981_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_1982_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_1983_power__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,N: nat] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A2)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,power_power(A,A2),N)) ) ).

% power_minus
tff(fact_1984_eval__nat__numeral_I3_J,axiom,
    ! [N: num] : aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,N)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,N))) ).

% eval_nat_numeral(3)
tff(fact_1985_cong__exp__iff__simps_I10_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q3: num,N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,M)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) ) ).

% cong_exp_iff_simps(10)
tff(fact_1986_cong__exp__iff__simps_I12_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q3: num,N: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) != modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) ) ).

% cong_exp_iff_simps(12)
tff(fact_1987_cong__exp__iff__simps_I13_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q3: num,N: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Q3)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q3)) ) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_1988_power__minus__Bit0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,K: num] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,K))) = aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,K))) ) ).

% power_minus_Bit0
tff(fact_1989_reals__Archimedean3,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ! [Y3: real] :
        ? [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y3),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),X))) ) ).

% reals_Archimedean3
tff(fact_1990_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% take_bit_Suc_bit1
tff(fact_1991_take__bit__Suc__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,suc,N))),one_one(A)) ) ).

% take_bit_Suc_minus_1_eq
tff(fact_1992_take__bit__numeral__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),K)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),K))),one_one(A)) ) ).

% take_bit_numeral_minus_1_eq
tff(fact_1993_real__0__less__add__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),X)),Y)) ) ).

% real_0_less_add_iff
tff(fact_1994_real__add__less__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),aa(real,real,uminus_uminus(real),X))) ) ).

% real_add_less_0_iff
tff(fact_1995_real__0__le__add__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),X)),Y)) ) ).

% real_0_le_add_iff
tff(fact_1996_real__add__le__0__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),aa(real,real,uminus_uminus(real),X))) ) ).

% real_add_le_0_iff
tff(fact_1997_real__of__nat__div__aux,axiom,
    ! [X: nat,D2: nat] : divide_divide(real,aa(nat,real,semiring_1_of_nat(real),X),aa(nat,real,semiring_1_of_nat(real),D2)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,X,D2))),divide_divide(real,aa(nat,real,semiring_1_of_nat(real),modulo_modulo(nat,X,D2)),aa(nat,real,semiring_1_of_nat(real),D2))) ).

% real_of_nat_div_aux
tff(fact_1998_zmod__zminus1__eq__if,axiom,
    ! [A2: int,B2: int] :
      ( ( ( modulo_modulo(int,A2,B2) = zero_zero(int) )
       => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),A2),B2) = zero_zero(int) ) )
      & ( ( modulo_modulo(int,A2,B2) != zero_zero(int) )
       => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),A2),B2) = aa(int,int,minus_minus(int,B2),modulo_modulo(int,A2,B2)) ) ) ) ).

% zmod_zminus1_eq_if
tff(fact_1999_zmod__zminus2__eq__if,axiom,
    ! [A2: int,B2: int] :
      ( ( ( modulo_modulo(int,A2,B2) = zero_zero(int) )
       => ( modulo_modulo(int,A2,aa(int,int,uminus_uminus(int),B2)) = zero_zero(int) ) )
      & ( ( modulo_modulo(int,A2,B2) != zero_zero(int) )
       => ( modulo_modulo(int,A2,aa(int,int,uminus_uminus(int),B2)) = aa(int,int,minus_minus(int,modulo_modulo(int,A2,B2)),B2) ) ) ) ).

% zmod_zminus2_eq_if
tff(fact_2000_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,M: nat,N: nat] : modulo_modulo(A,A2,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),modulo_modulo(A,divide_divide(A,A2,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)))),modulo_modulo(A,A2,aa(nat,A,semiring_1_of_nat(A),M))) ) ).

% mod_mult2_eq'
tff(fact_2001_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),K)) = aa(int,int,minus_minus(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_2002_numeral__Bit1__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: num] : divide_divide(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = aa(num,A,numeral_numeral(A),N) ) ).

% numeral_Bit1_div_2
tff(fact_2003_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),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_2004_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),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_2005_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),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_2006_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),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_2007_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ) ) ) ) ).

% minus_divide_less_eq
tff(fact_2008_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ) ) ) ) ).

% less_minus_divide_eq
tff(fact_2009_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) )
        <=> ( ( ( 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 ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) ) ) ) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_2010_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)) )
        <=> ( ( ( 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) ) )
            & ( ( C2 = zero_zero(A) )
             => ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A) ) ) ) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_2011_odd__numeral,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: num] : ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)))) ) ).

% odd_numeral
tff(fact_2012_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),divide_divide(A,X,Z))),Y) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)),Z) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_2013_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: 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,A2,Z))),B2) = B2 ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),divide_divide(A,A2,Z))),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_2014_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num,Q3: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) != zero_zero(A) ) ).

% cong_exp_iff_simps(3)
tff(fact_2015_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),divide_divide(A,A2,Z))),B2) = aa(A,A,uminus_uminus(A),B2) ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),divide_divide(A,A2,Z))),B2) = divide_divide(A,aa(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_2016_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,A2: A,B2: A] :
          ( ( ( Z = zero_zero(A) )
           => ( aa(A,A,minus_minus(A,divide_divide(A,A2,Z)),B2) = aa(A,A,uminus_uminus(A),B2) ) )
          & ( ( Z != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,divide_divide(A,A2,Z)),B2) = divide_divide(A,aa(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_2017_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,X: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),divide_divide(A,X,Z))),Y) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)),Z) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_2018_power3__eq__cube,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2)),A2) ) ).

% power3_eq_cube
tff(fact_2019_even__minus,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,uminus_uminus(A),A2)))
        <=> pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)) ) ) ).

% even_minus
tff(fact_2020_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_2021_even__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)))
        <=> ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
          <=> pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2)) ) ) ) ).

% even_xor_iff
tff(fact_2022_power2__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [X: A,Y: A] :
          ( ( aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) )
        <=> ( ( X = Y )
            | ( X = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% power2_eq_iff
tff(fact_2023_Suc3__eq__add__3,axiom,
    ! [N: nat] : aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N) ).

% Suc3_eq_add_3
tff(fact_2024_field__char__0__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),divide_divide(nat,M,N)) = divide_divide(A,aa(A,A,minus_minus(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),modulo_modulo(nat,M,N))),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% field_char_0_class.of_nat_div
tff(fact_2025_nat__less__real__le,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),N)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% nat_less_real_le
tff(fact_2026_nat__le__real__less,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),M)),one_one(real)))) ) ).

% nat_le_real_less
tff(fact_2027_verit__less__mono__div__int2,axiom,
    ! [A3: int,B3: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A3),B3))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,uminus_uminus(int),N)))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),divide_divide(int,B3,N)),divide_divide(int,A3,N))) ) ) ).

% verit_less_mono_div_int2
tff(fact_2028_div__eq__minus1,axiom,
    ! [B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),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_2029_take__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_Suc_bit0
tff(fact_2030_take__bit__eq__mod,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = modulo_modulo(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) ) ).

% take_bit_eq_mod
tff(fact_2031_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M) = M )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ) ).

% take_bit_nat_eq_self_iff
tff(fact_2032_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% take_bit_nat_less_exp
tff(fact_2033_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_2034_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_2035_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_2036_take__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M) = modulo_modulo(nat,M,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ).

% take_bit_nat_def
tff(fact_2037_of__nat__less__two__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))) ) ).

% of_nat_less_two_power
tff(fact_2038_inverse__of__nat__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => ( ( N != zero_zero(nat) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),M))),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),N)))) ) ) ) ).

% inverse_of_nat_le
tff(fact_2039_take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ).

% take_bit_int_less_exp
tff(fact_2040_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ) ) ) ) ).

% le_minus_divide_eq
tff(fact_2041_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ) ) ) ).

% minus_divide_le_eq
tff(fact_2042_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),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_2043_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),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_2044_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),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_2045_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),divide_divide(A,B2,C2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),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_2046_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),divide_divide(A,B2,C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),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)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),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))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),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_2047_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,B2,C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),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))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),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)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),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_2048_take__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = modulo_modulo(int,K,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ).

% take_bit_int_def
tff(fact_2049_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q3: num,N: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),N),aa(num,A,numeral_numeral(A),Q3)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(7)
tff(fact_2050_cong__exp__iff__simps_I11_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q3: num] :
          ( ( modulo_modulo(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) = modulo_modulo(A,aa(num,A,numeral_numeral(A),one2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Q3))) )
        <=> ( modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),Q3)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(11)
tff(fact_2051_power2__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A2: A] :
          ( ( aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,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_2052_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C2))
       => ( ! [M5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M5))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M5)),X)),C2)) )
         => ( X = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_2053_uminus__power__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat,A2: A] :
          ( ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
           => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A2)),N) = aa(nat,A,power_power(A,A2),N) ) )
          & ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
           => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A2)),N) = aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,A2),N)) ) ) ) ) ).

% uminus_power_if
tff(fact_2054_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: nat] : divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M))),N) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M),N) ).

% Suc_div_eq_add3_div
tff(fact_2055_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)) = aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,minus_minus(nat,N),K)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_2056_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: nat] : modulo_modulo(nat,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M))),N) = modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M),N) ).

% Suc_mod_eq_add3_mod
tff(fact_2057_realpow__square__minus__le,axiom,
    ! [U: real,X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(nat,real,power_power(real,U),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% realpow_square_minus_le
tff(fact_2058_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),K))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),K)) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_2059_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K))) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_2060_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_2061_take__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = zero_zero(A) )
        <=> pp(dvd_dvd(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N),A2)) ) ) ).

% take_bit_eq_0_iff
tff(fact_2062_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L))
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),K),L) = aa(int,int,minus_minus(int,aa(int,int,minus_minus(int,L),one_one(int))),modulo_modulo(int,aa(int,int,minus_minus(int,K),one_one(int)),L)) ) ) ).

% minus_mod_int_eq
tff(fact_2063_zmod__minus1,axiom,
    ! [B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),B2) = aa(int,int,minus_minus(int,B2),one_one(int)) ) ) ).

% zmod_minus1
tff(fact_2064_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( ( modulo_modulo(int,A2,B2) = zero_zero(int) )
         => ( divide_divide(int,aa(int,int,uminus_uminus(int),A2),B2) = aa(int,int,uminus_uminus(int),divide_divide(int,A2,B2)) ) )
        & ( ( modulo_modulo(int,A2,B2) != zero_zero(int) )
         => ( divide_divide(int,aa(int,int,uminus_uminus(int),A2),B2) = aa(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_2065_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( ( modulo_modulo(int,A2,B2) = zero_zero(int) )
         => ( divide_divide(int,A2,aa(int,int,uminus_uminus(int),B2)) = aa(int,int,uminus_uminus(int),divide_divide(int,A2,B2)) ) )
        & ( ( modulo_modulo(int,A2,B2) != zero_zero(int) )
         => ( divide_divide(int,A2,aa(int,int,uminus_uminus(int),B2)) = aa(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_2066_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)),M))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),M)) ) ).

% take_bit_nat_less_self_iff
tff(fact_2067_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q3: int,R3: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),product_Pair(int,int,Q3),R3))
     => ( ( 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,if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),R3),zero_zero(int)),aa(int,int,uminus_uminus(int),Q3),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),Q3)),one_one(int)))),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),R3),zero_zero(int)),zero_zero(int),aa(int,int,minus_minus(int,B2),R3)))) ) ) ).

% zminus1_lemma
tff(fact_2068_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),K))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),K)) ) ).

% take_bit_int_less_self_iff
tff(fact_2069_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_2070_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),divide_divide(A,B2,C2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),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)) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),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))) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),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_2071_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,B2,C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),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))) )
            & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
             => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),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)) )
                & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),zero_zero(A)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),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_2072_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A))) ) ) ) ).

% square_le_1
tff(fact_2073_minus__power__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A2: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A2)),N)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A2)),N)) = aa(nat,A,power_power(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ) ).

% minus_power_mult_self
tff(fact_2074_minus__one__power__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] :
          ( ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
           => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),N) = one_one(A) ) )
          & ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
           => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),N) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% minus_one_power_iff
tff(fact_2075_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))
       => ( aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_2076_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = K )
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),K))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_2077_minus__1__div__exp__eq__int,axiom,
    ! [N: nat] : divide_divide(int,aa(int,int,uminus_uminus(int),one_one(int)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% minus_1_div_exp_eq_int
tff(fact_2078_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),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_2079_take__bit__int__eq__self,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_2080_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = K )
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_2081_signed__take__bit__eq__take__bit__shift,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_2082_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) != aa(int,int,minus_minus(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),one_one(int)) )
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ) ) ).

% take_bit_incr_eq
tff(fact_2083_power__minus1__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] : aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% power_minus1_odd
tff(fact_2084_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% take_bit_Suc
tff(fact_2085_int__bit__induct,axiom,
    ! [P: fun(int,bool),K: int] :
      ( pp(aa(int,bool,P,zero_zero(int)))
     => ( pp(aa(int,bool,P,aa(int,int,uminus_uminus(int),one_one(int))))
       => ( ! [K2: int] :
              ( pp(aa(int,bool,P,K2))
             => ( ( K2 != zero_zero(int) )
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))) ) )
         => ( ! [K2: int] :
                ( pp(aa(int,bool,P,K2))
               => ( ( K2 != aa(int,int,uminus_uminus(int),one_one(int)) )
                 => pp(aa(int,bool,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),aa(num,num,bit0,one2)))))) ) )
           => pp(aa(int,bool,P,K)) ) ) ) ) ).

% int_bit_induct
tff(fact_2086_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(int,int,minus_minus(int,K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))) ) ) ).

% take_bit_int_less_eq
tff(fact_2087_xor__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = N ) )
      & ( ( M != zero_zero(nat) )
       => ( ( ( N = zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = M ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = 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,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ) ) ).

% xor_nat_unfold
tff(fact_2088_take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K))) ) ).

% take_bit_int_greater_eq
tff(fact_2089_xor__nat__rec,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,aa(bool,bool,aa(bool,fun(bool,bool),fequal(bool),aa(bool,bool,fNot,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),M))),aa(bool,bool,fNot,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% xor_nat_rec
tff(fact_2090_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( ( divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A2 )
         => ( ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
             => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = zero_zero(A) ) )
            & ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
             => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A)) ) ) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_2091_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,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(bool,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)))),aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)))) ) ).

% xor_one_eq
tff(fact_2092_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,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(bool,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)))),aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)))) ) ).

% one_xor_eq
tff(fact_2093_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(nat,nat,suc,N)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K))) ) ).

% signed_take_bit_int_greater_eq
tff(fact_2094_xor__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N)))) ).

% xor_Suc_0_eq
tff(fact_2095_real__average__minus__first,axiom,
    ! [A2: real,B2: real] : aa(real,real,minus_minus(real,divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),A2) = divide_divide(real,aa(real,real,minus_minus(real,B2),A2),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% real_average_minus_first
tff(fact_2096_real__average__minus__second,axiom,
    ! [B2: real,A2: real] : aa(real,real,minus_minus(real,divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),B2),A2),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),A2) = divide_divide(real,aa(real,real,minus_minus(real,B2),A2),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% real_average_minus_second
tff(fact_2097_linear__plus__1__le__power,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),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),N)),X)),one_one(real))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),one_one(real))),N))) ) ).

% linear_plus_1_le_power
tff(fact_2098_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = zero_zero(nat) )
      | ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = one_one(nat) )
      | ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
      | ( modulo_modulo(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) ) ).

% mod_exhaust_less_4
tff(fact_2099_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E2))
         => ~ ! [N2: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N2)))),E2)) ) ) ).

% nat_approx_posE
tff(fact_2100_compl__less__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X)) ) ) ).

% compl_less_compl_iff
tff(fact_2101_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% compl_le_compl_iff
tff(fact_2102_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_2103_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = aa(num,int,numeral_numeral(int),V) )
    <=> ( M = aa(num,nat,numeral_numeral(nat),V) ) ) ).

% int_eq_iff_numeral
tff(fact_2104_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) ) ) ).

% xor_nonnegative_int_iff
tff(fact_2105_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),zero_zero(int)))
    <=> ~ ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ).

% xor_negative_int_iff
tff(fact_2106_negative__zle,axiom,
    ! [N: nat,M: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))),aa(nat,int,semiring_1_of_nat(int),M))) ).

% negative_zle
tff(fact_2107_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_2108_negative__zless,axiom,
    ! [N: nat,M: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),aa(nat,int,semiring_1_of_nat(int),M))) ).

% negative_zless
tff(fact_2109_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_2110_take__bit__minus,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K))) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),K)) ).

% take_bit_minus
tff(fact_2111_int__cases,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] : Z != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))) ) ).

% int_cases
tff(fact_2112_int__of__nat__induct,axiom,
    ! [P: fun(int,bool),Z: int] :
      ( ! [N2: nat] : pp(aa(int,bool,P,aa(nat,int,semiring_1_of_nat(int),N2)))
     => ( ! [N2: nat] : pp(aa(int,bool,P,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2)))))
       => pp(aa(int,bool,P,Z)) ) ) ).

% int_of_nat_induct
tff(fact_2113_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M)))) ).

% not_int_zless_negative
tff(fact_2114_XOR__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),X),Y))) ) ) ).

% XOR_lower
tff(fact_2115_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] : M != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
           => ( M != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ) ) ).

% int_cases4
tff(fact_2116_int__ops_I3_J,axiom,
    ! [N: num] : aa(nat,int,semiring_1_of_nat(int),aa(num,nat,numeral_numeral(nat),N)) = aa(num,int,numeral_numeral(int),N) ).

% int_ops(3)
tff(fact_2117_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M))))
    <=> ( ( N = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% int_zle_neg
tff(fact_2118_nat__int__comparison_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A2),B2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),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_2119_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% zle_int
tff(fact_2120_nat__int__comparison_I3_J,axiom,
    ! [A2: nat,B2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),B2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),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_2121_nonneg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ~ ! [N2: nat] : K != aa(nat,int,semiring_1_of_nat(int),N2) ) ).

% nonneg_int_cases
tff(fact_2122_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ? [N2: nat] : K = aa(nat,int,semiring_1_of_nat(int),N2) ) ).

% zero_le_imp_eq_int
tff(fact_2123_negative__zle__0,axiom,
    ! [N: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))),zero_zero(int))) ).

% negative_zle_0
tff(fact_2124_nonpos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),zero_zero(int)))
     => ~ ! [N2: nat] : K != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ).

% nonpos_int_cases
tff(fact_2125_zadd__int__left,axiom,
    ! [M: nat,N: nat,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),Z)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))),Z) ).

% zadd_int_left
tff(fact_2126_int__plus,axiom,
    ! [N: nat,M: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),aa(nat,int,semiring_1_of_nat(int),M)) ).

% int_plus
tff(fact_2127_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_2128_int__ops_I2_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

% int_ops(2)
tff(fact_2129_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W),Z))
    <=> ? [N5: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),N5)) ) ).

% zle_iff_zadd
tff(fact_2130_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_2131_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_2132_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( ! [N2: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N2) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) )
       => ~ ! [N2: nat] :
              ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) )
             => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ) ) ).

% int_cases3
tff(fact_2133_not__zle__0__negative,axiom,
    ! [N: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),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,N))))) ).

% not_zle_0_negative
tff(fact_2134_negD,axiom,
    ! [X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),zero_zero(int)))
     => ? [N2: nat] : X = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))) ) ).

% negD
tff(fact_2135_negative__zless__0,axiom,
    ! [N: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),zero_zero(int))) ).

% negative_zless_0
tff(fact_2136_int__Suc,axiom,
    ! [N: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) ).

% int_Suc
tff(fact_2137_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_2138_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z))
    <=> ? [N5: 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,N5))) ) ).

% zless_iff_Suc_zadd
tff(fact_2139_pos__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ~ ! [N2: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N2) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ) ).

% pos_int_cases
tff(fact_2140_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),K))
     => ? [N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
          & ( K = aa(nat,int,semiring_1_of_nat(int),N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_2141_neg__int__cases,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
     => ~ ! [N2: nat] :
            ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) )
           => ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2)) ) ) ).

% neg_int_cases
tff(fact_2142_zmult__zless__mono2__lemma,axiom,
    ! [I2: int,J: int,K: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),J))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),I2)),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_2143_int__ops_I6_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,A2),B2)) = zero_zero(int) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))
       => ( aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,A2),B2)) = aa(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_2144_zdiff__int__split,axiom,
    ! [P: fun(int,bool),X: nat,Y: nat] :
      ( pp(aa(int,bool,P,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,X),Y))))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),X))
         => pp(aa(int,bool,P,aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),X)),aa(nat,int,semiring_1_of_nat(int),Y)))) )
        & ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
         => pp(aa(int,bool,P,zero_zero(int))) ) ) ) ).

% zdiff_int_split
tff(fact_2145_dbl__dec__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl_dec(A,X) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X)),one_one(A)) ) ).

% dbl_dec_def
tff(fact_2146_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).

% compl_le_swap2
tff(fact_2147_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,uminus_uminus(A),X)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y))) ) ) ).

% compl_le_swap1
tff(fact_2148_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),aa(A,A,uminus_uminus(A),X))) ) ) ).

% compl_mono
tff(fact_2149_compl__less__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),Y)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).

% compl_less_swap2
tff(fact_2150_compl__less__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(A,A,uminus_uminus(A),X)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,uminus_uminus(A),Y))) ) ) ).

% compl_less_swap1
tff(fact_2151_XOR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),X),Y)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ) ) ) ).

% XOR_upper
tff(fact_2152_real__arch__simple,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(nat,A,semiring_1_of_nat(A),N2))) ) ).

% real_arch_simple
tff(fact_2153_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(nat,A,semiring_1_of_nat(A),N2))) ) ).

% reals_Archimedean2
tff(fact_2154_exists__least__lemma,axiom,
    ! [P: fun(nat,bool)] :
      ( ~ pp(aa(nat,bool,P,zero_zero(nat)))
     => ( ? [X_12: nat] : pp(aa(nat,bool,P,X_12))
       => ? [N2: nat] :
            ( ~ pp(aa(nat,bool,P,N2))
            & pp(aa(nat,bool,P,aa(nat,nat,suc,N2))) ) ) ) ).

% exists_least_lemma
tff(fact_2155_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(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(bool,bool,aa(bool,fun(bool,bool),fequal(bool),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K))),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).

% xor_int_rec
tff(fact_2156_Bolzano,axiom,
    ! [A2: real,B2: real,P: fun(real,fun(real,bool))] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
     => ( ! [A5: real,B5: real,C4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),P,A5),B5))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),P,B5),C4))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A5),B5))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B5),C4))
                 => pp(aa(real,bool,aa(real,fun(real,bool),P,A5),C4)) ) ) ) )
       => ( ! [X3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
               => ? [D6: real] :
                    ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D6))
                    & ! [A5: real,B5: real] :
                        ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A5),X3))
                          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B5))
                          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,minus_minus(real,B5),A5)),D6)) )
                       => pp(aa(real,bool,aa(real,fun(real,bool),P,A5),B5)) ) ) ) )
         => pp(aa(real,bool,aa(real,fun(real,bool),P,A2),B2)) ) ) ) ).

% Bolzano
tff(fact_2157_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ? [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N2)),X))) ) ) ).

% ex_less_of_nat_mult
tff(fact_2158_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_2159_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit1,N)) = aa(A,product_prod(A,A),product_Pair(A,A,zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,M))) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit1,N)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit0,aa(num,num,bit1,N)))) ) ) ) ) ).

% divmod_algorithm_code(7)
tff(fact_2160_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit1,N)) = aa(A,product_prod(A,A),product_Pair(A,A,zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
           => ( unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit1,N)) = unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit0,aa(num,num,bit1,N)))) ) ) ) ) ).

% divmod_algorithm_code(8)
tff(fact_2161_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% take_bit_Suc_minus_bit1
tff(fact_2162_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [H: A,Z: A,K5: real,N: nat] :
          ( ( H != zero_zero(A) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),K5))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H))),K5))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,divide_divide(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H)),N)),aa(nat,A,power_power(A,Z),N)),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,power_power(A,Z),aa(nat,nat,minus_minus(nat,N),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),N)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))),aa(nat,real,power_power(real,K5),aa(nat,nat,minus_minus(nat,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),real_V7770717601297561774m_norm(A,H)))) ) ) ) ) ).

% lemma_termdiff3
tff(fact_2163_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),one_one(int)) ).

% signed_take_bit_numeral_bit1
tff(fact_2164_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_2165_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_2166_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( aa(nat,nat,suc,N) = aa(num,nat,numeral_numeral(nat),K) )
    <=> ( N = pred_numeral(K) ) ) ).

% Suc_eq_numeral
tff(fact_2167_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,N) )
    <=> ( pred_numeral(K) = N ) ) ).

% eq_numeral_Suc
tff(fact_2168_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_2169_pred__numeral__inc,axiom,
    ! [K: num] : pred_numeral(inc(K)) = aa(num,nat,numeral_numeral(nat),K) ).

% pred_numeral_inc
tff(fact_2170_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_2171_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_2172_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : pred_numeral(aa(num,num,bit1,K)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,K)) ).

% pred_numeral_simps(3)
tff(fact_2173_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),pred_numeral(K)),N)) ) ).

% less_numeral_Suc
tff(fact_2174_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),pred_numeral(K))) ) ).

% less_Suc_numeral
tff(fact_2175_le__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),pred_numeral(K))) ) ).

% le_Suc_numeral
tff(fact_2176_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),pred_numeral(K)),N)) ) ).

% le_numeral_Suc
tff(fact_2177_diff__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,minus_minus(nat,N),pred_numeral(K)) ).

% diff_Suc_numeral
tff(fact_2178_diff__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)) = aa(nat,nat,minus_minus(nat,pred_numeral(K)),N) ).

% diff_numeral_Suc
tff(fact_2179_minus__numeral__div__numeral,axiom,
    ! [M: num,N: num] : divide_divide(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M,N))) ).

% minus_numeral_div_numeral
tff(fact_2180_numeral__div__minus__numeral,axiom,
    ! [M: num,N: num] : divide_divide(int,aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M,N))) ).

% numeral_div_minus_numeral
tff(fact_2181_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_2182_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_2183_dvd__numeral__simp,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N)))
        <=> unique5940410009612947441es_aux(A,unique8689654367752047608divmod(A,N,M)) ) ) ).

% dvd_numeral_simp
tff(fact_2184_divmod__algorithm__code_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num] : unique8689654367752047608divmod(A,M,one2) = aa(A,product_prod(A,A),product_Pair(A,A,aa(num,A,numeral_numeral(A),M)),zero_zero(A)) ) ).

% divmod_algorithm_code(2)
tff(fact_2185_add__neg__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: 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),N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) ) ).

% add_neg_numeral_special(5)
tff(fact_2186_add__neg__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(M))) ) ).

% add_neg_numeral_special(6)
tff(fact_2187_diff__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),inc(M)) ) ).

% diff_numeral_special(6)
tff(fact_2188_diff__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) ) ).

% diff_numeral_special(5)
tff(fact_2189_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit0,N)) = 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_2190_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit1,N)) = 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_2191_one__div__minus__numeral,axiom,
    ! [N: num] : divide_divide(int,one_one(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,N))) ).

% one_div_minus_numeral
tff(fact_2192_minus__one__div__numeral,axiom,
    ! [N: num] : divide_divide(int,aa(int,int,uminus_uminus(int),one_one(int)),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,N))) ).

% minus_one_div_numeral
tff(fact_2193_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% signed_take_bit_numeral_bit0
tff(fact_2194_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_2195_num__induct,axiom,
    ! [P: fun(num,bool),X: num] :
      ( pp(aa(num,bool,P,one2))
     => ( ! [X3: num] :
            ( pp(aa(num,bool,P,X3))
           => pp(aa(num,bool,P,inc(X3))) )
       => pp(aa(num,bool,P,X)) ) ) ).

% num_induct
tff(fact_2196_add__inc,axiom,
    ! [X: num,Y: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),X),inc(Y)) = inc(aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y)) ).

% add_inc
tff(fact_2197_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_2198_inc_Osimps_I1_J,axiom,
    inc(one2) = aa(num,num,bit0,one2) ).

% inc.simps(1)
tff(fact_2199_inc_Osimps_I2_J,axiom,
    ! [X: num] : inc(aa(num,num,bit0,X)) = aa(num,num,bit1,X) ).

% inc.simps(2)
tff(fact_2200_inc_Osimps_I3_J,axiom,
    ! [X: num] : inc(aa(num,num,bit1,X)) = aa(num,num,bit0,inc(X)) ).

% inc.simps(3)
tff(fact_2201_add__One,axiom,
    ! [X: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),X),one2) = inc(X) ).

% add_One
tff(fact_2202_mult__inc,axiom,
    ! [X: num,Y: num] : aa(num,num,aa(num,fun(num,num),times_times(num),X),inc(Y)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),times_times(num),X),Y)),X) ).

% mult_inc
tff(fact_2203_pred__numeral__def,axiom,
    ! [K: num] : pred_numeral(K) = aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),K)),one_one(nat)) ).

% pred_numeral_def
tff(fact_2204_numeral__inc,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [X: num] : aa(num,A,numeral_numeral(A),inc(X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) ) ).

% numeral_inc
tff(fact_2205_dbl__inc__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [X: A] : neg_numeral_dbl_inc(A,X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X)),one_one(A)) ) ).

% dbl_inc_def
tff(fact_2206_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X6: fun(A,B)] :
          ( ? [K6: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K6))
              & ! [N5: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N5))),K6)) )
        <=> ? [N6: nat] :
            ! [N5: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N5))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6)))) ) ) ).

% lemma_NBseq_def2
tff(fact_2207_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X6: fun(A,B)] :
          ( ? [K6: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K6))
              & ! [N5: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N5))),K6)) )
        <=> ? [N6: nat] :
            ! [N5: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X6,N5))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6)))) ) ) ).

% lemma_NBseq_def
tff(fact_2208_divmod__int__def,axiom,
    ! [M: num,N: num] : unique8689654367752047608divmod(int,M,N) = aa(int,product_prod(int,int),product_Pair(int,int,divide_divide(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N))),modulo_modulo(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N))) ).

% divmod_int_def
tff(fact_2209_divmod__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,M,N) = aa(A,product_prod(A,A),product_Pair(A,A,divide_divide(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N))),modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N))) ) ).

% divmod_def
tff(fact_2210_divmod_H__nat__def,axiom,
    ! [M: num,N: num] : unique8689654367752047608divmod(nat,M,N) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,divide_divide(nat,aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N))),modulo_modulo(nat,aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N))) ).

% divmod'_nat_def
tff(fact_2211_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,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),aa(num,num,bit0,one2)))),one_one(int)) ).

% take_bit_numeral_minus_bit1
tff(fact_2212_take__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% take_bit_numeral_bit0
tff(fact_2213_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ).

% take_bit_numeral_minus_bit0
tff(fact_2214_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( ( pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
           => ( unique8689654367752047608divmod(A,M,N) = aa(A,product_prod(A,A),product_Pair(A,A,zero_zero(A)),aa(num,A,numeral_numeral(A),M)) ) )
          & ( ~ pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N))
           => ( unique8689654367752047608divmod(A,M,N) = unique1321980374590559556d_step(A,N,unique8689654367752047608divmod(A,M,aa(num,num,bit0,N))) ) ) ) ) ).

% divmod_divmod_step
tff(fact_2215_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% take_bit_numeral_bit1
tff(fact_2216_norm__divide__numeral,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,W: num] : real_V7770717601297561774m_norm(A,divide_divide(A,A2,aa(num,A,numeral_numeral(A),W))) = divide_divide(real,real_V7770717601297561774m_norm(A,A2),aa(num,real,numeral_numeral(real),W)) ) ).

% norm_divide_numeral
tff(fact_2217_norm__mult__numeral2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,W: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W))) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A2)),aa(num,real,numeral_numeral(real),W)) ) ).

% norm_mult_numeral2
tff(fact_2218_norm__mult__numeral1,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [W: num,A2: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),W)),real_V7770717601297561774m_norm(A,A2)) ) ).

% norm_mult_numeral1
tff(fact_2219_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_2220_norm__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,X)),zero_zero(real)))
        <=> ( X = zero_zero(A) ) ) ) ).

% norm_le_zero_iff
tff(fact_2221_zero__less__norm__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),real_V7770717601297561774m_norm(A,X)))
        <=> ( X != zero_zero(A) ) ) ) ).

% zero_less_norm_iff
tff(fact_2222_norm__numeral,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [W: num] : real_V7770717601297561774m_norm(A,aa(num,A,numeral_numeral(A),W)) = aa(num,real,numeral_numeral(real),W) ) ).

% norm_numeral
tff(fact_2223_norm__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( real_V7770717601297561774m_norm(A,one_one(A)) = one_one(real) ) ) ).

% norm_one
tff(fact_2224_norm__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),zero_zero(real))) ) ).

% norm_not_less_zero
tff(fact_2225_norm__ge__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),real_V7770717601297561774m_norm(A,X))) ) ).

% norm_ge_zero
tff(fact_2226_norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,B2: A] : real_V7770717601297561774m_norm(A,divide_divide(A,A2,B2)) = divide_divide(real,real_V7770717601297561774m_norm(A,A2),real_V7770717601297561774m_norm(A,B2)) ) ).

% norm_divide
tff(fact_2227_norm__power,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A,N: nat] : real_V7770717601297561774m_norm(A,aa(nat,A,power_power(A,X),N)) = aa(nat,real,power_power(real,real_V7770717601297561774m_norm(A,X)),N) ) ).

% norm_power
tff(fact_2228_norm__uminus__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A] : real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),X)),Y)) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) ) ).

% norm_uminus_minus
tff(fact_2229_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_2230_power__eq__imp__eq__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W: A,N: nat,Z: A] :
          ( ( aa(nat,A,power_power(A,W),N) = aa(nat,A,power_power(A,Z),N) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => ( real_V7770717601297561774m_norm(A,W) = real_V7770717601297561774m_norm(A,Z) ) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_2231_norm__mult__less,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [X: A,R3: real,Y: A,S: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),R3))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Y)),S))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),R3),S))) ) ) ) ).

% norm_mult_less
tff(fact_2232_norm__mult__ineq,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [X: A,Y: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y)))) ) ).

% norm_mult_ineq
tff(fact_2233_norm__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E2: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y))),E2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),E2)) ) ) ).

% norm_triangle_lt
tff(fact_2234_norm__add__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,R3: real,Y: A,S: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),R3))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Y)),S))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R3),S))) ) ) ) ).

% norm_add_less
tff(fact_2235_norm__add__leD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),C2))
         => pp(aa(real,bool,aa(real,fun(real,bool),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_2236_norm__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E2: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y))),E2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),E2)) ) ) ).

% norm_triangle_le
tff(fact_2237_norm__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y)))) ) ).

% norm_triangle_ineq
tff(fact_2238_norm__triangle__mono,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,R3: real,B2: A,S: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,A2)),R3))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,B2)),S))
           => pp(aa(real,bool,aa(real,fun(real,bool),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),R3),S))) ) ) ) ).

% norm_triangle_mono
tff(fact_2239_norm__diff__triangle__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E1: real,Z: A,E22: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Y))),E1))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Y),Z))),E22))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22))) ) ) ) ).

% norm_diff_triangle_less
tff(fact_2240_norm__power__ineq,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A,N: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,power_power(A,X),N))),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(A,X)),N))) ) ).

% norm_power_ineq
tff(fact_2241_norm__triangle__sub,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Y)),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Y))))) ) ).

% norm_triangle_sub
tff(fact_2242_norm__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(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_2243_norm__diff__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E1: real,Z: A,E22: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Y))),E1))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Y),Z))),E22))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22))) ) ) ) ).

% norm_diff_triangle_le
tff(fact_2244_norm__triangle__le__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A,Y: A,E2: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,Y))),E2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),Y))),E2)) ) ) ).

% norm_triangle_le_diff
tff(fact_2245_norm__diff__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(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_2246_norm__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A2),B2)))) ) ).

% norm_triangle_ineq2
tff(fact_2247_power__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W: A,N: nat] :
          ( ( aa(nat,A,power_power(A,W),N) = one_one(A) )
         => ( ( real_V7770717601297561774m_norm(A,W) = one_one(real) )
            | ( N = zero_zero(nat) ) ) ) ) ).

% power_eq_1_iff
tff(fact_2248_norm__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: A,D2: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D2)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A2),C2))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,B2),D2))))) ) ).

% norm_diff_triangle_ineq
tff(fact_2249_square__norm__one,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A] :
          ( ( aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(A) )
         => ( real_V7770717601297561774m_norm(A,X) = one_one(real) ) ) ) ).

% square_norm_one
tff(fact_2250_norm__power__diff,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A,W: A,M: nat] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),one_one(real)))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,W)),one_one(real)))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,Z),M)),aa(nat,A,power_power(A,W),M)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Z),W))))) ) ) ) ).

% norm_power_diff
tff(fact_2251_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arcosh(A),one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_2252_pochhammer__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Z),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))),comm_s3205402744901411588hammer(A,Z,N))),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),aa(num,num,bit0,one2)))),N)) ) ).

% pochhammer_double
tff(fact_2253_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,minus_minus(real,aa(real,real,uminus_uminus(real),X)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(real,real,ln_ln(real),aa(real,real,minus_minus(real,one_one(real)),X)))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_2254_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),divide_divide(real,aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))),N),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),N)))),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),aa(num,num,bit0,one2))),N)),N)))) ) ).

% central_binomial_lower_bound
tff(fact_2255_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique8689654367752047608divmod(int,bitM(M),aa(num,num,bit0,one2)) = aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),M)),one_one(int))),one_one(int)) ).

% divmod_BitM_2_eq
tff(fact_2256_pochhammer__1,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : comm_s3205402744901411588hammer(A,A2,one_one(nat)) = A2 ) ).

% pochhammer_1
tff(fact_2257_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,ln_ln(A),one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_2258_ln__inj__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => ( ( aa(real,real,ln_ln(real),X) = aa(real,real,ln_ln(real),Y) )
        <=> ( X = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_2259_ln__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ) ) ).

% ln_less_cancel_iff
tff(fact_2260_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_2261_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_2262_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_2263_ln__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y)) ) ) ) ).

% ln_le_cancel_iff
tff(fact_2264_ln__less__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,ln_ln(real),X)),zero_zero(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real))) ) ) ).

% ln_less_zero_iff
tff(fact_2265_ln__gt__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),X)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X)) ) ) ).

% ln_gt_zero_iff
tff(fact_2266_ln__eq__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( ( aa(real,real,ln_ln(real),X) = zero_zero(real) )
      <=> ( X = one_one(real) ) ) ) ).

% ln_eq_zero_iff
tff(fact_2267_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : pred_numeral(aa(num,num,bit0,K)) = aa(num,nat,numeral_numeral(nat),bitM(K)) ).

% pred_numeral_simps(2)
tff(fact_2268_ln__ge__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),X)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X)) ) ) ).

% ln_ge_zero_iff
tff(fact_2269_ln__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),X)),zero_zero(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real))) ) ) ).

% ln_le_zero_iff
tff(fact_2270_semiring__norm_I26_J,axiom,
    bitM(one2) = one2 ).

% semiring_norm(26)
tff(fact_2271_ln__less__self,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,ln_ln(real),X)),X)) ) ).

% ln_less_self
tff(fact_2272_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),comm_s3205402744901411588hammer(A,X,N))) ) ) ).

% pochhammer_pos
tff(fact_2273_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat,N: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,M) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( comm_s3205402744901411588hammer(A,A2,N) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_2274_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,N: nat,M: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,N) = zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
           => ( comm_s3205402744901411588hammer(A,A2,M) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_2275_semiring__norm_I27_J,axiom,
    ! [N: num] : bitM(aa(num,num,bit0,N)) = aa(num,num,bit1,bitM(N)) ).

% semiring_norm(27)
tff(fact_2276_semiring__norm_I28_J,axiom,
    ! [N: num] : bitM(aa(num,num,bit1,N)) = aa(num,num,bit1,aa(num,num,bit0,N)) ).

% semiring_norm(28)
tff(fact_2277_inc__BitM__eq,axiom,
    ! [N: num] : inc(bitM(N)) = aa(num,num,bit0,N) ).

% inc_BitM_eq
tff(fact_2278_BitM__inc__eq,axiom,
    ! [N: num] : bitM(inc(N)) = aa(num,num,bit1,N) ).

% BitM_inc_eq
tff(fact_2279_ln__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),X)),X)) ) ).

% ln_bound
tff(fact_2280_ln__gt__zero__imp__gt__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),X)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X)) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_2281_ln__less__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,ln_ln(real),X)),zero_zero(real))) ) ) ).

% ln_less_zero
tff(fact_2282_ln__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,ln_ln(real),X))) ) ).

% ln_gt_zero
tff(fact_2283_ln__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),X))) ) ).

% ln_ge_zero
tff(fact_2284_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [X: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),comm_s3205402744901411588hammer(A,X,N))) ) ) ).

% pochhammer_nonneg
tff(fact_2285_pochhammer__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,zero_zero(A),N) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,zero_zero(A),N) = zero_zero(A) ) ) ) ) ).

% pochhammer_0_left
tff(fact_2286_eval__nat__numeral_I2_J,axiom,
    ! [N: num] : aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,N)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bitM(N))) ).

% eval_nat_numeral(2)
tff(fact_2287_ln__ge__zero__imp__ge__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),X)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X)) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_2288_one__plus__BitM,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bitM(N)) = aa(num,num,bit0,N) ).

% one_plus_BitM
tff(fact_2289_BitM__plus__one,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bitM(N)),one2) = aa(num,num,bit0,N) ).

% BitM_plus_one
tff(fact_2290_ln__add__one__self__le__self,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X))),X)) ) ).

% ln_add_one_self_le_self
tff(fact_2291_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_mult
tff(fact_2292_ln__eq__minus__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( ( aa(real,real,ln_ln(real),X) = aa(real,real,minus_minus(real,X),one_one(real)) )
       => ( X = one_one(real) ) ) ) ).

% ln_eq_minus_one
tff(fact_2293_ln__div,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => ( aa(real,real,ln_ln(real),divide_divide(real,X,Y)) = aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_div
tff(fact_2294_pochhammer__rec,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,N)) = 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)),N)) ) ).

% pochhammer_rec
tff(fact_2295_pochhammer__rec_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,N: nat] : comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,N)) = 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),N))),comm_s3205402744901411588hammer(A,Z,N)) ) ).

% pochhammer_rec'
tff(fact_2296_pochhammer__Suc,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,A2,N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(nat,A,semiring_1_of_nat(A),N))) ) ).

% pochhammer_Suc
tff(fact_2297_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,K: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_2298_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [N: nat,K: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) = zero_zero(A) )
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K)) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_2299_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,N: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,N) = zero_zero(A) )
        <=> ? [K3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K3),N))
              & ( A2 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K3)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_2300_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_2301_pochhammer__product_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,N: nat,M: nat] : comm_s3205402744901411588hammer(A,Z,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,N)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),N)),M)) ) ).

% pochhammer_product'
tff(fact_2302_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),N))) ).

% binomial_maximum'
tff(fact_2303_binomial__mono,axiom,
    ! [K: nat,K7: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),K7))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K7)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),K7))) ) ) ).

% binomial_mono
tff(fact_2304_ln__2__less__1,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,ln_ln(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),one_one(real))) ).

% ln_2_less_1
tff(fact_2305_numeral__BitM,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),bitM(N)) = aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,N))),one_one(A)) ) ).

% numeral_BitM
tff(fact_2306_binomial__antimono,axiom,
    ! [K: nat,K7: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),K7))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),K))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K7),N))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(N),K7)),aa(nat,nat,binomial(N),K))) ) ) ) ).

% binomial_antimono
tff(fact_2307_binomial__maximum,axiom,
    ! [N: nat,K: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% binomial_maximum
tff(fact_2308_odd__numeral__BitM,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [W: num] : ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(num,A,numeral_numeral(A),bitM(W)))) ) ).

% odd_numeral_BitM
tff(fact_2309_ln__le__minus__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),X)),aa(real,real,minus_minus(real,X),one_one(real)))) ) ).

% ln_le_minus_one
tff(fact_2310_ln__diff__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),X)),aa(real,real,ln_ln(real),Y))),divide_divide(real,aa(real,real,minus_minus(real,X),Y),Y))) ) ) ).

% ln_diff_le
tff(fact_2311_ln__add__one__self__le__self2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X))),X)) ) ).

% ln_add_one_self_le_self2
tff(fact_2312_ln__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,ln_ln(real),aa(nat,real,power_power(real,X),N)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,ln_ln(real),X)) ) ) ).

% ln_realpow
tff(fact_2313_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,N: nat,Z: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( comm_s3205402744901411588hammer(A,Z,N) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,M)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,nat,minus_minus(nat,N),M))) ) ) ) ).

% pochhammer_product
tff(fact_2314_binomial__strict__antimono,axiom,
    ! [K: nat,K7: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),K7))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K7),N))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,binomial(N),K7)),aa(nat,nat,binomial(N),K))) ) ) ) ).

% binomial_strict_antimono
tff(fact_2315_binomial__strict__mono,axiom,
    ! [K: nat,K7: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),K7))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K7)),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),K7))) ) ) ).

% binomial_strict_mono
tff(fact_2316_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K)))) ) ).

% binomial_less_binomial_Suc
tff(fact_2317_central__binomial__odd,axiom,
    ! [N: nat] :
      ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
     => ( aa(nat,nat,binomial(N),aa(nat,nat,suc,divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) = aa(nat,nat,binomial(N),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% central_binomial_odd
tff(fact_2318_ln__one__minus__pos__upper__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),aa(real,real,minus_minus(real,one_one(real)),X))),aa(real,real,uminus_uminus(real),X))) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_2319_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,R3),aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R3),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R3),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R3)),one_one(A)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_2320_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,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)) ) ).

% pochhammer_minus
tff(fact_2321_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,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,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_2322_ln__one__plus__pos__lower__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,minus_minus(real,X),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_2323_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(N),K)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N)) ) ).

% zero_less_binomial_iff
tff(fact_2324_artanh__def,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [X: A] : aa(A,A,artanh(A),X) = 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)),X),aa(A,A,minus_minus(A,one_one(A)),X))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% artanh_def
tff(fact_2325_choose__two,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,minus_minus(nat,N),one_one(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% choose_two
tff(fact_2326_binomial__n__0,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_2327_binomial__Suc__Suc,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K))) ).

% binomial_Suc_Suc
tff(fact_2328_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa(nat,nat,binomial(N),K) = zero_zero(nat) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K)) ) ).

% binomial_eq_0_iff
tff(fact_2329_binomial__Suc__n,axiom,
    ! [N: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,N)),N) = aa(nat,nat,suc,N) ).

% binomial_Suc_n
tff(fact_2330_binomial__n__n,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),N) = one_one(nat) ).

% binomial_n_n
tff(fact_2331_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_2332_binomial__1,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),aa(nat,nat,suc,zero_zero(nat))) = N ).

% binomial_1
tff(fact_2333_choose__one,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),one_one(nat)) = N ).

% choose_one
tff(fact_2334_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
     => ( aa(nat,nat,binomial(N),K) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_2335_Suc__times__binomial,axiom,
    ! [K: nat,N: 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,N)),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,binomial(N),K)) ).

% Suc_times_binomial
tff(fact_2336_Suc__times__binomial__eq,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,binomial(N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K))),aa(nat,nat,suc,K)) ).

% Suc_times_binomial_eq
tff(fact_2337_binomial__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
     => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,binomial(N),aa(nat,nat,minus_minus(nat,N),K)) ) ) ).

% binomial_symmetric
tff(fact_2338_choose__mult__lemma,axiom,
    ! [M: nat,R3: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R3)),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R3)),K)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R3)),M)) ).

% choose_mult_lemma
tff(fact_2339_binomial__le__pow,axiom,
    ! [R3: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R3),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(N),R3)),aa(nat,nat,power_power(nat,N),R3))) ) ).

% binomial_le_pow
tff(fact_2340_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,binomial(N),K))) ) ).

% zero_less_binomial
tff(fact_2341_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_2342_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N),M)),aa(nat,nat,binomial(M),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),K)),aa(nat,nat,minus_minus(nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_2343_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,binomial(N),K)),aa(nat,nat,suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_2344_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,N),K)),aa(nat,nat,binomial(N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),K)) ).

% binomial_absorb_comp
tff(fact_2345_binomial__absorption,axiom,
    ! [K: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),K)) ).

% binomial_absorption
tff(fact_2346_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,divide_divide(A,aa(nat,A,semiring_1_of_nat(A),N),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K)))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_2347_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% binomial_le_pow2
tff(fact_2348_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_2349_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,binomial(N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_2350_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,binomial(N),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),aa(nat,nat,suc,K))),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),K)) ) ) ).

% binomial_addition_formula
tff(fact_2351_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X))),X))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_2352_tanh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,tanh(real),aa(real,real,ln_ln(real),X)) = divide_divide(real,aa(real,real,minus_minus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real))) ) ) ).

% tanh_ln_real
tff(fact_2353_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X))),X))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_2354_signed__take__bit__eq__take__bit__minus,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),K)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(nat,nat,suc,N))),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_2355_fact__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))),comm_s3205402744901411588hammer(A,divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N))),semiring_char_0_fact(A,N)) ) ).

% fact_double
tff(fact_2356_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X))),X))),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_2357_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_2358_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_2359_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_2360_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_2361_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_2362_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_2363_abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : aa(A,A,abs_abs(A),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),N) ) ).

% abs_numeral
tff(fact_2364_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_2365_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_2366_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_2367_abs__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A,B2: 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)) ) ).

% abs_divide
tff(fact_2368_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_2369_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_2370_abs__dvd__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: A,K: A] :
          ( pp(dvd_dvd(A,aa(A,A,abs_abs(A),M),K))
        <=> pp(dvd_dvd(A,M,K)) ) ) ).

% abs_dvd_iff
tff(fact_2371_dvd__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: A,K: A] :
          ( pp(dvd_dvd(A,M,aa(A,A,abs_abs(A),K)))
        <=> pp(dvd_dvd(A,M,K)) ) ) ).

% dvd_abs_iff
tff(fact_2372_abs__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : aa(A,A,abs_abs(A),aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ).

% abs_of_nat
tff(fact_2373_abs__bool__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: bool] : aa(A,A,abs_abs(A),aa(bool,A,zero_neq_one_of_bool(A),P)) = aa(bool,A,zero_neq_one_of_bool(A),P) ) ).

% abs_bool_eq
tff(fact_2374_tanh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tanh(real),X)),aa(real,real,tanh(real),Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ).

% tanh_real_less_iff
tff(fact_2375_tanh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,tanh(real),X)),aa(real,real,tanh(real),Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y)) ) ).

% tanh_real_le_iff
tff(fact_2376_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),zero_zero(A)))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_2377_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),A2))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% abs_le_self_iff
tff(fact_2378_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ).

% abs_of_nonneg
tff(fact_2379_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,abs_abs(A),A2)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_2380_abs__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),N) ) ).

% abs_neg_numeral
tff(fact_2381_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_2382_abs__power__minus,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] : aa(A,A,abs_abs(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),A2)),N)) = aa(A,A,abs_abs(A),aa(nat,A,power_power(A,A2),N)) ) ).

% abs_power_minus
tff(fact_2383_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_2384_fact__1,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).

% fact_1
tff(fact_2385_tanh__real__neg__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tanh(real),X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real))) ) ).

% tanh_real_neg_iff
tff(fact_2386_tanh__real__pos__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,tanh(real),X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) ) ).

% tanh_real_pos_iff
tff(fact_2387_tanh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,tanh(real),X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) ) ).

% tanh_real_nonneg_iff
tff(fact_2388_tanh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,tanh(real),X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) ) ).

% tanh_real_nonpos_iff
tff(fact_2389_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,A2,aa(A,A,abs_abs(A),B2))),zero_zero(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A)))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_2390_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),divide_divide(A,A2,aa(A,A,abs_abs(A),B2))))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_2391_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2392_bit__numeral__Bit0__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,M))),aa(nat,nat,suc,N)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),M)),N)) ) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_2393_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_2394_bit__numeral__Bit1__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))),aa(nat,nat,suc,N)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),M)),N)) ) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_2395_fact__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N))),semiring_char_0_fact(A,N)) ) ).

% fact_Suc
tff(fact_2396_artanh__minus__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( aa(real,real,artanh(real),aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,artanh(real),X)) ) ) ).

% artanh_minus_real
tff(fact_2397_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)))
    <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_2398_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),zero_zero(int)))
    <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) ) ).

% signed_take_bit_negative_iff
tff(fact_2399_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A2)),N)))
        <=> ( ( A2 != zero_zero(A) )
            | ( N = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_2400_power2__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% power2_abs
tff(fact_2401_abs__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% abs_power2
tff(fact_2402_fact__2,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)) ) ) ).

% fact_2
tff(fact_2403_bit__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,N: num] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,W))),aa(num,nat,numeral_numeral(nat),N)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W)),pred_numeral(N))) ) ) ).

% bit_numeral_simps(2)
tff(fact_2404_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W)))),aa(nat,nat,suc,N)))
    <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),N)) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_2405_bit__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,N: num] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,W))),aa(num,nat,numeral_numeral(nat),N)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W)),pred_numeral(N))) ) ) ).

% bit_numeral_simps(3)
tff(fact_2406_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( pp(aa(nat,bool,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,N)))
    <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W)),N)) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_2407_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),zero_zero(nat)))
        <=> ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)) ) ) ).

% bit_0
tff(fact_2408_power__even__abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: num,A2: A] :
          ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(num,nat,numeral_numeral(nat),W)))
         => ( aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A2)),aa(num,nat,numeral_numeral(nat),W)) = aa(nat,A,power_power(A,A2),aa(num,nat,numeral_numeral(nat),W)) ) ) ) ).

% power_even_abs_numeral
tff(fact_2409_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N: num] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,W)))),aa(num,nat,numeral_numeral(nat),N)))
    <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),pred_numeral(N))) ) ).

% bit_minus_numeral_int(1)
tff(fact_2410_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N: num] :
      ( pp(aa(nat,bool,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),N)))
    <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(num,int,numeral_numeral(int),W)),pred_numeral(N))) ) ).

% bit_minus_numeral_int(2)
tff(fact_2411_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),N))
        <=> ( ( N = zero_zero(nat) )
            & ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)) ) ) ) ).

% bit_mod_2_iff
tff(fact_2412_bit__numeral__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),M)),N))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(num,nat,numeral_numeral(nat),M)),N)) ) ) ).

% bit_numeral_iff
tff(fact_2413_bit__of__nat__iff__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(nat,A,semiring_1_of_nat(A),M)),N))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,M),N)) ) ) ).

% bit_of_nat_iff_bit
tff(fact_2414_bit__disjunctive__add__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A,N: nat] :
          ( ! [N2: nat] :
              ( ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N2))
              | ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N2)) )
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),N))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
              | pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N)) ) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_2415_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,abs_abs(A),A2))) ) ).

% abs_ge_self
tff(fact_2416_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% abs_le_D1
tff(fact_2417_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_2418_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_2419_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_2420_abs__minus__commute,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A2),B2)) = aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,B2),A2)) ) ).

% abs_minus_commute
tff(fact_2421_abs__eq__iff,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,abs_abs(A),X) = aa(A,A,abs_abs(A),Y) )
        <=> ( ( X = Y )
            | ( X = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% abs_eq_iff
tff(fact_2422_fact__ge__self,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),semiring_char_0_fact(nat,N))) ).

% fact_ge_self
tff(fact_2423_fact__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N))) ) ).

% fact_mono_nat
tff(fact_2424_power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] : aa(A,A,abs_abs(A),aa(nat,A,power_power(A,A2),N)) = aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A2)),N) ) ).

% power_abs
tff(fact_2425_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) )
         => pp(dvd_dvd(A,L,K)) ) ) ).

% dvd_if_abs_eq
tff(fact_2426_bit__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)),N))
        <=> ~ ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
            <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N)) ) ) ) ).

% bit_xor_iff
tff(fact_2427_bit__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A2)),N))
        <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
            & ( M != N ) ) ) ) ).

% bit_unset_bit_iff
tff(fact_2428_bit__xor__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),N))
    <=> ~ ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,L),N)) ) ) ).

% bit_xor_int_iff
tff(fact_2429_not__bit__1__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(nat,nat,suc,N))) ) ).

% not_bit_1_Suc
tff(fact_2430_bit__1__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,one_one(A)),N))
        <=> ( N = zero_zero(nat) ) ) ) ).

% bit_1_iff
tff(fact_2431_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,abs_abs(A),A2))) ) ).

% abs_ge_zero
tff(fact_2432_bit__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: num] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,one_one(A)),aa(num,nat,numeral_numeral(nat),N))) ) ).

% bit_numeral_simps(1)
tff(fact_2433_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ).

% abs_of_pos
tff(fact_2434_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A2)),zero_zero(A))) ) ).

% abs_not_less_zero
tff(fact_2435_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),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_2436_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A2)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),B2)),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2))) ) ) ) ).

% abs_mult_less
tff(fact_2437_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(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,minus_minus(A,B2),A2)))) ) ).

% abs_triangle_ineq2_sym
tff(fact_2438_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(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,minus_minus(A,A2),B2)))) ) ).

% abs_triangle_ineq3
tff(fact_2439_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(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,minus_minus(A,A2),B2)))) ) ).

% abs_triangle_ineq2
tff(fact_2440_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,abs_abs(A),A2))) ) ).

% abs_ge_minus_self
tff(fact_2441_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2)) ) ) ) ).

% abs_le_iff
tff(fact_2442_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2)) ) ) ).

% abs_le_D2
tff(fact_2443_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)) ) ) ) ).

% abs_leI
tff(fact_2444_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_2445_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),A2)),B2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),A2)),B2)) ) ) ) ).

% abs_less_iff
tff(fact_2446_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N))) ) ) ).

% fact_less_mono_nat
tff(fact_2447_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se2584673776208193580ke_bit(A,M),A2)),N))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ) ).

% bit_take_bit_iff
tff(fact_2448_bit__of__bool__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [B2: bool,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(bool,A,zero_neq_one_of_bool(A),B2)),N))
        <=> ( pp(B2)
            & ( N = zero_zero(nat) ) ) ) ) ).

% bit_of_bool_iff
tff(fact_2449_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),semiring_char_0_fact(A,N))) ) ).

% fact_ge_zero
tff(fact_2450_fact__not__neg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),semiring_char_0_fact(A,N)),zero_zero(A))) ) ).

% fact_not_neg
tff(fact_2451_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),semiring_char_0_fact(A,N))) ) ).

% fact_gt_zero
tff(fact_2452_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,N))) ) ).

% fact_ge_1
tff(fact_2453_fact__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,N))) ) ) ).

% fact_mono
tff(fact_2454_signed__take__bit__eq__if__positive,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,N: nat] :
          ( ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
         => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) ) ) ) ).

% signed_take_bit_eq_if_positive
tff(fact_2455_fact__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,M: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
         => pp(dvd_dvd(A,semiring_char_0_fact(A,N),semiring_char_0_fact(A,M))) ) ) ).

% fact_dvd
tff(fact_2456_pochhammer__fact,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_semiring_1(A) )
     => ! [N: nat] : semiring_char_0_fact(A,N) = comm_s3205402744901411588hammer(A,one_one(A),N) ) ).

% pochhammer_fact
tff(fact_2457_tanh__real__lt__1,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tanh(real),X)),one_one(real))) ).

% tanh_real_lt_1
tff(fact_2458_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [X: A] :
          ( ! [E: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),E))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),E)) )
         => ( X = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_2459_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A2: A,B2: A] :
          ( ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
              | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
              | pp(aa(A,bool,aa(A,fun(A,bool),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_2460_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Y)),X) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),X)) ) ) ) ).

% abs_mult_pos
tff(fact_2461_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),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_2462_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,abs_abs(A),B2) )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
            & ( ( B2 = A2 )
              | ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_2463_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = B2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
            & ( ( A2 = B2 )
              | ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_2464_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A2)),N))) ) ).

% zero_le_power_abs
tff(fact_2465_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Y))
         => ( divide_divide(A,aa(A,A,abs_abs(A),X),Y) = aa(A,A,abs_abs(A),divide_divide(A,X,Y)) ) ) ) ).

% abs_div_pos
tff(fact_2466_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),A2) = aa(A,A,uminus_uminus(A),A2) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ) ).

% abs_if
tff(fact_2467_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_2468_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),X2) = aa(A,A,uminus_uminus(A),X2) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),zero_zero(A)))
           => ( aa(A,A,abs_abs(A),X2) = X2 ) ) ) ) ).

% abs_if_raw
tff(fact_2469_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A2: A,R3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X),A2))),R3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,minus_minus(A,A2),R3)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R3))) ) ) ) ).

% abs_diff_le_iff
tff(fact_2470_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(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_2471_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A,C2: A,D2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A2),C2))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,B2),D2))))) ) ).

% abs_diff_triangle_ineq
tff(fact_2472_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,A2: A,R3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X),A2))),R3))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,minus_minus(A,A2),R3)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R3))) ) ) ) ).

% abs_diff_less_iff
tff(fact_2473_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),semiring_char_0_fact(nat,N))) ).

% fact_ge_Suc_0_nat
tff(fact_2474_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => pp(dvd_dvd(nat,M,semiring_char_0_fact(nat,N))) ) ) ).

% dvd_fact
tff(fact_2475_bit__not__int__iff_H,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),K)),one_one(int))),N))
    <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) ) ).

% bit_not_int_iff'
tff(fact_2476_abs__real__def,axiom,
    ! [A2: real] :
      ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),zero_zero(real)))
       => ( aa(real,real,abs_abs(real),A2) = aa(real,real,uminus_uminus(real),A2) ) )
      & ( ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),zero_zero(real)))
       => ( aa(real,real,abs_abs(real),A2) = A2 ) ) ) ).

% abs_real_def
tff(fact_2477_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,N))) ) ) ) ).

% fact_less_mono
tff(fact_2478_lemma__interval__lt,axiom,
    ! [A2: real,X: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),B2))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [Y3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y3))),D3))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Y3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y3),B2)) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_2479_fact__fact__dvd__fact,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,N: nat] : pp(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,N)),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)))) ) ).

% fact_fact_dvd_fact
tff(fact_2480_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( modulo_modulo(A,semiring_char_0_fact(A,N),semiring_char_0_fact(A,M)) = zero_zero(A) ) ) ) ).

% fact_mod
tff(fact_2481_fact__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),semiring_char_0_fact(A,N)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,power_power(nat,N),N)))) ) ).

% fact_le_power
tff(fact_2482_sin__bound__lemma,axiom,
    ! [X: real,Y: real,U: real,V: real] :
      ( ( X = Y )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),U)),V))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),U)),Y))),V)) ) ) ).

% sin_bound_lemma
tff(fact_2483_flip__bit__eq__if,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : bit_se8732182000553998342ip_bit(A,N,A2) = aa(A,A,aa(nat,fun(A,A),if(fun(nat,fun(A,A)),aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N),bit_se2638667681897837118et_bit(A),bit_se5668285175392031749et_bit(A)),N),A2) ) ).

% flip_bit_eq_if
tff(fact_2484_tanh__real__gt__neg1,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(real,real,tanh(real),X))) ).

% tanh_real_gt_neg1
tff(fact_2485_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,abs_abs(A),X)))) ) ).

% abs_add_one_gt_zero
tff(fact_2486_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,suc,M)))
     => ( semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),N)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,M),N))) ) ) ).

% fact_diff_Suc
tff(fact_2487_fact__div__fact__le__pow,axiom,
    ! [R3: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),R3),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),divide_divide(nat,semiring_char_0_fact(nat,N),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,N),R3)))),aa(nat,nat,power_power(nat,N),R3))) ) ).

% fact_div_fact_le_pow
tff(fact_2488_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,N),K)))),aa(nat,nat,binomial(N),K)) = semiring_char_0_fact(nat,N) ) ) ).

% binomial_fact_lemma
tff(fact_2489_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K))) ) ) ).

% bit_imp_take_bit_positive
tff(fact_2490_lemma__interval,axiom,
    ! [A2: real,X: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),B2))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [Y3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y3))),D3))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),Y3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y3),B2)) ) ) ) ) ) ).

% lemma_interval
tff(fact_2491_norm__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2)))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A2),B2)))) ) ).

% norm_triangle_ineq3
tff(fact_2492_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_concat_bit(M,K),L)),N))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
          & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) )
        | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
          & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,L),aa(nat,nat,minus_minus(nat,N),M))) ) ) ) ).

% bit_concat_bit_iff
tff(fact_2493_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => pp(dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,N),K))),semiring_char_0_fact(A,N))) ) ) ).

% choose_dvd
tff(fact_2494_fact__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: num] : semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K)),semiring_char_0_fact(A,pred_numeral(K))) ) ).

% fact_numeral
tff(fact_2495_signed__take__bit__eq__concat__bit,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = aa(int,int,bit_concat_bit(N,K),aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_2496_exp__eq__0__imp__not__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat,A2: A] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) = zero_zero(A) )
         => ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_2497_bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,suc,N)))
        <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),N)) ) ) ).

% bit_Suc
tff(fact_2498_stable__imp__bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( ( divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A2 )
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
          <=> ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)) ) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_2499_bit__iff__idd__imp__stable,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( ! [N2: nat] :
              ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N2))
            <=> ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)) )
         => ( divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A2 ) ) ) ).

% bit_iff_idd_imp_stable
tff(fact_2500_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),aa(A,A,abs_abs(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% abs_le_square_iff
tff(fact_2501_abs__square__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( ( aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = one_one(A) )
        <=> ( aa(A,A,abs_abs(A),X) = one_one(A) ) ) ) ).

% abs_square_eq_1
tff(fact_2502_power__even__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: A] :
          ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
         => ( aa(nat,A,power_power(A,aa(A,A,abs_abs(A),A2)),N) = aa(nat,A,power_power(A,A2),N) ) ) ) ).

% power_even_abs
tff(fact_2503_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),M2))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),M2))
              <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N2)) ) )
         => ~ ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
             => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,minus_minus(nat,N2),one_one(nat))))
              <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N2)) ) ) ) ).

% int_bit_bound
tff(fact_2504_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
     => ( aa(nat,nat,binomial(N),K) = divide_divide(nat,semiring_char_0_fact(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,N),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_2505_bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
        <=> ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),divide_divide(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)))) ) ) ).

% bit_iff_odd
tff(fact_2506_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,power_power(A,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),Y)) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_2507_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X)),one_one(A))) ) ) ).

% abs_square_le_1
tff(fact_2508_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A))) ) ) ).

% abs_square_less_1
tff(fact_2509_square__fact__le__2__fact,axiom,
    ! [N: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),semiring_char_0_fact(real,N)),semiring_char_0_fact(real,N))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) ).

% square_fact_le_2_fact
tff(fact_2510_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: A,B2: A] :
          ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,A2),N)),aa(nat,A,power_power(A,B2),N))) ) ) ) ).

% power_mono_even
tff(fact_2511_bit__int__def,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N))
    <=> ~ pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),divide_divide(int,K,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))) ) ).

% bit_int_def
tff(fact_2512_fact__num__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat] :
          ( ( ( M = zero_zero(nat) )
           => ( semiring_char_0_fact(A,M) = one_one(A) ) )
          & ( ( M != zero_zero(nat) )
           => ( semiring_char_0_fact(A,M) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,M),one_one(nat)))) ) ) ) ) ).

% fact_num_eq_if
tff(fact_2513_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( semiring_char_0_fact(A,N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,N),one_one(nat)))) ) ) ) ).

% fact_reduce
tff(fact_2514_pochhammer__same,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_ring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),N)),semiring_char_0_fact(A,N)) ) ).

% pochhammer_same
tff(fact_2515_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K)) = divide_divide(A,semiring_char_0_fact(A,N),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,N),K)))) ) ) ) ).

% binomial_fact
tff(fact_2516_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( 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(N),K))) = divide_divide(A,semiring_char_0_fact(A,N),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,N),K))) ) ) ) ).

% fact_binomial
tff(fact_2517_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A2)),N))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
              | ( N = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_2518_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
         => ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
          <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,minus_minus(A,A2),one_one(A))),N))
              | ( N = zero_zero(nat) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_2519_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,N: nat] :
          ( ! [J2: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),aa(nat,nat,suc,J2)))
         => ( pp(aa(nat,bool,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),aa(num,num,bit0,one2))),B2))),N))
          <=> ( ( ( N = zero_zero(nat) )
               => ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)) )
              & ( ( N != zero_zero(nat) )
               => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),B2)),N)) ) ) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_2520_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
        <=> ( ( ( N = zero_zero(nat) )
             => ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)) )
            & ( ( N != zero_zero(nat) )
             => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,nat,minus_minus(nat,N),one_one(nat)))) ) ) ) ) ).

% bit_rec
tff(fact_2521_set__bit__eq,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),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(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ).

% set_bit_eq
tff(fact_2522_unset__bit__eq,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K) = aa(int,int,minus_minus(int,K),aa(int,int,aa(int,fun(int,int),times_times(int),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N))),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ).

% unset_bit_eq
tff(fact_2523_take__bit__Suc__from__most,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),aa(bool,int,zero_neq_one_of_bool(int),aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)))),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ).

% take_bit_Suc_from_most
tff(fact_2524_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: fun(A,fun(A,bool)),X: A] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X3))
             => pp(aa(A,bool,aa(A,fun(A,bool),P,X3),aa(nat,A,power_power(A,X3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),P,aa(A,A,abs_abs(A),X)),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% abs_sqrt_wlog
tff(fact_2525_sin__coeff__def,axiom,
    ! [X2: nat] :
      ( ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),X2))
       => ( sin_coeff(X2) = zero_zero(real) ) )
      & ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),X2))
       => ( sin_coeff(X2) = divide_divide(real,aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),divide_divide(nat,aa(nat,nat,minus_minus(nat,X2),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),semiring_char_0_fact(real,X2)) ) ) ) ).

% sin_coeff_def
tff(fact_2526_binomial__code,axiom,
    ! [N: nat,K: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
       => ( aa(nat,nat,binomial(N),K) = zero_zero(nat) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K))
       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)))
           => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,binomial(N),aa(nat,nat,minus_minus(nat,N),K)) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)))
           => ( aa(nat,nat,binomial(N),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,minus_minus(nat,N),K)),one_one(nat)),N,one_one(nat)),semiring_char_0_fact(nat,K)) ) ) ) ) ) ).

% binomial_code
tff(fact_2527_cos__coeff__def,axiom,
    ! [X2: nat] :
      ( ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),X2))
       => ( cos_coeff(X2) = divide_divide(real,aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),divide_divide(nat,X2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),semiring_char_0_fact(real,X2)) ) )
      & ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),X2))
       => ( cos_coeff(X2) = zero_zero(real) ) ) ) ).

% cos_coeff_def
tff(fact_2528_arctan__double,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,arctan,X)) = aa(real,real,arctan,divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),X),aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% arctan_double
tff(fact_2529_fact__code,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N,one_one(nat))) ) ).

% fact_code
tff(fact_2530_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),Z)),one_one(int)))
    <=> ( Z = zero_zero(int) ) ) ).

% zabs_less_one_iff
tff(fact_2531_zero__less__arctan__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,arctan,X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) ) ).

% zero_less_arctan_iff
tff(fact_2532_arctan__less__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arctan,X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real))) ) ).

% arctan_less_zero_iff
tff(fact_2533_zero__le__arctan__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,arctan,X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) ) ).

% zero_le_arctan_iff
tff(fact_2534_arctan__le__zero__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arctan,X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) ) ).

% arctan_le_zero_iff
tff(fact_2535_cos__coeff__0,axiom,
    cos_coeff(zero_zero(nat)) = one_one(real) ).

% cos_coeff_0
tff(fact_2536_arctan__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arctan,X)),aa(real,real,arctan,Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ).

% arctan_less_iff
tff(fact_2537_arctan__monotone,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arctan,X)),aa(real,real,arctan,Y))) ) ).

% arctan_monotone
tff(fact_2538_arctan__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arctan,X)),aa(real,real,arctan,Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y)) ) ).

% arctan_le_iff
tff(fact_2539_arctan__monotone_H,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arctan,X)),aa(real,real,arctan,Y))) ) ).

% arctan_monotone'
tff(fact_2540_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),N))
    <=> ( N = zero_zero(nat) ) ) ).

% bit_Suc_0_iff
tff(fact_2541_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,suc,N))) ).

% not_bit_Suc_0_Suc
tff(fact_2542_abs__div,axiom,
    ! [Y: int,X: int] :
      ( pp(dvd_dvd(int,Y,X))
     => ( aa(int,int,abs_abs(int),divide_divide(int,X,Y)) = divide_divide(int,aa(int,int,abs_abs(int),X),aa(int,int,abs_abs(int),Y)) ) ) ).

% abs_div
tff(fact_2543_sin__coeff__Suc,axiom,
    ! [N: nat] : sin_coeff(aa(nat,nat,suc,N)) = divide_divide(real,cos_coeff(N),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))) ).

% sin_coeff_Suc
tff(fact_2544_not__bit__Suc__0__numeral,axiom,
    ! [N: num] : ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),N))) ).

% not_bit_Suc_0_numeral
tff(fact_2545_zabs__def,axiom,
    ! [I2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),zero_zero(int)))
       => ( aa(int,int,abs_abs(int),I2) = aa(int,int,uminus_uminus(int),I2) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),zero_zero(int)))
       => ( aa(int,int,abs_abs(int),I2) = I2 ) ) ) ).

% zabs_def
tff(fact_2546_dvd__imp__le__int,axiom,
    ! [I2: int,D2: int] :
      ( ( I2 != zero_zero(int) )
     => ( pp(dvd_dvd(int,D2,I2))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),D2)),aa(int,int,abs_abs(int),I2))) ) ) ).

% dvd_imp_le_int
tff(fact_2547_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => pp(aa(int,bool,aa(int,fun(int,bool),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_2548_cos__coeff__Suc,axiom,
    ! [N: nat] : cos_coeff(aa(nat,nat,suc,N)) = divide_divide(real,aa(real,real,uminus_uminus(real),sin_coeff(N)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))) ).

% cos_coeff_Suc
tff(fact_2549_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,X: fun(nat,fun(A,A)),Xa: nat,Xb: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,X,Xa,Xb,Xc) = Y )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb),Xa))
         => ( Y = Xc ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb),Xa))
         => ( Y = set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),X,Xa),Xc)) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_2550_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,B2: nat,A2: nat,F3: fun(nat,fun(A,A)),Acc2: A] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A2))
       => ( set_fo6178422350223883121st_nat(A,F3,A2,B2,Acc2) = Acc2 ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A2))
       => ( set_fo6178422350223883121st_nat(A,F3,A2,B2,Acc2) = set_fo6178422350223883121st_nat(A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F3,A2),Acc2)) ) ) ) ).

% fold_atLeastAtMost_nat.simps
tff(fact_2551_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(int,int,abs_abs(int),L))))
    <=> pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L))) ) ).

% even_add_abs_iff
tff(fact_2552_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),K)),L)))
    <=> pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L))) ) ).

% even_abs_add_iff
tff(fact_2553_bit__nat__def,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,M),N))
    <=> ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),divide_divide(nat,M,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) ) ).

% bit_nat_def
tff(fact_2554_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,F3: fun(nat,int),K: int] :
      ( ! [I4: nat] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),I4))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N)) )
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F3,aa(nat,nat,suc,I4))),aa(nat,int,F3,I4)))),one_one(int))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F3,M)),K))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,F3,N)))
           => ? [I4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),I4))
                & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),N))
                & ( aa(nat,int,F3,I4) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_2555_incr__lemma,axiom,
    ! [D2: int,Z: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),X),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,X),Z))),one_one(int))),D2)))) ) ).

% incr_lemma
tff(fact_2556_decr__lemma,axiom,
    ! [D2: int,X: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,minus_minus(int,X),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,X),Z))),one_one(int))),D2))),Z)) ) ).

% decr_lemma
tff(fact_2557_nat__ivt__aux,axiom,
    ! [N: nat,F3: fun(nat,int),K: int] :
      ( ! [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F3,aa(nat,nat,suc,I4))),aa(nat,int,F3,I4)))),one_one(int))) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F3,zero_zero(nat))),K))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,F3,N)))
         => ? [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),N))
              & ( aa(nat,int,F3,I4) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_2558_complex__mod__minus__le__complex__mod,axiom,
    ! [X: complex] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),real_V7770717601297561774m_norm(complex,X))),real_V7770717601297561774m_norm(complex,X))) ).

% complex_mod_minus_le_complex_mod
tff(fact_2559_complex__mod__triangle__ineq2,axiom,
    ! [B2: complex,A2: complex] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),B2),A2))),real_V7770717601297561774m_norm(complex,B2))),real_V7770717601297561774m_norm(complex,A2))) ).

% complex_mod_triangle_ineq2
tff(fact_2560_nat0__intermed__int__val,axiom,
    ! [N: nat,F3: fun(nat,int),K: int] :
      ( ! [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I4),one_one(nat)))),aa(nat,int,F3,I4)))),one_one(int))) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,F3,zero_zero(nat))),K))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(nat,int,F3,N)))
         => ? [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),N))
              & ( aa(nat,int,F3,I4) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_2561_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),Y)),one_one(real)))
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,arctan,X)),aa(real,real,arctan,Y)) = aa(real,real,arctan,divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y),aa(real,real,minus_minus(real,one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)))) ) ) ) ).

% arctan_add
tff(fact_2562_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
       => ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,bit_ri4277139882892585799ns_not(int),L) ) )
      & ( ( K != aa(int,int,uminus_uminus(int),one_one(int)) )
       => ( ( ( L = aa(int,int,uminus_uminus(int),one_one(int)) )
           => ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,bit_ri4277139882892585799ns_not(int),K) ) )
          & ( ( L != aa(int,int,uminus_uminus(int),one_one(int)) )
           => ( ( ( K = zero_zero(int) )
               => ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = L ) )
              & ( ( K != zero_zero(int) )
               => ( ( ( L = zero_zero(int) )
                   => ( aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = K ) )
                  & ( ( L != zero_zero(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(int,int,abs_abs(int),aa(int,int,minus_minus(int,modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
tff(fact_2563_mask__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: num] : bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),N)) = 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),aa(num,num,bit0,one2))),bit_se2239418461657761734s_mask(A,pred_numeral(N)))) ) ).

% mask_numeral
tff(fact_2564_modulo__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( sgn_sgn(int,L) = zero_zero(int) )
          | ( sgn_sgn(int,K) = zero_zero(int) )
          | ( N = zero_zero(nat) ) )
       => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M)) ) )
      & ( ~ ( ( sgn_sgn(int,L) = zero_zero(int) )
            | ( sgn_sgn(int,K) = zero_zero(int) )
            | ( N = zero_zero(nat) ) )
       => ( ( ( sgn_sgn(int,K) = sgn_sgn(int,L) )
           => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M,N))) ) )
          & ( ( sgn_sgn(int,K) != sgn_sgn(int,L) )
           => ( modulo_modulo(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,dvd_dvd(nat,N,M)))))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,M,N)))) ) ) ) ) ) ).

% modulo_int_unfold
tff(fact_2565_divide__int__unfold,axiom,
    ! [L: int,K: int,N: nat,M: nat] :
      ( ( ( ( sgn_sgn(int,L) = zero_zero(int) )
          | ( sgn_sgn(int,K) = zero_zero(int) )
          | ( N = zero_zero(nat) ) )
       => ( divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),N))) = zero_zero(int) ) )
      & ( ~ ( ( sgn_sgn(int,L) = zero_zero(int) )
            | ( sgn_sgn(int,K) = zero_zero(int) )
            | ( N = zero_zero(nat) ) )
       => ( ( ( sgn_sgn(int,K) = sgn_sgn(int,L) )
           => ( divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),N))) = aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,M,N)) ) )
          & ( ( sgn_sgn(int,K) != sgn_sgn(int,L) )
           => ( divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),N))) = 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,M,N)),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,dvd_dvd(nat,N,M)))))) ) ) ) ) ) ).

% divide_int_unfold
tff(fact_2566_tanh__real__altdef,axiom,
    ! [X: real] : aa(real,real,tanh(real),X) = divide_divide(real,aa(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),aa(num,num,bit0,one2)))),X))),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),aa(num,num,bit0,one2)))),X)))) ).

% tanh_real_altdef
tff(fact_2567_and__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( ( K = zero_zero(int) )
          | ( L = zero_zero(int) ) )
       => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = zero_zero(int) ) )
      & ( ~ ( ( K = zero_zero(int) )
            | ( L = zero_zero(int) ) )
       => ( ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
           => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = L ) )
          & ( ( K != aa(int,int,uminus_uminus(int),one_one(int)) )
           => ( ( ( L = aa(int,int,uminus_uminus(int),one_one(int)) )
               => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = K ) )
              & ( ( L != aa(int,int,uminus_uminus(int),one_one(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(int,int,aa(int,fun(int,int),times_times(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ) ) ) ) ).

% and_int_unfold
tff(fact_2568_and_Oidem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),A2) = A2 ) ).

% and.idem
tff(fact_2569_and_Oleft__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) ) ).

% and.left_idem
tff(fact_2570_and_Oright__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) ) ).

% and.right_idem
tff(fact_2571_sgn__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : sgn_sgn(A,sgn_sgn(A,A2)) = sgn_sgn(A,A2) ) ).

% sgn_sgn
tff(fact_2572_bit_Odouble__compl,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = X ) ).

% bit.double_compl
tff(fact_2573_bit_Ocompl__eq__compl__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,bit_ri4277139882892585799ns_not(A),X) = aa(A,A,bit_ri4277139882892585799ns_not(A),Y) )
        <=> ( X = Y ) ) ) ).

% bit.compl_eq_compl_iff
tff(fact_2574_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),bit_se2239418461657761734s_mask(nat,N)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% mask_nat_positive_iff
tff(fact_2575_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_2576_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_2577_bit_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),X) = zero_zero(A) ) ).

% bit.conj_zero_left
tff(fact_2578_bit_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),zero_zero(A)) = zero_zero(A) ) ).

% bit.conj_zero_right
tff(fact_2579_sgn__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_0
tff(fact_2580_sgn__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( sgn_sgn(A,one_one(A)) = one_one(A) ) ) ).

% sgn_1
tff(fact_2581_sgn__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( sgn_sgn(A,one_one(A)) = one_one(A) ) ) ).

% sgn_one
tff(fact_2582_sgn__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A,B2: A] : sgn_sgn(A,divide_divide(A,A2,B2)) = divide_divide(A,sgn_sgn(A,A2),sgn_sgn(A,B2)) ) ).

% sgn_divide
tff(fact_2583_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : sgn_sgn(A,aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),sgn_sgn(A,A2)) ) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_2584_power__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,N: nat] : sgn_sgn(A,aa(nat,A,power_power(A,A2),N)) = aa(nat,A,power_power(A,sgn_sgn(A,A2)),N) ) ).

% power_sgn
tff(fact_2585_exp__less__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),exp(real,X)),exp(real,Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ).

% exp_less_cancel_iff
tff(fact_2586_exp__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),exp(real,X)),exp(real,Y))) ) ).

% exp_less_mono
tff(fact_2587_take__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2)) ) ).

% take_bit_and
tff(fact_2588_exp__le__cancel__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),exp(real,X)),exp(real,Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y)) ) ).

% exp_le_cancel_iff
tff(fact_2589_bit_Oxor__compl__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),Y) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y)) ) ).

% bit.xor_compl_left
tff(fact_2590_bit_Oxor__compl__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y)) ) ).

% bit.xor_compl_right
tff(fact_2591_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),sgn_sgn(A,A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% sgn_less
tff(fact_2592_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),sgn_sgn(A,A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% sgn_greater
tff(fact_2593_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_2594_bit_Oconj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = X ) ).

% bit.conj_one_right
tff(fact_2595_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_2596_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_2597_divide__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] : divide_divide(A,A2,sgn_sgn(A,B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),sgn_sgn(A,B2)) ) ).

% divide_sgn
tff(fact_2598_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_2599_mask__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] :
          ( ( bit_se2239418461657761734s_mask(A,N) = zero_zero(A) )
        <=> ( N = zero_zero(nat) ) ) ) ).

% mask_eq_0_iff
tff(fact_2600_bit_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = zero_zero(A) ) ).

% bit.conj_cancel_left
tff(fact_2601_bit_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = zero_zero(A) ) ).

% bit.conj_cancel_right
tff(fact_2602_exp__eq__one__iff,axiom,
    ! [X: real] :
      ( ( exp(real,X) = one_one(real) )
    <=> ( X = zero_zero(real) ) ) ).

% exp_eq_one_iff
tff(fact_2603_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) ) ) ).

% and_nonnegative_int_iff
tff(fact_2604_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),zero_zero(int)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ).

% and_negative_int_iff
tff(fact_2605_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( sgn_sgn(A,A2) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_2606_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_2607_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_2608_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_2609_and__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = one_one(A) ) ).

% and_numerals(8)
tff(fact_2610_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),sgn_sgn(A,A2)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_2611_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A2)),sgn_sgn(A,A2)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A2),zero_zero(A)))) ) ).

% sgn_mult_self_eq
tff(fact_2612_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_2613_bit_Oxor__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,uminus_uminus(A),one_one(A))),X) = aa(A,A,bit_ri4277139882892585799ns_not(A),X) ) ).

% bit.xor_one_left
tff(fact_2614_bit_Oxor__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),X) ) ).

% bit.xor_one_right
tff(fact_2615_bit_Oxor__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_left
tff(fact_2616_bit_Oxor__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_right
tff(fact_2617_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : sgn_sgn(A,aa(A,A,abs_abs(A),A2)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A2),zero_zero(A)))) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_2618_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),sgn_sgn(A,A2)) = aa(bool,A,zero_neq_one_of_bool(A),aa(bool,bool,fNot,aa(A,bool,aa(A,fun(A,bool),fequal(A),A2),zero_zero(A)))) ) ).

% sgn_abs
tff(fact_2619_one__less__exp__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),exp(real,X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) ) ).

% one_less_exp_iff
tff(fact_2620_exp__less__one__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),exp(real,X)),one_one(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real))) ) ).

% exp_less_one_iff
tff(fact_2621_exp__le__one__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),exp(real,X)),one_one(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) ) ).

% exp_le_one_iff
tff(fact_2622_one__le__exp__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),exp(real,X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) ) ).

% one_le_exp_iff
tff(fact_2623_take__bit__minus__one__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,uminus_uminus(A),one_one(A))) = bit_se2239418461657761734s_mask(A,N) ) ).

% take_bit_minus_one_eq_mask
tff(fact_2624_not__negative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% not_negative_int_iff
tff(fact_2625_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% not_nonnegative_int_iff
tff(fact_2626_exp__ln,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( exp(real,aa(real,real,ln_ln(real),X)) = X ) ) ).

% exp_ln
tff(fact_2627_exp__ln__iff,axiom,
    ! [X: real] :
      ( ( exp(real,aa(real,real,ln_ln(real),X)) = X )
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) ) ).

% exp_ln_iff
tff(fact_2628_minus__not__numeral__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,uminus_uminus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),inc(N)) ) ).

% minus_not_numeral_eq
tff(fact_2629_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),aa(num,num,bit0,Y))) = zero_zero(A) ) ).

% and_numerals(1)
tff(fact_2630_and__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X))),one_one(A)) = zero_zero(A) ) ).

% and_numerals(5)
tff(fact_2631_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( sgn_sgn(A,A2) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_2632_and__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(3)
tff(fact_2633_even__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)))
        <=> ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)) ) ) ).

% even_not_iff
tff(fact_2634_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : sgn_sgn(A,aa(nat,A,semiring_1_of_nat(A),N)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ).

% sgn_of_nat
tff(fact_2635_and__minus__numerals_I2_J,axiom,
    ! [N: 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,N)))) = one_one(int) ).

% and_minus_numerals(2)
tff(fact_2636_and__minus__numerals_I6_J,axiom,
    ! [N: 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,N)))),one_one(int)) = one_one(int) ).

% and_minus_numerals(6)
tff(fact_2637_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),aa(num,num,bit0,one2))) ) ) ).

% not_one_eq
tff(fact_2638_and__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(4)
tff(fact_2639_and__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(6)
tff(fact_2640_and__minus__numerals_I1_J,axiom,
    ! [N: 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,bit0,N)))) = zero_zero(int) ).

% and_minus_numerals(1)
tff(fact_2641_and__minus__numerals_I5_J,axiom,
    ! [N: 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,bit0,N)))),one_one(int)) = zero_zero(int) ).

% and_minus_numerals(5)
tff(fact_2642_and__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),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),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% and_numerals(7)
tff(fact_2643_disjunctive__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [B2: A,A2: A] :
          ( ! [N2: nat] :
              ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N2))
             => pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N2)) )
         => ( aa(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_2644_norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,exp(A,X))),exp(real,real_V7770717601297561774m_norm(A,X)))) ) ).

% norm_exp
tff(fact_2645_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_2646_take__bit__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),bit_se2239418461657761734s_mask(A,N)) ) ).

% take_bit_eq_mask
tff(fact_2647_and_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),C2)) ) ).

% and.assoc
tff(fact_2648_and_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),A2) ) ).

% and.commute
tff(fact_2649_and_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B2),C2)) ) ).

% and.left_commute
tff(fact_2650_of__nat__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),bit_se2239418461657761734s_mask(nat,N)) = bit_se2239418461657761734s_mask(A,N) ) ).

% of_nat_mask_eq
tff(fact_2651_of__nat__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_and_eq
tff(fact_2652_exp__less__cancel,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),exp(real,X)),exp(real,Y)))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ).

% exp_less_cancel
tff(fact_2653_sgn__0__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( sgn_sgn(A,A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% sgn_0_0
tff(fact_2654_sgn__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] :
          ( ( sgn_sgn(A,A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% sgn_eq_0_iff
tff(fact_2655_sgn__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A,B2: A] : sgn_sgn(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A2)),sgn_sgn(A,B2)) ) ).

% sgn_mult
tff(fact_2656_same__sgn__sgn__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( sgn_sgn(A,B2) = sgn_sgn(A,A2) )
         => ( sgn_sgn(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = sgn_sgn(A,A2) ) ) ) ).

% same_sgn_sgn_add
tff(fact_2657_take__bit__not__eq__mask__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) = aa(A,A,minus_minus(A,bit_se2239418461657761734s_mask(A,N)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) ) ).

% take_bit_not_eq_mask_diff
tff(fact_2658_bit__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)),N))
        <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
            & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N)) ) ) ) ).

% bit_and_iff
tff(fact_2659_bit_Oconj__xor__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Z)) ) ).

% bit.conj_xor_distrib
tff(fact_2660_bit_Oconj__xor__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Y),Z)),X) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),X)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Z),X)) ) ).

% bit.conj_xor_distrib2
tff(fact_2661_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( sgn_sgn(int,K) = sgn_sgn(int,L) )
     => ( divide_divide(int,K,L) = divide_divide(int,aa(int,int,abs_abs(int),K),aa(int,int,abs_abs(int),L)) ) ) ).

% div_eq_sgn_abs
tff(fact_2662_take__bit__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A,B2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,bit_ri4277139882892585799ns_not(A),B2)) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2) ) ) ) ).

% take_bit_not_iff
tff(fact_2663_take__bit__not__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2))) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) ) ).

% take_bit_not_take_bit
tff(fact_2664_bit__not__int__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_ri4277139882892585799ns_not(int),K)),N))
    <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) ) ).

% bit_not_int_iff
tff(fact_2665_bit__and__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),N))
    <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,L),N)) ) ) ).

% bit_and_int_iff
tff(fact_2666_less__eq__mask,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),bit_se2239418461657761734s_mask(nat,N))) ).

% less_eq_mask
tff(fact_2667_and__not__numerals_I2_J,axiom,
    ! [N: 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,bit0,N)))) = one_one(int) ).

% and_not_numerals(2)
tff(fact_2668_and__not__numerals_I4_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),aa(num,num,bit0,M)) ).

% and_not_numerals(4)
tff(fact_2669_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N))) = zero_zero(A) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_2670_and__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% and_not_numerals(5)
tff(fact_2671_and__not__numerals_I7_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),aa(num,num,bit0,M)) ).

% and_not_numerals(7)
tff(fact_2672_and__not__numerals_I3_J,axiom,
    ! [N: 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,N)))) = zero_zero(int) ).

% and_not_numerals(3)
tff(fact_2673_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_2674_exp__total,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
     => ? [X3: real] : exp(real,X3) = Y ) ).

% exp_total
tff(fact_2675_exp__gt__zero,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),exp(real,X))) ).

% exp_gt_zero
tff(fact_2676_not__exp__less__zero,axiom,
    ! [X: real] : ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),exp(real,X)),zero_zero(real))) ).

% not_exp_less_zero
tff(fact_2677_exp__ge__zero,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),exp(real,X))) ).

% exp_ge_zero
tff(fact_2678_not__exp__le__zero,axiom,
    ! [X: real] : ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),exp(real,X)),zero_zero(real))) ).

% not_exp_le_zero
tff(fact_2679_sgn__not__eq__imp,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( sgn_sgn(A,B2) != sgn_sgn(A,A2) )
         => ( ( sgn_sgn(A,A2) != zero_zero(A) )
           => ( ( sgn_sgn(A,B2) != zero_zero(A) )
             => ( sgn_sgn(A,A2) = aa(A,A,uminus_uminus(A),sgn_sgn(A,B2)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_2680_sgn__minus__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(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_2681_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y),Z))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Z)) ) ) ).

% AND_upper2'
tff(fact_2682_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y),Z))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z)) ) ) ).

% AND_upper1'
tff(fact_2683_AND__upper2,axiom,
    ! [Y: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Y)) ) ).

% AND_upper2
tff(fact_2684_AND__upper1,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),X)) ) ).

% AND_upper1
tff(fact_2685_AND__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y))) ) ).

% AND_lower
tff(fact_2686_not__diff__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(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_2687_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,minus_minus(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),B2) ) ).

% not_add_distrib
tff(fact_2688_linordered__idom__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [K: A] : aa(A,A,abs_abs(A),K) = aa(A,A,aa(A,fun(A,A),times_times(A),K),sgn_sgn(A,K)) ) ).

% linordered_idom_class.abs_sgn
tff(fact_2689_abs__mult__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),sgn_sgn(A,A2)) = A2 ) ).

% abs_mult_sgn
tff(fact_2690_sgn__mult__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,A2)),aa(A,A,abs_abs(A),A2)) = A2 ) ).

% sgn_mult_abs
tff(fact_2691_mult__sgn__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sgn_sgn(A,X)),aa(A,A,abs_abs(A),X)) = X ) ).

% mult_sgn_abs
tff(fact_2692_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( sgn_sgn(A,B2) = sgn_sgn(A,A2) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% same_sgn_abs_add
tff(fact_2693_exp__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),X) )
         => ( exp(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,X)),exp(A,Y)) ) ) ) ).

% exp_add_commuting
tff(fact_2694_mult__exp__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,X)),exp(A,Y)) = exp(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) ) ).

% mult_exp_exp
tff(fact_2695_exp__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : exp(A,aa(A,A,minus_minus(A,X),Y)) = divide_divide(A,exp(A,X),exp(A,Y)) ) ).

% exp_diff
tff(fact_2696_and__not__numerals_I6_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% and_not_numerals(6)
tff(fact_2697_and__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% and_not_numerals(9)
tff(fact_2698_mask__nonnegative__int,axiom,
    ! [N: nat] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),bit_se2239418461657761734s_mask(int,N))) ).

% mask_nonnegative_int
tff(fact_2699_not__mask__negative__int,axiom,
    ! [N: nat] : ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),bit_se2239418461657761734s_mask(int,N)),zero_zero(int))) ).

% not_mask_negative_int
tff(fact_2700_minus__exp__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)) ) ).

% minus_exp_eq_not_mask
tff(fact_2701_exp__gt__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),exp(real,X))) ) ).

% exp_gt_one
tff(fact_2702_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( sgn_sgn(A,A2) = one_one(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% sgn_1_pos
tff(fact_2703_exp__ge__add__one__self,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),exp(real,X))) ).

% exp_ge_add_one_self
tff(fact_2704_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( ( A2 = zero_zero(A) )
           => ( aa(A,A,abs_abs(A),sgn_sgn(A,A2)) = zero_zero(A) ) )
          & ( ( A2 != zero_zero(A) )
           => ( aa(A,A,abs_abs(A),sgn_sgn(A,A2)) = one_one(A) ) ) ) ) ).

% abs_sgn_eq
tff(fact_2705_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int)))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),K)) ) ).

% and_less_eq
tff(fact_2706_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y),Z))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z)) ) ) ).

% AND_upper1''
tff(fact_2707_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y),Z))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),Z)) ) ) ).

% AND_upper2''
tff(fact_2708_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_2709_not__eq__complement,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),A2) = aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A2)),one_one(A)) ) ).

% not_eq_complement
tff(fact_2710_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,minus_minus(A,A2),one_one(A))) ) ).

% minus_eq_not_minus_1
tff(fact_2711_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero(int) )
     => ( divide_divide(int,aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,V)),aa(int,int,abs_abs(int),K)),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,V)),aa(int,int,abs_abs(int),L))) = divide_divide(int,aa(int,int,abs_abs(int),K),aa(int,int,abs_abs(int),L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_2712_not__int__def,axiom,
    ! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),K)),one_one(int)) ).

% not_int_def
tff(fact_2713_exp__minus__inverse,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,X)),exp(A,aa(A,A,uminus_uminus(A),X))) = one_one(A) ) ).

% exp_minus_inverse
tff(fact_2714_exp__of__nat2__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,N: nat] : exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(nat,A,semiring_1_of_nat(A),N))) = aa(nat,A,power_power(A,exp(A,X)),N) ) ).

% exp_of_nat2_mult
tff(fact_2715_exp__of__nat__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [N: nat,X: A] : exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),X)) = aa(nat,A,power_power(A,exp(A,X)),N) ) ).

% exp_of_nat_mult
tff(fact_2716_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( pp(dvd_dvd(int,L,K))
     => ( divide_divide(int,K,L) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,K)),sgn_sgn(int,L))),divide_divide(int,aa(int,int,abs_abs(int),K),aa(int,int,abs_abs(int),L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_2717_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ~ pp(dvd_dvd(int,L,K))
       => ( sgn_sgn(int,modulo_modulo(int,K,L)) = sgn_sgn(int,L) ) ) ) ).

% sgn_mod
tff(fact_2718_and__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = 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),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% and_not_numerals(8)
tff(fact_2719_minus__numeral__inc__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),N)) ) ).

% minus_numeral_inc_eq
tff(fact_2720_less__mask,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),N))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),bit_se2239418461657761734s_mask(nat,N))) ) ).

% less_mask
tff(fact_2721_even__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)))
        <=> ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
            | pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2)) ) ) ) ).

% even_and_iff
tff(fact_2722_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( sgn_sgn(A,A2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% sgn_1_neg
tff(fact_2723_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( ( ( X = zero_zero(A) )
           => ( sgn_sgn(A,X) = zero_zero(A) ) )
          & ( ( X != zero_zero(A) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
               => ( sgn_sgn(A,X) = one_one(A) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
               => ( sgn_sgn(A,X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ) ) ).

% sgn_if
tff(fact_2724_not__int__div__2,axiom,
    ! [K: int] : divide_divide(int,aa(int,int,bit_ri4277139882892585799ns_not(int),K),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) = aa(int,int,bit_ri4277139882892585799ns_not(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))) ).

% not_int_div_2
tff(fact_2725_even__not__iff__int,axiom,
    ! [K: int] :
      ( pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),aa(int,int,bit_ri4277139882892585799ns_not(int),K)))
    <=> ~ pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)) ) ).

% even_not_iff_int
tff(fact_2726_exp__ge__add__one__self__aux,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),exp(real,X))) ) ).

% exp_ge_add_one_self_aux
tff(fact_2727_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)))
    <=> ( pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K))
        | pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L)) ) ) ).

% even_and_iff_int
tff(fact_2728_lemma__exp__total,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),Y))
     => ? [X3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X3))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),aa(real,real,minus_minus(real,Y),one_one(real))))
          & ( exp(real,X3) = Y ) ) ) ).

% lemma_exp_total
tff(fact_2729_ln__ge__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),aa(real,real,ln_ln(real),X)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),exp(real,Y)),X)) ) ) ).

% ln_ge_iff
tff(fact_2730_zsgn__def,axiom,
    ! [I2: int] :
      ( ( ( I2 = zero_zero(int) )
       => ( sgn_sgn(int,I2) = zero_zero(int) ) )
      & ( ( I2 != zero_zero(int) )
       => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),I2))
           => ( sgn_sgn(int,I2) = one_one(int) ) )
          & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),I2))
           => ( sgn_sgn(int,I2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ) ).

% zsgn_def
tff(fact_2731_ln__x__over__x__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),exp(real,one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),divide_divide(real,aa(real,real,ln_ln(real),Y),Y)),divide_divide(real,aa(real,real,ln_ln(real),X),X))) ) ) ).

% ln_x_over_x_mono
tff(fact_2732_not__numeral__Bit0__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,N))) ) ).

% not_numeral_Bit0_eq
tff(fact_2733_norm__sgn,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X: A] :
          ( ( ( X = zero_zero(A) )
           => ( real_V7770717601297561774m_norm(A,sgn_sgn(A,X)) = zero_zero(real) ) )
          & ( ( X != zero_zero(A) )
           => ( real_V7770717601297561774m_norm(A,sgn_sgn(A,X)) = one_one(real) ) ) ) ) ).

% norm_sgn
tff(fact_2734_bit__minus__int__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,uminus_uminus(int),K)),N))
    <=> pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,minus_minus(int,K),one_one(int)))),N)) ) ).

% bit_minus_int_iff
tff(fact_2735_not__numeral__BitM__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),bitM(N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,N))) ) ).

% not_numeral_BitM_eq
tff(fact_2736_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),aa(num,num,bit0,one2))) ) ).

% one_and_eq
tff(fact_2737_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),aa(num,num,bit0,one2))) ) ).

% and_one_eq
tff(fact_2738_exp__le,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),exp(real,one_one(real))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) ).

% exp_le
tff(fact_2739_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = bit_se2239418461657761734s_mask(int,N) )
    <=> ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),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_2740_exp__divide__power__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [N: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(nat,A,power_power(A,exp(A,divide_divide(A,X,aa(nat,A,semiring_1_of_nat(A),N)))),N) = exp(A,X) ) ) ) ).

% exp_divide_power_eq
tff(fact_2741_eucl__rel__int__remainderI,axiom,
    ! [R3: int,L: int,K: int,Q3: int] :
      ( ( sgn_sgn(int,R3) = sgn_sgn(int,L) )
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R3)),aa(int,int,abs_abs(int),L)))
       => ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q3),L)),R3) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),product_Pair(int,int,Q3),R3)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_2742_tanh__altdef,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,tanh(A),X) = divide_divide(A,aa(A,A,minus_minus(A,exp(A,X)),exp(A,aa(A,A,uminus_uminus(A),X))),aa(A,A,aa(A,fun(A,A),plus_plus(A),exp(A,X)),exp(A,aa(A,A,uminus_uminus(A),X)))) ) ).

% tanh_altdef
tff(fact_2743_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = zero_zero(A) )
        <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_2744_exp__half__le2,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),exp(real,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ).

% exp_half_le2
tff(fact_2745_Suc__mask__eq__exp,axiom,
    ! [N: nat] : aa(nat,nat,suc,bit_se2239418461657761734s_mask(nat,N)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) ).

% Suc_mask_eq_exp
tff(fact_2746_mask__nat__less__exp,axiom,
    ! [N: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),bit_se2239418461657761734s_mask(nat,N)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) ).

% mask_nat_less_exp
tff(fact_2747_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),N))
        <=> ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) != zero_zero(A) )
            & ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ) ).

% bit_not_iff_eq
tff(fact_2748_exp__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] : exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Z)) = aa(nat,A,power_power(A,exp(A,Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ).

% exp_double
tff(fact_2749_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) ) )
        | ? [L2: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L2 )
            & ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q5),zero_zero(int)) )
            & ( L2 != zero_zero(int) )
            & ( K3 = aa(int,int,aa(int,fun(int,int),times_times(int),Q5),L2) ) )
        | ? [R5: int,L2: int,K3: int,Q5: int] :
            ( ( A1 = K3 )
            & ( A22 = L2 )
            & ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q5),R5) )
            & ( sgn_sgn(int,R5) = sgn_sgn(int,L2) )
            & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R5)),aa(int,int,abs_abs(int),L2)))
            & ( K3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q5),L2)),R5) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_2750_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) ) ) )
         => ~ ! [R2: int,Q2: int] :
                ( ( A32 = aa(int,product_prod(int,int),product_Pair(int,int,Q2),R2) )
               => ( ( sgn_sgn(int,R2) = sgn_sgn(int,A22) )
                 => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,abs_abs(int),R2)),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)),R2) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_2751_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ( sgn_sgn(int,K) != sgn_sgn(int,L) )
       => ( divide_divide(int,K,L) = aa(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(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,dvd_dvd(int,L,K)))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_2752_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),bit_se2239418461657761734s_mask(A,N)))
        <=> ( N = zero_zero(nat) ) ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_2753_mask__half__int,axiom,
    ! [N: nat] : divide_divide(int,bit_se2239418461657761734s_mask(int,N),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) = bit_se2239418461657761734s_mask(int,aa(nat,nat,minus_minus(nat,N),one_one(nat))) ).

% mask_half_int
tff(fact_2754_mask__nat__def,axiom,
    ! [N: nat] : bit_se2239418461657761734s_mask(nat,N) = aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),one_one(nat)) ).

% mask_nat_def
tff(fact_2755_mask__int__def,axiom,
    ! [N: nat] : bit_se2239418461657761734s_mask(int,N) = aa(int,int,minus_minus(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)),one_one(int)) ).

% mask_int_def
tff(fact_2756_exp__bound__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,exp(A,Z))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ).

% exp_bound_half
tff(fact_2757_mask__eq__exp__minus__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(A,N) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A)) ) ).

% mask_eq_exp_minus_1
tff(fact_2758_exp__bound,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),exp(real,X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% exp_bound
tff(fact_2759_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(bool,int,zero_neq_one_of_bool(int),dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,bit_ri4277139882892585799ns_not(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).

% not_int_rec
tff(fact_2760_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(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).

% and_int_rec
tff(fact_2761_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),exp(real,X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),X)))) ) ) ).

% real_exp_bound_lemma
tff(fact_2762_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),N))),X))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),divide_divide(real,X,aa(nat,real,semiring_1_of_nat(real),N)))),N)),exp(real,X))) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_2763_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(nat,real,semiring_1_of_nat(real),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,one_one(real)),divide_divide(real,X,aa(nat,real,semiring_1_of_nat(real),N)))),N)),exp(real,aa(real,real,uminus_uminus(real),X)))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_2764_exp__bound__lemma,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),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),aa(num,num,bit0,one2))),real_V7770717601297561774m_norm(A,Z))))) ) ) ).

% exp_bound_lemma
tff(fact_2765_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = bit_se2239418461657761734s_mask(int,N) )
    <=> pp(dvd_dvd(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int)))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_2766_exp__lower__Taylor__quadratic,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),X)),divide_divide(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),exp(real,X))) ) ).

% exp_lower_Taylor_quadratic
tff(fact_2767_log__base__10__eq1,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),X) = 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),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))))),aa(real,real,ln_ln(real),X)) ) ) ).

% log_base_10_eq1
tff(fact_2768_modulo__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero(int) )
       => ( modulo_modulo(int,K,L) = K ) )
      & ( ( L != zero_zero(int) )
       => ( ( ( sgn_sgn(int,K) = sgn_sgn(int,L) )
           => ( modulo_modulo(int,K,L) = aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,nat2(aa(int,int,abs_abs(int),K)),nat2(aa(int,int,abs_abs(int),L))))) ) )
          & ( ( sgn_sgn(int,K) != sgn_sgn(int,L) )
           => ( modulo_modulo(int,K,L) = aa(int,int,aa(int,fun(int,int),times_times(int),sgn_sgn(int,L)),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),L)),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,dvd_dvd(int,L,K))))),aa(nat,int,semiring_1_of_nat(int),modulo_modulo(nat,nat2(aa(int,int,abs_abs(int),K)),nat2(aa(int,int,abs_abs(int),L)))))) ) ) ) ) ) ).

% modulo_int_def
tff(fact_2769_divide__int__def,axiom,
    ! [L: int,K: int] :
      ( ( ( L = zero_zero(int) )
       => ( divide_divide(int,K,L) = zero_zero(int) ) )
      & ( ( L != zero_zero(int) )
       => ( ( ( sgn_sgn(int,K) = sgn_sgn(int,L) )
           => ( divide_divide(int,K,L) = aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,nat2(aa(int,int,abs_abs(int),K)),nat2(aa(int,int,abs_abs(int),L)))) ) )
          & ( ( sgn_sgn(int,K) != sgn_sgn(int,L) )
           => ( divide_divide(int,K,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,nat2(aa(int,int,abs_abs(int),K)),nat2(aa(int,int,abs_abs(int),L)))),aa(bool,nat,zero_neq_one_of_bool(nat),aa(bool,bool,fNot,dvd_dvd(int,L,K)))))) ) ) ) ) ) ).

% divide_int_def
tff(fact_2770_arctan__half,axiom,
    ! [X: real] : aa(real,real,arctan,X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,arctan,divide_divide(real,X,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))))) ).

% arctan_half
tff(fact_2771_log__base__10__eq2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit0,one2))))),exp(real,one_one(real)))),aa(real,real,ln_ln(real),X)) ) ) ).

% log_base_10_eq2
tff(fact_2772_machin,axiom,
    divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,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,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,aa(num,num,bit0,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))) ).

% machin
tff(fact_2773_zero__le__sgn__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),sgn_sgn(real,X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) ) ).

% zero_le_sgn_iff
tff(fact_2774_sgn__le__0__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),sgn_sgn(real,X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) ) ).

% sgn_le_0_iff
tff(fact_2775_real__sqrt__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ).

% real_sqrt_less_iff
tff(fact_2776_real__sqrt__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y)) ) ).

% real_sqrt_le_iff
tff(fact_2777_real__sqrt__one,axiom,
    aa(real,real,sqrt,one_one(real)) = one_one(real) ).

% real_sqrt_one
tff(fact_2778_real__sqrt__eq__1__iff,axiom,
    ! [X: real] :
      ( ( aa(real,real,sqrt,X) = one_one(real) )
    <=> ( X = one_one(real) ) ) ).

% real_sqrt_eq_1_iff
tff(fact_2779_real__sqrt__lt__0__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real))) ) ).

% real_sqrt_lt_0_iff
tff(fact_2780_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,sqrt,Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y)) ) ).

% real_sqrt_gt_0_iff
tff(fact_2781_nat__numeral,axiom,
    ! [K: num] : nat2(aa(num,int,numeral_numeral(int),K)) = aa(num,nat,numeral_numeral(nat),K) ).

% nat_numeral
tff(fact_2782_real__sqrt__le__0__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sqrt,X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) ) ).

% real_sqrt_le_0_iff
tff(fact_2783_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y)) ) ).

% real_sqrt_ge_0_iff
tff(fact_2784_real__sqrt__lt__1__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,X)),one_one(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real))) ) ).

% real_sqrt_lt_1_iff
tff(fact_2785_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),aa(real,real,sqrt,Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),Y)) ) ).

% real_sqrt_gt_1_iff
tff(fact_2786_real__sqrt__le__1__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sqrt,X)),one_one(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real))) ) ).

% real_sqrt_le_1_iff
tff(fact_2787_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(real,real,sqrt,Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),Y)) ) ).

% real_sqrt_ge_1_iff
tff(fact_2788_log__one,axiom,
    ! [A2: real] : aa(real,real,log(A2),one_one(real)) = zero_zero(real) ).

% log_one
tff(fact_2789_real__sqrt__four,axiom,
    aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))) = aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)) ).

% real_sqrt_four
tff(fact_2790_nat__1,axiom,
    nat2(one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_2791_nat__0__iff,axiom,
    ! [I2: int] :
      ( ( nat2(I2) = zero_zero(nat) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),zero_zero(int))) ) ).

% nat_0_iff
tff(fact_2792_nat__le__0,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),zero_zero(int)))
     => ( nat2(Z) = zero_zero(nat) ) ) ).

% nat_le_0
tff(fact_2793_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),nat2(W)),nat2(Z)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z)) ) ) ).

% zless_nat_conj
tff(fact_2794_log__eq__one,axiom,
    ! [A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),A2) = one_one(real) ) ) ) ).

% log_eq_one
tff(fact_2795_log__less__cancel__iff,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log(A2),X)),aa(real,real,log(A2),Y)))
          <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_2796_log__less__one__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log(A2),X)),one_one(real)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),A2)) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_2797_one__less__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),aa(real,real,log(A2),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X)) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_2798_log__less__zero__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log(A2),X)),zero_zero(real)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real))) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_2799_zero__less__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,log(A2),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X)) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_2800_int__nat__eq,axiom,
    ! [Z: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
       => ( aa(nat,int,semiring_1_of_nat(int),nat2(Z)) = Z ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
       => ( aa(nat,int,semiring_1_of_nat(int),nat2(Z)) = zero_zero(int) ) ) ) ).

% int_nat_eq
tff(fact_2801_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),nat2(Z)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)) ) ).

% zero_less_nat_eq
tff(fact_2802_zero__le__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,log(A2),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X)) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_2803_log__le__zero__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log(A2),X)),zero_zero(real)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real))) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_2804_one__le__log__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(real,real,log(A2),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X)) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_2805_log__le__one__cancel__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log(A2),X)),one_one(real)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),A2)) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_2806_log__le__cancel__iff,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log(A2),X)),aa(real,real,log(A2),Y)))
          <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y)) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_2807_diff__nat__numeral,axiom,
    ! [V: num,V4: num] : aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V)),aa(num,nat,numeral_numeral(nat),V4)) = nat2(aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),V4))) ).

% diff_nat_numeral
tff(fact_2808_and__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,X))),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% and_nat_numerals(3)
tff(fact_2809_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),aa(num,num,bit0,Y))) = zero_zero(nat) ).

% and_nat_numerals(1)
tff(fact_2810_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X: num,N: nat] :
      ( ( nat2(Y) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),N) )
    <=> ( Y = aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_2811_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X: num,N: nat,Y: int] :
      ( ( aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),N) = nat2(Y) )
    <=> ( aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_2812_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,zero_zero(nat))),nat2(Z)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),Z)) ) ).

% one_less_nat_eq
tff(fact_2813_real__sqrt__abs,axiom,
    ! [X: real] : aa(real,real,sqrt,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(real,real,abs_abs(real),X) ).

% real_sqrt_abs
tff(fact_2814_log__pow__cancel,axiom,
    ! [A2: real,B2: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),aa(nat,real,power_power(real,A2),B2)) = aa(nat,real,semiring_1_of_nat(real),B2) ) ) ) ).

% log_pow_cancel
tff(fact_2815_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_2816_and__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% and_nat_numerals(4)
tff(fact_2817_real__sqrt__pow2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( aa(nat,real,power_power(real,aa(real,real,sqrt,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = X ) ) ).

% real_sqrt_pow2
tff(fact_2818_real__sqrt__pow2__iff,axiom,
    ! [X: real] :
      ( ( aa(nat,real,power_power(real,aa(real,real,sqrt,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = X )
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) ) ).

% real_sqrt_pow2_iff
tff(fact_2819_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa: real,Ya: real] : aa(nat,real,power_power(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_2820_nat__numeral__diff__1,axiom,
    ! [V: num] : aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V)),one_one(nat)) = nat2(aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),V)),one_one(int))) ).

% nat_numeral_diff_1
tff(fact_2821_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),nat2(A2)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),N)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N))) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_2822_numeral__power__less__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),N)),nat2(A2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N)),A2)) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_2823_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A2: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),N)),nat2(A2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N)),A2)) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_2824_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,X: num,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),nat2(A2)),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),X)),N)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N))) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_2825_Suc__0__and__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),N) = modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Suc_0_and_eq
tff(fact_2826_and__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),N),aa(nat,nat,suc,zero_zero(nat))) = modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% and_Suc_0_eq
tff(fact_2827_nat__mask__eq,axiom,
    ! [N: nat] : nat2(bit_se2239418461657761734s_mask(int,N)) = bit_se2239418461657761734s_mask(nat,N) ).

% nat_mask_eq
tff(fact_2828_and__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = nat2(aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N))) ).

% and_nat_def
tff(fact_2829_real__sqrt__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y))) ) ).

% real_sqrt_less_mono
tff(fact_2830_real__sqrt__le__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y))) ) ).

% real_sqrt_le_mono
tff(fact_2831_real__sqrt__power,axiom,
    ! [X: real,K: nat] : aa(real,real,sqrt,aa(nat,real,power_power(real,X),K)) = aa(nat,real,power_power(real,aa(real,real,sqrt,X)),K) ).

% real_sqrt_power
tff(fact_2832_real__sqrt__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,sqrt,X))) ) ).

% real_sqrt_gt_zero
tff(fact_2833_real__sqrt__eq__zero__cancel,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( ( aa(real,real,sqrt,X) = zero_zero(real) )
       => ( X = zero_zero(real) ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_2834_real__sqrt__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,X))) ) ).

% real_sqrt_ge_zero
tff(fact_2835_real__sqrt__ge__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(real,real,sqrt,X))) ) ).

% real_sqrt_ge_one
tff(fact_2836_nat__numeral__as__int,axiom,
    ! [X2: num] : aa(num,nat,numeral_numeral(nat),X2) = nat2(aa(num,int,numeral_numeral(int),X2)) ).

% nat_numeral_as_int
tff(fact_2837_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Y))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),nat2(X)),nat2(Y))) ) ).

% nat_mono
tff(fact_2838_ex__nat,axiom,
    ! [P: fun(nat,bool)] :
      ( ? [X_1: nat] : pp(aa(nat,bool,P,X_1))
    <=> ? [X4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X4))
          & pp(aa(nat,bool,P,nat2(X4))) ) ) ).

% ex_nat
tff(fact_2839_all__nat,axiom,
    ! [P: fun(nat,bool)] :
      ( ! [X_1: nat] : pp(aa(nat,bool,P,X_1))
    <=> ! [X4: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X4))
         => pp(aa(nat,bool,P,nat2(X4))) ) ) ).

% all_nat
tff(fact_2840_eq__nat__nat__iff,axiom,
    ! [Z: int,Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z4))
       => ( ( nat2(Z) = nat2(Z4) )
        <=> ( Z = Z4 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_2841_nat__one__as__int,axiom,
    one_one(nat) = nat2(one_one(int)) ).

% nat_one_as_int
tff(fact_2842_pi__not__less__zero,axiom,
    ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),pi),zero_zero(real))) ).

% pi_not_less_zero
tff(fact_2843_pi__gt__zero,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),pi)) ).

% pi_gt_zero
tff(fact_2844_pi__ge__zero,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),pi)) ).

% pi_ge_zero
tff(fact_2845_unset__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se2638667681897837118et_bit(nat),M),N) = nat2(aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),M),aa(nat,int,semiring_1_of_nat(int),N))) ).

% unset_bit_nat_def
tff(fact_2846_real__div__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( divide_divide(real,X,aa(real,real,sqrt,X)) = aa(real,real,sqrt,X) ) ) ).

% real_div_sqrt
tff(fact_2847_sqrt__add__le__add__sqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,X)),aa(real,real,sqrt,Y)))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_2848_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),X),X)),aa(real,real,aa(real,fun(real,real),times_times(real),Y),Y))))) ).

% le_real_sqrt_sumsq
tff(fact_2849_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),nat2(W)),nat2(Z)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z)) ) ) ).

% nat_mono_iff
tff(fact_2850_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),nat2(Z)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),M)),Z)) ) ).

% zless_nat_eq_int_zless
tff(fact_2851_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),nat2(X)),N))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),aa(nat,int,semiring_1_of_nat(int),N))) ) ).

% nat_le_iff
tff(fact_2852_nat__0__le,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => ( aa(nat,int,semiring_1_of_nat(int),nat2(Z)) = Z ) ) ).

% nat_0_le
tff(fact_2853_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = Z )
    <=> ( ( M = nat2(Z) )
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) ) ) ).

% int_eq_iff
tff(fact_2854_nat__int__add,axiom,
    ! [A2: nat,B2: 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_2855_log__ln,axiom,
    ! [X: real] : aa(real,real,ln_ln(real),X) = aa(real,real,log(exp(real,one_one(real))),X) ).

% log_ln
tff(fact_2856_xor__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = nat2(aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N))) ).

% xor_nat_def
tff(fact_2857_sqrt2__less__2,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ).

% sqrt2_less_2
tff(fact_2858_sgn__real__def,axiom,
    ! [A2: real] :
      ( ( ( A2 = zero_zero(real) )
       => ( sgn_sgn(real,A2) = zero_zero(real) ) )
      & ( ( A2 != zero_zero(real) )
       => ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
           => ( sgn_sgn(real,A2) = one_one(real) ) )
          & ( ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
           => ( sgn_sgn(real,A2) = aa(real,real,uminus_uminus(real),one_one(real)) ) ) ) ) ) ).

% sgn_real_def
tff(fact_2859_log__base__change,axiom,
    ! [A2: real,B2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(B2),X) = divide_divide(real,aa(real,real,log(A2),X),aa(real,real,log(A2),B2)) ) ) ) ).

% log_base_change
tff(fact_2860_less__log__of__power,axiom,
    ! [B2: real,N: nat,M: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,power_power(real,B2),N)),M))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(B2),M))) ) ) ).

% less_log_of_power
tff(fact_2861_log__of__power__eq,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( ( aa(nat,real,semiring_1_of_nat(real),M) = aa(nat,real,power_power(real,B2),N) )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => ( aa(nat,real,semiring_1_of_nat(real),N) = aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M)) ) ) ) ).

% log_of_power_eq
tff(fact_2862_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),nat2(W)),nat2(Z)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z)) ) ) ).

% nat_less_eq_zless
tff(fact_2863_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),W))
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),nat2(W)),nat2(Z)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W),Z)) ) ) ).

% nat_le_eq_zle
tff(fact_2864_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( nat2(W) = M )
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W))
         => ( W = aa(nat,int,semiring_1_of_nat(int),M) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_eq_iff
tff(fact_2865_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = nat2(W) )
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W))
         => ( W = aa(nat,int,semiring_1_of_nat(int),M) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_eq_iff2
tff(fact_2866_split__nat,axiom,
    ! [P: fun(nat,bool),I2: int] :
      ( pp(aa(nat,bool,P,nat2(I2)))
    <=> ( ! [N5: nat] :
            ( ( I2 = aa(nat,int,semiring_1_of_nat(int),N5) )
           => pp(aa(nat,bool,P,N5)) )
        & ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),I2),zero_zero(int)))
         => pp(aa(nat,bool,P,zero_zero(nat))) ) ) ) ).

% split_nat
tff(fact_2867_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),nat2(K)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),N)),K)) ) ) ).

% le_nat_iff
tff(fact_2868_nat__add__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z4))
       => ( nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat2(Z)),nat2(Z4)) ) ) ) ).

% nat_add_distrib
tff(fact_2869_nat__mult__distrib,axiom,
    ! [Z: int,Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => ( nat2(aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(Z)),nat2(Z4)) ) ) ).

% nat_mult_distrib
tff(fact_2870_Suc__as__int,axiom,
    ! [X2: nat] : aa(nat,nat,suc,X2) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X2)),one_one(int))) ).

% Suc_as_int
tff(fact_2871_nat__diff__distrib_H,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => ( nat2(aa(int,int,minus_minus(int,X),Y)) = aa(nat,nat,minus_minus(nat,nat2(X)),nat2(Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_2872_nat__diff__distrib,axiom,
    ! [Z4: int,Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z4))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z4),Z))
       => ( nat2(aa(int,int,minus_minus(int,Z),Z4)) = aa(nat,nat,minus_minus(nat,nat2(Z)),nat2(Z4)) ) ) ) ).

% nat_diff_distrib
tff(fact_2873_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(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),nat2(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L))))) ).

% nat_abs_triangle_ineq
tff(fact_2874_nat__div__distrib,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( nat2(divide_divide(int,X,Y)) = divide_divide(nat,nat2(X),nat2(Y)) ) ) ).

% nat_div_distrib
tff(fact_2875_nat__div__distrib_H,axiom,
    ! [Y: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
     => ( nat2(divide_divide(int,X,Y)) = divide_divide(nat,nat2(X),nat2(Y)) ) ) ).

% nat_div_distrib'
tff(fact_2876_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => ( nat2(aa(nat,int,power_power(int,Z),N)) = aa(nat,nat,power_power(nat,nat2(Z)),N) ) ) ).

% nat_power_eq
tff(fact_2877_nat__mod__distrib,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => ( nat2(modulo_modulo(int,X,Y)) = modulo_modulo(nat,nat2(X),nat2(Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_2878_pi__less__4,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2))))) ).

% pi_less_4
tff(fact_2879_pi__ge__two,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)) ).

% pi_ge_two
tff(fact_2880_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,nat2(aa(int,int,abs_abs(int),K)),nat2(aa(int,int,abs_abs(int),L)))) ).

% div_abs_eq_div_nat
tff(fact_2881_pi__half__neq__two,axiom,
    divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) != aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)) ).

% pi_half_neq_two
tff(fact_2882_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,nat2(aa(int,int,abs_abs(int),K)),nat2(aa(int,int,abs_abs(int),L)))) ).

% mod_abs_eq_div_nat
tff(fact_2883_take__bit__nat__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),nat2(K)) = nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ) ) ).

% take_bit_nat_eq
tff(fact_2884_nat__take__bit__eq,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),nat2(K)) ) ) ).

% nat_take_bit_eq
tff(fact_2885_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero(real) )
     => ( aa(real,real,arctan,divide_divide(real,one_one(real),X)) = aa(real,real,minus_minus(real,divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,X)),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(real,real,arctan,X)) ) ) ).

% arctan_inverse
tff(fact_2886_bit__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,nat2(K)),N))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N)) ) ) ).

% bit_nat_iff
tff(fact_2887_real__less__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),Y))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,sqrt,Y))) ) ).

% real_less_rsqrt
tff(fact_2888_sqrt__le__D,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sqrt,X)),Y))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% sqrt_le_D
tff(fact_2889_real__le__rsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),Y))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(real,real,sqrt,Y))) ) ).

% real_le_rsqrt
tff(fact_2890_nat__2,axiom,
    nat2(aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% nat_2
tff(fact_2891_log__mult,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
           => ( aa(real,real,log(A2),aa(real,real,aa(real,fun(real,real),times_times(real),X),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(A2),X)),aa(real,real,log(A2),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_2892_sgn__power__injE,axiom,
    ! [A2: real,N: nat,X: real,B2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,A2)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),A2)),N)) = X )
     => ( ( X = aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,B2)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),B2)),N)) )
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( A2 = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_2893_log__divide,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
           => ( aa(real,real,log(A2),divide_divide(real,X,Y)) = aa(real,real,minus_minus(real,aa(real,real,log(A2),X)),aa(real,real,log(A2),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_2894_le__log__of__power,axiom,
    ! [B2: real,N: nat,M: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,power_power(real,B2),N)),M))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(B2),M))) ) ) ).

% le_log_of_power
tff(fact_2895_log__base__pow,axiom,
    ! [A2: real,N: nat,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( aa(real,real,log(aa(nat,real,power_power(real,A2),N)),X) = divide_divide(real,aa(real,real,log(A2),X),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ).

% log_base_pow
tff(fact_2896_log__nat__power,axiom,
    ! [X: real,B2: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,log(B2),aa(nat,real,power_power(real,X),N)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(B2),X)) ) ) ).

% log_nat_power
tff(fact_2897_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
     => ( aa(nat,nat,suc,nat2(Z)) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_2898_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),W))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),nat2(W)),M))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),aa(nat,int,semiring_1_of_nat(int),M))) ) ) ).

% nat_less_iff
tff(fact_2899_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z4: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),zero_zero(int)))
     => ( nat2(aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z4)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(aa(int,int,uminus_uminus(int),Z))),nat2(aa(int,int,uminus_uminus(int),Z4))) ) ) ).

% nat_mult_distrib_neg
tff(fact_2900_nat__abs__int__diff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),B2))
       => ( nat2(aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))) = aa(nat,nat,minus_minus(nat,B2),A2) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),B2))
       => ( nat2(aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))) = aa(nat,nat,minus_minus(nat,A2),B2) ) ) ) ).

% nat_abs_int_diff
tff(fact_2901_pi__half__neq__zero,axiom,
    divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) != zero_zero(real) ).

% pi_half_neq_zero
tff(fact_2902_pi__half__less__two,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ).

% pi_half_less_two
tff(fact_2903_pi__half__le__two,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ).

% pi_half_le_two
tff(fact_2904_real__le__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sqrt,X)),Y)) ) ) ) ).

% real_le_lsqrt
tff(fact_2905_real__sqrt__unique,axiom,
    ! [Y: real,X: real] :
      ( ( aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = X )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => ( aa(real,real,sqrt,X) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_2906_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),U))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),divide_divide(real,U,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),U)) ) ).

% lemma_real_divide_sqrt_less
tff(fact_2907_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) = X )
     => ( Y = zero_zero(real) ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_2908_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) = Y )
     => ( X = zero_zero(real) ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_2909_real__sqrt__sum__squares__ge1,axiom,
    ! [X: real,Y: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_2910_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_2911_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),C2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),B2),D2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,B2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,C2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,D2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_2912_sqrt__ge__absD,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),aa(real,real,sqrt,Y)))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),Y)) ) ).

% sqrt_ge_absD
tff(fact_2913_log__of__power__less,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,power_power(real,B2),N)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ) ).

% log_of_power_less
tff(fact_2914_log2__of__power__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) )
     => ( aa(nat,real,semiring_1_of_nat(real),N) = aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),M)) ) ) ).

% log2_of_power_eq
tff(fact_2915_log__eq__div__ln__mult__log,axiom,
    ! [A2: real,B2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
         => ( ( B2 != one_one(real) )
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
             => ( aa(real,real,log(A2),X) = 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,log(B2),X)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_2916_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( pp(dvd_dvd(nat,nat2(Z),M))
    <=> ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
         => pp(dvd_dvd(int,Z,aa(nat,int,semiring_1_of_nat(int),M))) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
         => ( M = zero_zero(nat) ) ) ) ) ).

% nat_dvd_iff
tff(fact_2917_pi__half__gt__zero,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ).

% pi_half_gt_zero
tff(fact_2918_pi__half__ge__zero,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ).

% pi_half_ge_zero
tff(fact_2919_m2pi__less__pi,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),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),aa(num,num,bit0,one2))),pi))),pi)) ).

% m2pi_less_pi
tff(fact_2920_arctan__ubound,axiom,
    ! [Y: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arctan,Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ).

% arctan_ubound
tff(fact_2921_arctan__one,axiom,
    aa(real,real,arctan,one_one(real)) = divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))) ).

% arctan_one
tff(fact_2922_real__less__lsqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,X)),Y)) ) ) ) ).

% real_less_lsqrt
tff(fact_2923_sqrt__sum__squares__le__sum,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y))) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_2924_log__of__power__le,axiom,
    ! [M: nat,B2: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,power_power(real,B2),N)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log(B2),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ) ).

% log_of_power_le
tff(fact_2925_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X: real,Y: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,abs_abs(real),X)),aa(real,real,abs_abs(real),Y)))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_2926_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ).

% real_sqrt_ge_abs2
tff(fact_2927_real__sqrt__ge__abs1,axiom,
    ! [X: real,Y: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ).

% real_sqrt_ge_abs1
tff(fact_2928_ln__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,ln_ln(real),aa(real,real,sqrt,X)) = divide_divide(real,aa(real,real,ln_ln(real),X),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ).

% ln_sqrt
tff(fact_2929_sqrt__even__pow2,axiom,
    ! [N: nat] :
      ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
     => ( aa(real,real,sqrt,aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),N)) = aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% sqrt_even_pow2
tff(fact_2930_minus__pi__half__less__zero,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),zero_zero(real))) ).

% minus_pi_half_less_zero
tff(fact_2931_arctan__bounded,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arctan,Y)))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arctan,Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ).

% arctan_bounded
tff(fact_2932_arctan__lbound,axiom,
    ! [Y: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arctan,Y))) ).

% arctan_lbound
tff(fact_2933_and__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ( M = zero_zero(nat) )
          | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = zero_zero(nat) ) )
      & ( ~ ( ( M = zero_zero(nat) )
            | ( N = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = 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,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ).

% and_nat_unfold
tff(fact_2934_arsinh__real__aux,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real)))))) ).

% arsinh_real_aux
tff(fact_2935_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa: real,Ya: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,Xa),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Ya),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_2936_real__sqrt__power__even,axiom,
    ! [N: nat,X: real] :
      ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => ( aa(nat,real,power_power(real,aa(real,real,sqrt,X)),N) = aa(nat,real,power_power(real,X),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_2937_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y))),divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_2938_less__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),M))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),M)))) ) ).

% less_log2_of_power
tff(fact_2939_le__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),M))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),M)))) ) ).

% le_log2_of_power
tff(fact_2940_and__nat__rec,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),fconj(aa(bool,bool,fNot,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),M)),aa(bool,bool,fNot,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% and_nat_rec
tff(fact_2941_even__nat__iff,axiom,
    ! [K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
     => ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),nat2(K)))
      <=> pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)) ) ) ).

% even_nat_iff
tff(fact_2942_log2__of__power__less,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ).

% log2_of_power_less
tff(fact_2943_cos__x__y__le__one,axiom,
    ! [X: real,Y: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),divide_divide(real,X,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))),one_one(real))) ).

% cos_x_y_le_one
tff(fact_2944_real__sqrt__sum__squares__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),divide_divide(real,U,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),Y)),divide_divide(real,U,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),U)) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_2945_arcosh__real__def,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
     => ( aa(real,real,arcosh(real),X) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,sqrt,aa(real,real,minus_minus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real))))) ) ) ).

% arcosh_real_def
tff(fact_2946_sqrt__sum__squares__half__less,axiom,
    ! [X: real,U: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,U,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),divide_divide(real,U,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),U)) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_2947_log2__of__power__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ).

% log2_of_power_le
tff(fact_2948_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,aa(num,num,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),aa(num,num,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,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))) = divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))) ).

% machin_Euler
tff(fact_2949_sin__cos__npi,axiom,
    ! [N: 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),aa(num,num,bit0,one2))),N)))),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),N) ).

% sin_cos_npi
tff(fact_2950_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => ( ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) )
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,power_power(nat,B2),N)),K))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(nat,nat,power_power(nat,B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_2951_arsinh__real__def,axiom,
    ! [X: real] : aa(real,real,arsinh(real),X) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real))))) ).

% arsinh_real_def
tff(fact_2952_cos__pi__eq__zero,axiom,
    ! [M: nat] : 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),aa(num,num,bit0,one2))),M)))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = zero_zero(real) ).

% cos_pi_eq_zero
tff(fact_2953_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,power_power(nat,B2),N)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(nat,nat,power_power(nat,B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
         => ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_2954_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),N))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,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,minus_minus(nat,N),one_one(nat)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(nat)))))),one_one(int)) ) ) ).

% ceiling_log2_div2
tff(fact_2955_cos__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,zero_zero(A)) = one_one(A) ) ) ).

% cos_zero
tff(fact_2956_ceiling__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archimedean_ceiling(A,aa(num,A,numeral_numeral(A),V)) = aa(num,int,numeral_numeral(int),V) ) ).

% ceiling_numeral
tff(fact_2957_ceiling__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).

% ceiling_one
tff(fact_2958_cos__pi,axiom,
    cos(real,pi) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% cos_pi
tff(fact_2959_sin__cos__squared__add3,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,X))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,X))) = one_one(A) ) ).

% sin_cos_squared_add3
tff(fact_2960_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A))) ) ) ).

% ceiling_le_zero
tff(fact_2961_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X)) ) ) ).

% zero_less_ceiling
tff(fact_2962_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(num,A,numeral_numeral(A),V))) ) ) ).

% ceiling_le_numeral
tff(fact_2963_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A))) ) ) ).

% ceiling_less_one
tff(fact_2964_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X)) ) ) ).

% one_le_ceiling
tff(fact_2965_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),V)),X)) ) ) ).

% numeral_less_ceiling
tff(fact_2966_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A))) ) ) ).

% ceiling_le_one
tff(fact_2967_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ).

% one_less_ceiling
tff(fact_2968_ceiling__add__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(num,A,numeral_numeral(A),V))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_add_numeral
tff(fact_2969_ceiling__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archimedean_ceiling(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_neg_numeral
tff(fact_2970_ceiling__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),one_one(int)) ) ).

% ceiling_add_one
tff(fact_2971_ceiling__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] : archimedean_ceiling(A,aa(A,A,minus_minus(A,X),aa(num,A,numeral_numeral(A),V))) = aa(int,int,minus_minus(int,archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_diff_numeral
tff(fact_2972_ceiling__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_ceiling(A,aa(A,A,minus_minus(A,X),one_one(A))) = aa(int,int,minus_minus(int,archimedean_ceiling(A,X)),one_one(int)) ) ).

% ceiling_diff_one
tff(fact_2973_ceiling__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: num,N: nat] : archimedean_ceiling(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),N)) = aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N) ) ).

% ceiling_numeral_power
tff(fact_2974_nat__ceiling__le__eq,axiom,
    ! [X: real,A2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),nat2(archimedean_ceiling(real,X))),A2))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(nat,real,semiring_1_of_nat(real),A2))) ) ).

% nat_ceiling_le_eq
tff(fact_2975_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),one_one(A)))) ) ) ).

% ceiling_less_zero
tff(fact_2976_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),one_one(A))),X)) ) ) ).

% zero_le_ceiling
tff(fact_2977_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_2978_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),aa(num,int,numeral_numeral(int),V)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),V)),one_one(A)))) ) ) ).

% ceiling_less_numeral
tff(fact_2979_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),V)),one_one(A))),X)) ) ) ).

% numeral_le_ceiling
tff(fact_2980_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_2981_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),X)) ) ) ).

% neg_numeral_less_ceiling
tff(fact_2982_cos__pi__half,axiom,
    cos(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = zero_zero(real) ).

% cos_pi_half
tff(fact_2983_sin__two__pi,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)) = zero_zero(real) ).

% sin_two_pi
tff(fact_2984_sin__pi__half,axiom,
    sin(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = one_one(real) ).

% sin_pi_half
tff(fact_2985_cos__two__pi,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)) = one_one(real) ).

% cos_two_pi
tff(fact_2986_cos__periodic,axiom,
    ! [X: real] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))) = cos(real,X) ).

% cos_periodic
tff(fact_2987_sin__periodic,axiom,
    ! [X: real] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))) = sin(real,X) ).

% sin_periodic
tff(fact_2988_cos__2pi__minus,axiom,
    ! [X: real] : cos(real,aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)),X)) = cos(real,X) ).

% cos_2pi_minus
tff(fact_2989_cos__npi,axiom,
    ! [N: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),pi)) = aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),N) ).

% cos_npi
tff(fact_2990_cos__npi2,axiom,
    ! [N: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),N))) = aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),N) ).

% cos_npi2
tff(fact_2991_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_2992_sin__cos__squared__add2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,cos(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,power_power(A,sin(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(A) ) ).

% sin_cos_squared_add2
tff(fact_2993_sin__cos__squared__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,sin(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,power_power(A,cos(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(A) ) ).

% sin_cos_squared_add
tff(fact_2994_sin__2npi,axiom,
    ! [N: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),N))),pi)) = zero_zero(real) ).

% sin_2npi
tff(fact_2995_cos__2npi,axiom,
    ! [N: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),N))),pi)) = one_one(real) ).

% cos_2npi
tff(fact_2996_sin__2pi__minus,axiom,
    ! [X: real] : sin(real,aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)),X)) = aa(real,real,uminus_uminus(real),sin(real,X)) ).

% sin_2pi_minus
tff(fact_2997_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A)))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_2998_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),X)) ) ) ).

% neg_numeral_le_ceiling
tff(fact_2999_cos__3over2__pi,axiom,
    cos(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),aa(num,num,bit0,one2)))),pi)) = zero_zero(real) ).

% cos_3over2_pi
tff(fact_3000_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),aa(num,num,bit0,one2)))),pi)) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% sin_3over2_pi
tff(fact_3001_cos__one__sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) = one_one(A) )
         => ( sin(A,X) = zero_zero(A) ) ) ) ).

% cos_one_sin_zero
tff(fact_3002_sin__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),sin(A,Y))) ) ).

% sin_add
tff(fact_3003_cos__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : cos(A,aa(A,A,minus_minus(A,X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,Y))) ) ).

% cos_diff
tff(fact_3004_cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,Y))) ) ).

% cos_add
tff(fact_3005_sin__zero__norm__cos__one,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sin(A,X) = zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,cos(A,X)) = one_one(real) ) ) ) ).

% sin_zero_norm_cos_one
tff(fact_3006_sin__zero__abs__cos__one,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
     => ( aa(real,real,abs_abs(real),cos(real,X)) = one_one(real) ) ) ).

% sin_zero_abs_cos_one
tff(fact_3007_sin__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),sin(A,X))),cos(A,X)) ) ).

% sin_double
tff(fact_3008_sincos__principal__value,axiom,
    ! [X: real] :
    ? [Y4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Y4))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y4),pi))
      & ( sin(real,Y4) = sin(real,X) )
      & ( cos(real,Y4) = cos(real,X) ) ) ).

% sincos_principal_value
tff(fact_3009_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,Y)),archimedean_ceiling(A,X))) ) ) ).

% ceiling_mono
tff(fact_3010_sin__x__le__x,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),sin(real,X)),X)) ) ).

% sin_x_le_x
tff(fact_3011_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),archimedean_ceiling(A,Y)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% ceiling_less_cancel
tff(fact_3012_sin__le__one,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),sin(real,X)),one_one(real))) ).

% sin_le_one
tff(fact_3013_cos__le__one,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),cos(real,X)),one_one(real))) ).

% cos_le_one
tff(fact_3014_abs__sin__x__le__abs__x,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),sin(real,X))),aa(real,real,abs_abs(real),X))) ).

% abs_sin_x_le_abs_x
tff(fact_3015_sin__cos__le1,axiom,
    ! [X: real,Y: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),sin(real,X)),sin(real,Y))),aa(real,real,aa(real,fun(real,real),times_times(real),cos(real,X)),cos(real,Y))))),one_one(real))) ).

% sin_cos_le1
tff(fact_3016_sin__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,power_power(A,sin(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,cos(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% sin_squared_eq
tff(fact_3017_cos__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,power_power(A,cos(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,sin(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% cos_squared_eq
tff(fact_3018_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),R3),aa(nat,A,semiring_1_of_nat(A),nat2(archimedean_ceiling(A,R3))))) ) ).

% of_nat_ceiling
tff(fact_3019_sin__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),pi))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),sin(real,X))) ) ) ).

% sin_gt_zero
tff(fact_3020_sin__x__ge__neg__x,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),X)),sin(real,X))) ) ).

% sin_x_ge_neg_x
tff(fact_3021_sin__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),pi))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),sin(real,X))) ) ) ).

% sin_ge_zero
tff(fact_3022_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),archimedean_ceiling(A,Y)))) ) ).

% ceiling_add_le
tff(fact_3023_real__nat__ceiling__ge,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),aa(nat,real,semiring_1_of_nat(real),nat2(archimedean_ceiling(real,X))))) ).

% real_nat_ceiling_ge
tff(fact_3024_sin__ge__minus__one,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),sin(real,X))) ).

% sin_ge_minus_one
tff(fact_3025_cos__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),pi))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),pi))
           => ( ( cos(real,X) = cos(real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_3026_cos__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),pi))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),pi))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),cos(real,X)),cos(real,Y)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),X)) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_3027_cos__monotone__0__pi__le,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),pi))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),cos(real,X)),cos(real,Y))) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_3028_cos__ge__minus__one,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),cos(real,X))) ).

% cos_ge_minus_one
tff(fact_3029_abs__sin__le__one,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),sin(real,X))),one_one(real))) ).

% abs_sin_le_one
tff(fact_3030_abs__cos__le__one,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),cos(real,X))),one_one(real))) ).

% abs_cos_le_one
tff(fact_3031_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,minus_minus(A,cos(A,aa(A,A,minus_minus(A,W),Z))),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% sin_times_sin
tff(fact_3032_sin__times__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,W)),cos(A,Z)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z))),sin(A,aa(A,A,minus_minus(A,W),Z))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% sin_times_cos
tff(fact_3033_cos__times__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,W)),sin(A,Z)) = divide_divide(A,aa(A,A,minus_minus(A,sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z))),sin(A,aa(A,A,minus_minus(A,W),Z))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% cos_times_sin
tff(fact_3034_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),aa(num,num,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),aa(num,num,bit0,one2)))))),cos(A,divide_divide(A,aa(A,A,minus_minus(A,W),Z),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% sin_plus_sin
tff(fact_3035_sin__diff__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,minus_minus(A,sin(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),sin(A,divide_divide(A,aa(A,A,minus_minus(A,W),Z),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))),cos(A,divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% sin_diff_sin
tff(fact_3036_cos__diff__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,minus_minus(A,cos(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2)))))),sin(A,divide_divide(A,aa(A,A,minus_minus(A,Z),W),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% cos_diff_cos
tff(fact_3037_cos__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X)) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,cos(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,power_power(A,sin(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% cos_double
tff(fact_3038_cos__double__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),W)) = aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,power_power(A,sin(A,W)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% cos_double_sin
tff(fact_3039_cos__two__neq__zero,axiom,
    cos(real,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) != zero_zero(real) ).

% cos_two_neq_zero
tff(fact_3040_cos__monotone__0__pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),pi))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cos(real,X)),cos(real,Y))) ) ) ) ).

% cos_monotone_0_pi
tff(fact_3041_cos__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),pi))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),pi))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cos(real,X)),cos(real,Y)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),X)) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_3042_sin__eq__0__pi,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),pi)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),pi))
       => ( ( sin(real,X) = zero_zero(real) )
         => ( X = zero_zero(real) ) ) ) ) ).

% sin_eq_0_pi
tff(fact_3043_sin__zero__pi__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),pi))
     => ( ( sin(real,X) = zero_zero(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% sin_zero_pi_iff
tff(fact_3044_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),cos(real,Y)),cos(real,X))) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_3045_sincos__total__pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
     => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(real) )
       => ? [T4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T4))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),pi))
            & ( X = cos(real,T4) )
            & ( Y = sin(real,T4) ) ) ) ) ).

% sincos_total_pi
tff(fact_3046_sin__cos__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),sin(real,X)))
     => ( sin(real,X) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,cos(real,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% sin_cos_sqrt
tff(fact_3047_sin__expansion__lemma,axiom,
    ! [X: real,M: nat] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,M))),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) = cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ).

% sin_expansion_lemma
tff(fact_3048_cos__expansion__lemma,axiom,
    ! [X: real,M: nat] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,M))),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) = aa(real,real,uminus_uminus(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))) ).

% cos_expansion_lemma
tff(fact_3049_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(int,bool,aa(int,fun(int,bool),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_3050_sin__gt__zero__02,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),sin(real,X))) ) ) ).

% sin_gt_zero_02
tff(fact_3051_cos__two__less__zero,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cos(real,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),zero_zero(real))) ).

% cos_two_less_zero
tff(fact_3052_cos__is__zero,axiom,
    ? [X3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X3))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
      & ( cos(real,X3) = zero_zero(real) )
      & ! [Y3: real] :
          ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y3))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y3),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
            & ( cos(real,Y3) = zero_zero(real) ) )
         => ( Y3 = X3 ) ) ) ).

% cos_is_zero
tff(fact_3053_cos__two__le__zero,axiom,
    pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),cos(real,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),zero_zero(real))) ).

% cos_two_le_zero
tff(fact_3054_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cos(real,Y)),cos(real,X))) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_3055_cos__total,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => ? [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X3))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),pi))
            & ( cos(real,X3) = Y )
            & ! [Y3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y3),pi))
                  & ( cos(real,Y3) = Y ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% cos_total
tff(fact_3056_sincos__total__pi__half,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(real) )
         => ? [T4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T4))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
              & ( X = cos(real,T4) )
              & ( Y = sin(real,T4) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_3057_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(real) )
     => ? [T4: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T4))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)))
          & ( X = cos(real,T4) )
          & ( Y = sin(real,T4) ) ) ) ).

% sincos_total_2pi_le
tff(fact_3058_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(real) )
     => ~ ! [T4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T4))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)))
             => ( ( X = cos(real,T4) )
               => ( Y != sin(real,T4) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_3059_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),sin(real,divide_divide(real,pi,aa(nat,real,semiring_1_of_nat(real),N))))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_3060_sin__45,axiom,
    sin(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2))))) = divide_divide(real,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% sin_45
tff(fact_3061_cos__45,axiom,
    cos(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2))))) = divide_divide(real,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% cos_45
tff(fact_3062_cos__times__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,W)),cos(A,Z)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,aa(A,A,minus_minus(A,W),Z))),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% cos_times_cos
tff(fact_3063_cos__plus__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),cos(A,divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))),cos(A,divide_divide(A,aa(A,A,minus_minus(A,W),Z),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% cos_plus_cos
tff(fact_3064_sin__gt__zero2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),sin(real,X))) ) ) ).

% sin_gt_zero2
tff(fact_3065_sin__lt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),pi),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),sin(real,X)),zero_zero(real))) ) ) ).

% sin_lt_zero
tff(fact_3066_cos__double__less__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),X))),one_one(real))) ) ) ).

% cos_double_less_one
tff(fact_3067_sin__30,axiom,
    sin(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,one2))))) = divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% sin_30
tff(fact_3068_cos__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),cos(real,X))) ) ) ).

% cos_gt_zero
tff(fact_3069_sin__inj__pi,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( ( sin(real,X) = sin(real,Y) )
             => ( X = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_3070_sin__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),sin(real,X)),sin(real,Y)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y)) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_3071_sin__monotone__2pi__le,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),sin(real,Y)),sin(real,X))) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_3072_cos__60,axiom,
    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),aa(num,num,bit0,one2))) ).

% cos_60
tff(fact_3073_sin__60,axiom,
    sin(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = divide_divide(real,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% sin_60
tff(fact_3074_cos__30,axiom,
    cos(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit1,one2))))) = divide_divide(real,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% cos_30
tff(fact_3075_cos__double__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),W)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,power_power(A,cos(A,W)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),one_one(A)) ) ).

% cos_double_cos
tff(fact_3076_cos__treble__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2))),X)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,aa(num,num,bit0,one2)))),aa(nat,A,power_power(A,cos(A,X)),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))),cos(A,X))) ) ).

% cos_treble_cos
tff(fact_3077_sin__le__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),pi),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),sin(real,X)),zero_zero(real))) ) ) ).

% sin_le_zero
tff(fact_3078_sin__less__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),sin(real,X)),zero_zero(real))) ) ) ).

% sin_less_zero
tff(fact_3079_sin__mono__less__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),sin(real,X)),sin(real,Y)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_3080_sin__monotone__2pi,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),sin(real,Y)),sin(real,X))) ) ) ) ).

% sin_monotone_2pi
tff(fact_3081_sin__total,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => ? [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X3))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
            & ( sin(real,X3) = Y )
            & ! [Y3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
                  & ( sin(real,Y3) = Y ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% sin_total
tff(fact_3082_cos__gt__zero__pi,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),cos(real,X))) ) ) ).

% cos_gt_zero_pi
tff(fact_3083_cos__ge__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),cos(real,X))) ) ) ).

% cos_ge_zero
tff(fact_3084_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( cos(real,X) = one_one(real) )
    <=> ( ? [X4: nat] : X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),X4)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi)
        | ? [X4: nat] : X = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),X4)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_3085_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),sin(real,divide_divide(real,pi,aa(nat,real,semiring_1_of_nat(real),N))))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_3086_sin__arctan,axiom,
    ! [X: real] : sin(real,aa(real,real,arctan,X)) = divide_divide(real,X,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% sin_arctan
tff(fact_3087_cos__arctan,axiom,
    ! [X: real] : cos(real,aa(real,real,arctan,X)) = divide_divide(real,one_one(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% cos_arctan
tff(fact_3088_sin__zero__lemma,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( ( sin(real,X) = zero_zero(real) )
       => ? [N2: nat] :
            ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N2))
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_3089_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
    <=> ( ? [N5: nat] :
            ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N5))
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N5)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) )
        | ? [N5: nat] :
            ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N5))
            & ( X = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N5)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_3090_cos__zero__lemma,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( ( cos(real,X) = zero_zero(real) )
       => ? [N2: nat] :
            ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N2))
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_3091_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( cos(real,X) = zero_zero(real) )
    <=> ( ? [N5: nat] :
            ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N5))
            & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N5)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) )
        | ? [N5: nat] :
            ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N5))
            & ( X = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N5)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_3092_tan__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X)) != zero_zero(A) )
           => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,tan(A),X)),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,aa(A,A,tan(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ) ) ).

% tan_double
tff(fact_3093_sin__tan,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
     => ( sin(real,X) = divide_divide(real,aa(real,real,tan(real),X),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,power_power(real,aa(real,real,tan(real),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% sin_tan
tff(fact_3094_cos__tan,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
     => ( cos(real,X) = divide_divide(real,one_one(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,power_power(real,aa(real,real,tan(real),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ).

% cos_tan
tff(fact_3095_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z) = one_one(real) )
     => ~ ! [T4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T4))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)))
             => ( Z != complex2(cos(real,T4),sin(real,T4)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_3096_ceiling__log__eq__powr__iff,axiom,
    ! [X: real,B2: real,K: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => ( ( archimedean_ceiling(real,aa(real,real,log(B2),X)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),K)),one_one(int)) )
        <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,B2,aa(nat,real,semiring_1_of_nat(real),K))),X))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),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_3097_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_3098_powr__zero__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] :
          ( ( ( X = zero_zero(A) )
           => ( powr(A,X,zero_zero(A)) = zero_zero(A) ) )
          & ( ( X != zero_zero(A) )
           => ( powr(A,X,zero_zero(A)) = one_one(A) ) ) ) ) ).

% powr_zero_eq_one
tff(fact_3099_powr__gt__zero,axiom,
    ! [X: real,A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),powr(real,X,A2)))
    <=> ( X != zero_zero(real) ) ) ).

% powr_gt_zero
tff(fact_3100_powr__nonneg__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,A2,X)),zero_zero(real)))
    <=> ( A2 = zero_zero(real) ) ) ).

% powr_nonneg_iff
tff(fact_3101_powr__less__cancel__iff,axiom,
    ! [X: real,A2: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,X,A2)),powr(real,X,B2)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2)) ) ) ).

% powr_less_cancel_iff
tff(fact_3102_powr__eq__one__iff,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
     => ( ( powr(real,A2,X) = one_one(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% powr_eq_one_iff
tff(fact_3103_powr__one__gt__zero__iff,axiom,
    ! [X: real] :
      ( ( powr(real,X,one_one(real)) = X )
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) ) ).

% powr_one_gt_zero_iff
tff(fact_3104_powr__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( powr(real,X,one_one(real)) = X ) ) ).

% powr_one
tff(fact_3105_powr__le__cancel__iff,axiom,
    ! [X: real,A2: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,X,A2)),powr(real,X,B2)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2)) ) ) ).

% powr_le_cancel_iff
tff(fact_3106_numeral__powr__numeral__real,axiom,
    ! [M: num,N: num] : powr(real,aa(num,real,numeral_numeral(real),M),aa(num,real,numeral_numeral(real),N)) = aa(nat,real,power_power(real,aa(num,real,numeral_numeral(real),M)),aa(num,nat,numeral_numeral(nat),N)) ).

% numeral_powr_numeral_real
tff(fact_3107_log__powr__cancel,axiom,
    ! [A2: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( aa(real,real,log(A2),powr(real,A2,Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_3108_powr__log__cancel,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( powr(real,A2,aa(real,real,log(A2),X)) = X ) ) ) ) ).

% powr_log_cancel
tff(fact_3109_tan__periodic__n,axiom,
    ! [X: real,N: num] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),N)),pi))) = aa(real,real,tan(real),X) ).

% tan_periodic_n
tff(fact_3110_norm__cos__sin,axiom,
    ! [T2: real] : real_V7770717601297561774m_norm(complex,complex2(cos(real,T2),sin(real,T2))) = one_one(real) ).

% norm_cos_sin
tff(fact_3111_powr__numeral,axiom,
    ! [X: real,N: num] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( powr(real,X,aa(num,real,numeral_numeral(real),N)) = aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),N)) ) ) ).

% powr_numeral
tff(fact_3112_tan__periodic,axiom,
    ! [X: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))) = aa(real,real,tan(real),X) ).

% tan_periodic
tff(fact_3113_square__powr__half,axiom,
    ! [X: real] : powr(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = aa(real,real,abs_abs(real),X) ).

% square_powr_half
tff(fact_3114_powr__non__neg,axiom,
    ! [A2: real,X: real] : ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,A2,X)),zero_zero(real))) ).

% powr_non_neg
tff(fact_3115_powr__less__mono2__neg,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),zero_zero(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,Y,A2)),powr(real,X,A2))) ) ) ) ).

% powr_less_mono2_neg
tff(fact_3116_powr__ge__pzero,axiom,
    ! [X: real,Y: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),powr(real,X,Y))) ).

% powr_ge_pzero
tff(fact_3117_powr__mono2,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,X,A2)),powr(real,Y,A2))) ) ) ) ).

% powr_mono2
tff(fact_3118_powr__less__cancel,axiom,
    ! [X: real,A2: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,X,A2)),powr(real,X,B2)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2)) ) ) ).

% powr_less_cancel
tff(fact_3119_powr__less__mono,axiom,
    ! [A2: real,B2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,X,A2)),powr(real,X,B2))) ) ) ).

% powr_less_mono
tff(fact_3120_powr__mono,axiom,
    ! [A2: real,B2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,X,A2)),powr(real,X,B2))) ) ) ).

% powr_mono
tff(fact_3121_one__complex_Ocode,axiom,
    one_one(complex) = complex2(one_one(real),zero_zero(real)) ).

% one_complex.code
tff(fact_3122_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_3123_Complex__eq__numeral,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( complex2(A2,B2) = aa(num,complex,numeral_numeral(complex),W) )
    <=> ( ( A2 = aa(num,real,numeral_numeral(real),W) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_numeral
tff(fact_3124_powr__mono2_H,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),zero_zero(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,Y,A2)),powr(real,X,A2))) ) ) ) ).

% powr_mono2'
tff(fact_3125_powr__less__mono2,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,X,A2)),powr(real,Y,A2))) ) ) ) ).

% powr_less_mono2
tff(fact_3126_powr__inj,axiom,
    ! [A2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( ( powr(real,A2,X) = powr(real,A2,Y) )
        <=> ( X = Y ) ) ) ) ).

% powr_inj
tff(fact_3127_gr__one__powr,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),powr(real,X,Y))) ) ) ).

% gr_one_powr
tff(fact_3128_ge__one__powr__ge__zero,axiom,
    ! [X: real,A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),powr(real,X,A2))) ) ) ).

% ge_one_powr_ge_zero
tff(fact_3129_powr__mono__both,axiom,
    ! [A2: real,B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,X,A2)),powr(real,Y,B2))) ) ) ) ) ).

% powr_mono_both
tff(fact_3130_powr__le1,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,X,A2)),one_one(real))) ) ) ) ).

% powr_le1
tff(fact_3131_powr__divide,axiom,
    ! [X: real,Y: real,A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => ( powr(real,divide_divide(real,X,Y),A2) = divide_divide(real,powr(real,X,A2),powr(real,Y,A2)) ) ) ) ).

% powr_divide
tff(fact_3132_powr__mult,axiom,
    ! [X: real,Y: real,A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => ( powr(real,aa(real,real,aa(real,fun(real,real),times_times(real),X),Y),A2) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,X,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_mult
tff(fact_3133_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_3134_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_3135_powr__add,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [X: A,A2: A,B2: A] : powr(A,X,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,X,A2)),powr(A,X,B2)) ) ).

% powr_add
tff(fact_3136_powr__diff,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [W: A,Z1: A,Z22: A] : powr(A,W,aa(A,A,minus_minus(A,Z1),Z22)) = divide_divide(A,powr(A,W,Z1),powr(A,W,Z22)) ) ).

% powr_diff
tff(fact_3137_tan__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X2: A] : aa(A,A,tan(A),X2) = divide_divide(A,sin(A,X2),cos(A,X2)) ) ).

% tan_def
tff(fact_3138_powr__realpow,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( powr(real,X,aa(nat,real,semiring_1_of_nat(real),N)) = aa(nat,real,power_power(real,X),N) ) ) ).

% powr_realpow
tff(fact_3139_less__log__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),aa(real,real,log(B2),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,B2,Y)),X)) ) ) ) ).

% less_log_iff
tff(fact_3140_log__less__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log(B2),X)),Y))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),powr(real,B2,Y))) ) ) ) ).

% log_less_iff
tff(fact_3141_less__powr__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),powr(real,B2,Y)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,log(B2),X)),Y)) ) ) ) ).

% less_powr_iff
tff(fact_3142_powr__less__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),powr(real,B2,Y)),X))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),aa(real,real,log(B2),X))) ) ) ) ).

% powr_less_iff
tff(fact_3143_powr__minus__divide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [X: A,A2: A] : powr(A,X,aa(A,A,uminus_uminus(A),A2)) = divide_divide(A,one_one(A),powr(A,X,A2)) ) ).

% powr_minus_divide
tff(fact_3144_powr__neg__one,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( powr(real,X,aa(real,real,uminus_uminus(real),one_one(real))) = divide_divide(real,one_one(real),X) ) ) ).

% powr_neg_one
tff(fact_3145_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),X),powr(real,X,Y)) = powr(real,X,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Y)) ) ) ).

% powr_mult_base
tff(fact_3146_le__log__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),aa(real,real,log(B2),X)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,B2,Y)),X)) ) ) ) ).

% le_log_iff
tff(fact_3147_log__le__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log(B2),X)),Y))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),powr(real,B2,Y))) ) ) ) ).

% log_le_iff
tff(fact_3148_le__powr__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),powr(real,B2,Y)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,log(B2),X)),Y)) ) ) ) ).

% le_powr_iff
tff(fact_3149_powr__le__iff,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,B2,Y)),X))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),aa(real,real,log(B2),X))) ) ) ) ).

% powr_le_iff
tff(fact_3150_ln__powr__bound,axiom,
    ! [X: real,A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,ln_ln(real),X)),divide_divide(real,powr(real,X,A2),A2))) ) ) ).

% ln_powr_bound
tff(fact_3151_ln__powr__bound2,axiom,
    ! [X: real,A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,aa(real,real,ln_ln(real),X),A2)),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,A2,A2)),X))) ) ) ).

% ln_powr_bound2
tff(fact_3152_tan__45,axiom,
    aa(real,real,tan(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2))))) = one_one(real) ).

% tan_45
tff(fact_3153_log__add__eq__powr,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
     => ( ( B2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(B2),X)),Y) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),X),powr(real,B2,Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_3154_add__log__eq__powr,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
     => ( ( B2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Y),aa(real,real,log(B2),X)) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,B2,Y)),X)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_3155_minus__log__eq__powr,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
     => ( ( B2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( aa(real,real,minus_minus(real,Y),aa(real,real,log(B2),X)) = aa(real,real,log(B2),divide_divide(real,powr(real,B2,Y),X)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_3156_tan__60,axiom,
    aa(real,real,tan(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))) ).

% tan_60
tff(fact_3157_lemma__tan__total,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y))
     => ? [X3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X3))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),aa(real,real,tan(real),X3))) ) ) ).

% lemma_tan_total
tff(fact_3158_tan__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,tan(real),X))) ) ) ).

% tan_gt_zero
tff(fact_3159_tan__total,axiom,
    ! [Y: real] :
    ? [X3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X3))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
      & ( aa(real,real,tan(real),X3) = Y )
      & ! [Y3: real] :
          ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y3))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
            & ( aa(real,real,tan(real),Y3) = Y ) )
         => ( Y3 = X3 ) ) ) ).

% tan_total
tff(fact_3160_tan__monotone,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tan(real),Y)),aa(real,real,tan(real),X))) ) ) ) ).

% tan_monotone
tff(fact_3161_tan__monotone_H,axiom,
    ! [Y: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),X))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tan(real),Y)),aa(real,real,tan(real),X))) ) ) ) ) ) ).

% tan_monotone'
tff(fact_3162_tan__mono__lt__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tan(real),X)),aa(real,real,tan(real),Y)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_3163_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X3))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
      & ( aa(real,real,tan(real),X3) = Y ) ) ).

% lemma_tan_total1
tff(fact_3164_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),aa(num,num,bit0,aa(num,num,bit0,one2)))))) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% tan_minus_45
tff(fact_3165_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,minus_minus(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),Y)) ).

% tan_inverse
tff(fact_3166_log__minus__eq__powr,axiom,
    ! [B2: real,X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
     => ( ( B2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( aa(real,real,minus_minus(real,aa(real,real,log(B2),X)),Y) = aa(real,real,log(B2),aa(real,real,aa(real,fun(real,real),times_times(real),X),powr(real,B2,aa(real,real,uminus_uminus(real),Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_3167_complex__norm,axiom,
    ! [X: real,Y: real] : real_V7770717601297561774m_norm(complex,complex2(X,Y)) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% complex_norm
tff(fact_3168_add__tan__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)) = divide_divide(A,sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))) ) ) ) ) ).

% add_tan_eq
tff(fact_3169_powr__half__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( powr(real,X,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = aa(real,real,sqrt,X) ) ) ).

% powr_half_sqrt
tff(fact_3170_powr__neg__numeral,axiom,
    ! [X: real,N: num] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( powr(real,X,aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),N))) = divide_divide(real,one_one(real),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),N))) ) ) ).

% powr_neg_numeral
tff(fact_3171_tan__total__pos,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
     => ? [X3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X3))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
          & ( aa(real,real,tan(real),X3) = Y ) ) ) ).

% tan_total_pos
tff(fact_3172_tan__pos__pi2__le,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,tan(real),X))) ) ) ).

% tan_pos_pi2_le
tff(fact_3173_tan__less__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,tan(real),X)),zero_zero(real))) ) ) ).

% tan_less_zero
tff(fact_3174_tan__mono__le__eq,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,tan(real),X)),aa(real,real,tan(real),Y)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y)) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_3175_tan__mono__le,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,tan(real),X)),aa(real,real,tan(real),Y))) ) ) ) ).

% tan_mono_le
tff(fact_3176_tan__bound__pi2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2))))))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,tan(real),X))),one_one(real))) ) ).

% tan_bound_pi2
tff(fact_3177_tan__30,axiom,
    aa(real,real,tan(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,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_3178_arctan__unique,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( ( aa(real,real,tan(real),X) = Y )
         => ( aa(real,real,arctan,Y) = X ) ) ) ) ).

% arctan_unique
tff(fact_3179_arctan__tan,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( aa(real,real,arctan,aa(real,real,tan(real),X)) = X ) ) ) ).

% arctan_tan
tff(fact_3180_arctan,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arctan,Y)))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arctan,Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
      & ( aa(real,real,tan(real),aa(real,real,arctan,Y)) = Y ) ) ).

% arctan
tff(fact_3181_lemma__tan__add1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y))) = divide_divide(A,cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))) ) ) ) ) ).

% lemma_tan_add1
tff(fact_3182_tan__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,minus_minus(A,X),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,minus_minus(A,X),Y)) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_diff
tff(fact_3183_tan__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)),aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),X)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_add
tff(fact_3184_tan__total__pi4,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ? [Z2: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))))),Z2))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z2),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2))))))
          & ( aa(real,real,tan(real),Z2) = X ) ) ) ).

% tan_total_pi4
tff(fact_3185_tan__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,tan(A),X) = divide_divide(A,sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X)),aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X))),one_one(A))) ) ).

% tan_half
tff(fact_3186_arcosh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] : aa(A,A,arcosh(A),X) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),powr(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)),real_Vector_of_real(A,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))))) ) ).

% arcosh_def
tff(fact_3187_cos__arcsin,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => ( cos(real,aa(real,real,arcsin,X)) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ) ).

% cos_arcsin
tff(fact_3188_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real)))
     => ( sin(real,aa(real,real,arccos,Y)) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,Y),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% sin_arccos_abs
tff(fact_3189_sin__arccos,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => ( sin(real,aa(real,real,arccos,X)) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ) ).

% sin_arccos
tff(fact_3190_arsinh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [X: A] : aa(A,A,arsinh(A),X) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),powr(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)),real_Vector_of_real(A,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))))) ) ).

% arsinh_def
tff(fact_3191_of__real__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real] :
          ( ( real_Vector_of_real(A,X) = one_one(A) )
        <=> ( X = one_one(real) ) ) ) ).

% of_real_eq_1_iff
tff(fact_3192_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_3193_of__real__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W: num] : real_Vector_of_real(A,aa(num,real,numeral_numeral(real),W)) = aa(num,A,numeral_numeral(A),W) ) ).

% of_real_numeral
tff(fact_3194_of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [X: real,Y: real] : real_Vector_of_real(A,divide_divide(real,X,Y)) = divide_divide(A,real_Vector_of_real(A,X),real_Vector_of_real(A,Y)) ) ).

% of_real_divide
tff(fact_3195_of__real__add,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real,Y: real] : real_Vector_of_real(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,X)),real_Vector_of_real(A,Y)) ) ).

% of_real_add
tff(fact_3196_of__real__power,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [X: real,N: nat] : real_Vector_of_real(A,aa(nat,real,power_power(real,X),N)) = aa(nat,A,power_power(A,real_Vector_of_real(A,X)),N) ) ).

% of_real_power
tff(fact_3197_arccos__1,axiom,
    aa(real,real,arccos,one_one(real)) = zero_zero(real) ).

% arccos_1
tff(fact_3198_arccos__minus__1,axiom,
    aa(real,real,arccos,aa(real,real,uminus_uminus(real),one_one(real))) = pi ).

% arccos_minus_1
tff(fact_3199_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_3200_cos__of__real__pi,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,real_Vector_of_real(A,pi)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% cos_of_real_pi
tff(fact_3201_cos__arccos,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_3202_sin__arcsin,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_3203_norm__of__real__add1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: real] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,X)),one_one(A))) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),one_one(real))) ) ).

% norm_of_real_add1
tff(fact_3204_norm__of__real__addn,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: real,B2: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,X)),aa(num,A,numeral_numeral(A),B2))) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(num,real,numeral_numeral(real),B2))) ) ).

% norm_of_real_addn
tff(fact_3205_arccos__0,axiom,
    aa(real,real,arccos,zero_zero(real)) = divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% arccos_0
tff(fact_3206_arcsin__1,axiom,
    aa(real,real,arcsin,one_one(real)) = divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% arcsin_1
tff(fact_3207_cos__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,divide_divide(A,real_Vector_of_real(A,pi),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = zero_zero(A) ) ) ).

% cos_of_real_pi_half
tff(fact_3208_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),aa(num,num,bit0,one2)))) = one_one(A) ) ) ).

% sin_of_real_pi_half
tff(fact_3209_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),aa(num,num,bit0,one2)))) ).

% arcsin_minus_1
tff(fact_3210_nonzero__of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [Y: real,X: real] :
          ( ( Y != zero_zero(real) )
         => ( real_Vector_of_real(A,divide_divide(real,X,Y)) = divide_divide(A,real_Vector_of_real(A,X),real_Vector_of_real(A,Y)) ) ) ) ).

% nonzero_of_real_divide
tff(fact_3211_arccos__le__arccos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arccos,Y)),aa(real,real,arccos,X))) ) ) ) ).

% arccos_le_arccos
tff(fact_3212_arccos__le__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arccos,X)),aa(real,real,arccos,Y)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),X)) ) ) ) ).

% arccos_le_mono
tff(fact_3213_arccos__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))) )
     => ( ( aa(real,real,arccos,X) = aa(real,real,arccos,Y) )
      <=> ( X = Y ) ) ) ).

% arccos_eq_iff
tff(fact_3214_arcsin__minus,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => ( aa(real,real,arcsin,aa(real,real,uminus_uminus(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,arcsin,X)) ) ) ) ).

% arcsin_minus
tff(fact_3215_arcsin__le__arcsin,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arcsin,X)),aa(real,real,arcsin,Y))) ) ) ) ).

% arcsin_le_arcsin
tff(fact_3216_arcsin__le__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arcsin,X)),aa(real,real,arcsin,Y)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y)) ) ) ) ).

% arcsin_le_mono
tff(fact_3217_arcsin__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real)))
       => ( ( aa(real,real,arcsin,X) = aa(real,real,arcsin,Y) )
        <=> ( X = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_3218_norm__less__p1,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,real_V7770717601297561774m_norm(A,X))),one_one(A))))) ) ).

% norm_less_p1
tff(fact_3219_arccos__lbound,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Y))) ) ) ).

% arccos_lbound
tff(fact_3220_arccos__less__arccos,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arccos,Y)),aa(real,real,arccos,X))) ) ) ) ).

% arccos_less_arccos
tff(fact_3221_arccos__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arccos,X)),aa(real,real,arccos,Y)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),X)) ) ) ) ).

% arccos_less_mono
tff(fact_3222_arccos__ubound,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arccos,Y)),pi)) ) ) ).

% arccos_ubound
tff(fact_3223_arccos__cos,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),pi))
       => ( aa(real,real,arccos,cos(real,X)) = X ) ) ) ).

% arccos_cos
tff(fact_3224_arcsin__less__arcsin,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arcsin,X)),aa(real,real,arcsin,Y))) ) ) ) ).

% arcsin_less_arcsin
tff(fact_3225_arcsin__less__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arcsin,X)),aa(real,real,arcsin,Y)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ) ) ).

% arcsin_less_mono
tff(fact_3226_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real)))
     => ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ).

% cos_arccos_abs
tff(fact_3227_norm__of__real__diff,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [B2: real,A2: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(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,minus_minus(real,B2),A2)))) ) ).

% norm_of_real_diff
tff(fact_3228_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),Theta)),pi))
     => ( aa(real,real,arccos,cos(real,Theta)) = aa(real,real,abs_abs(real),Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_3229_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,arccos,Y)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arccos,Y)),pi)) ) ) ) ).

% arccos_lt_bounded
tff(fact_3230_arccos__bounded,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Y)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arccos,Y)),pi)) ) ) ) ).

% arccos_bounded
tff(fact_3231_sin__arccos__nonzero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => ( sin(real,aa(real,real,arccos,X)) != zero_zero(real) ) ) ) ).

% sin_arccos_nonzero
tff(fact_3232_arccos__cos2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),X))
       => ( aa(real,real,arccos,cos(real,X)) = aa(real,real,uminus_uminus(real),X) ) ) ) ).

% arccos_cos2
tff(fact_3233_arccos__minus,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),X)) = aa(real,real,minus_minus(real,pi),aa(real,real,arccos,X)) ) ) ) ).

% arccos_minus
tff(fact_3234_cos__arcsin__nonzero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => ( cos(real,aa(real,real,arcsin,X)) != zero_zero(real) ) ) ) ).

% cos_arcsin_nonzero
tff(fact_3235_arccos,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,arccos,Y)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arccos,Y)),pi))
          & ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_3236_arccos__minus__abs,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( aa(real,real,arccos,aa(real,real,uminus_uminus(real),X)) = aa(real,real,minus_minus(real,pi),aa(real,real,arccos,X)) ) ) ).

% arccos_minus_abs
tff(fact_3237_cos__sin__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : cos(A,X) = sin(A,aa(A,A,minus_minus(A,divide_divide(A,real_Vector_of_real(A,pi),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),X)) ) ).

% cos_sin_eq
tff(fact_3238_sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : sin(A,X) = cos(A,aa(A,A,minus_minus(A,divide_divide(A,real_Vector_of_real(A,pi),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),X)) ) ).

% sin_cos_eq
tff(fact_3239_minus__sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,uminus_uminus(A),sin(A,X)) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),divide_divide(A,real_Vector_of_real(A,pi),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% minus_sin_cos_eq
tff(fact_3240_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arccos,Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ) ).

% arccos_le_pi2
tff(fact_3241_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,arcsin,Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_3242_arcsin__bounded,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arcsin,Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ) ) ).

% arcsin_bounded
tff(fact_3243_arcsin__ubound,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arcsin,Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ) ).

% arcsin_ubound
tff(fact_3244_arcsin__lbound,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y))) ) ) ).

% arcsin_lbound
tff(fact_3245_arcsin__sin,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => ( aa(real,real,arcsin,sin(real,X)) = X ) ) ) ).

% arcsin_sin
tff(fact_3246_arcsin,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arcsin,Y)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_3247_arcsin__pi,axiom,
    ! [Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),aa(real,real,arcsin,Y)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arcsin,Y)),pi))
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_3248_arcsin__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,arcsin,X)),Y))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),sin(real,Y))) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_3249_le__arcsin__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),aa(real,real,arcsin,X)))
            <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),sin(real,Y)),X)) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_3250_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),N) )
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,power_power(nat,B2),N)),K))
            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(nat,nat,power_power(nat,B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_3251_cos__npi__int,axiom,
    ! [N: int] :
      ( ( pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),N))
       => ( cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),ring_1_of_int(real,N))) = one_one(real) ) )
      & ( ~ pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),N))
       => ( cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),ring_1_of_int(real,N))) = aa(real,real,uminus_uminus(real),one_one(real)) ) ) ) ).

% cos_npi_int
tff(fact_3252_cot__less__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),divide_divide(real,aa(real,real,uminus_uminus(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,cot(real),X)),zero_zero(real))) ) ) ).

% cot_less_zero
tff(fact_3253_floor__log__nat__eq__if,axiom,
    ! [B2: nat,N: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,power_power(nat,B2),N)),K))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(nat,nat,power_power(nat,B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),B2))
         => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B2)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),N) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_3254_cot__periodic,axiom,
    ! [X: real] : aa(real,real,cot(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi))) = aa(real,real,cot(real),X) ).

% cot_periodic
tff(fact_3255_of__int__floor__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( ring_1_of_int(A,archim6421214686448440834_floor(A,X)) = X )
        <=> ? [N5: int] : X = ring_1_of_int(A,N5) ) ) ).

% of_int_floor_cancel
tff(fact_3256_of__int__ceiling__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( ring_1_of_int(A,archimedean_ceiling(A,X)) = X )
        <=> ? [N5: int] : X = ring_1_of_int(A,N5) ) ) ).

% of_int_ceiling_cancel
tff(fact_3257_of__int__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: int,Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,W)),ring_1_of_int(A,Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),W),Z)) ) ) ).

% of_int_le_iff
tff(fact_3258_of__int__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : ring_1_of_int(A,aa(num,int,numeral_numeral(int),K)) = aa(num,A,numeral_numeral(A),K) ) ).

% of_int_numeral
tff(fact_3259_of__int__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int,N: num] :
          ( ( ring_1_of_int(A,Z) = aa(num,A,numeral_numeral(A),N) )
        <=> ( Z = aa(num,int,numeral_numeral(int),N) ) ) ) ).

% of_int_eq_numeral_iff
tff(fact_3260_of__int__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: int,Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),ring_1_of_int(A,W)),ring_1_of_int(A,Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),W),Z)) ) ) ).

% of_int_less_iff
tff(fact_3261_of__int__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( ring_1_of_int(A,one_one(int)) = one_one(A) ) ) ).

% of_int_1
tff(fact_3262_of__int__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int] :
          ( ( ring_1_of_int(A,Z) = one_one(A) )
        <=> ( Z = one_one(int) ) ) ) ).

% of_int_eq_1_iff
tff(fact_3263_of__int__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : 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),ring_1_of_int(A,W)),ring_1_of_int(A,Z)) ) ).

% of_int_add
tff(fact_3264_floor__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archim6421214686448440834_floor(A,aa(num,A,numeral_numeral(A),V)) = aa(num,int,numeral_numeral(int),V) ) ).

% floor_numeral
tff(fact_3265_floor__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).

% floor_one
tff(fact_3266_of__int__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int,N: nat] : ring_1_of_int(A,aa(nat,int,power_power(int,Z),N)) = aa(nat,A,power_power(A,ring_1_of_int(A,Z)),N) ) ).

% of_int_power
tff(fact_3267_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [B2: int,W: nat,X: int] :
          ( ( aa(nat,A,power_power(A,ring_1_of_int(A,B2)),W) = ring_1_of_int(A,X) )
        <=> ( aa(nat,int,power_power(int,B2),W) = X ) ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_3268_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: int,B2: int,W: nat] :
          ( ( ring_1_of_int(A,X) = aa(nat,A,power_power(A,ring_1_of_int(A,B2)),W) )
        <=> ( X = aa(nat,int,power_power(int,B2),W) ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_3269_ceiling__add__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),ring_1_of_int(A,Z))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,X)),Z) ) ).

% ceiling_add_of_int
tff(fact_3270_of__nat__nat__take__bit__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat,K: int] : aa(nat,A,semiring_1_of_nat(A),nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,N),K))) = ring_1_of_int(A,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ) ).

% of_nat_nat_take_bit_eq
tff(fact_3271_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),ring_1_of_int(A,Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z)) ) ) ).

% of_int_0_le_iff
tff(fact_3272_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,Z)),zero_zero(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),zero_zero(int))) ) ) ).

% of_int_le_0_iff
tff(fact_3273_of__int__0__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),ring_1_of_int(A,Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z)) ) ) ).

% of_int_0_less_iff
tff(fact_3274_of__int__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),ring_1_of_int(A,Z)),zero_zero(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),zero_zero(int))) ) ) ).

% of_int_less_0_iff
tff(fact_3275_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),N)),ring_1_of_int(A,Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),N)),Z)) ) ) ).

% of_int_numeral_le_iff
tff(fact_3276_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,Z)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),aa(num,int,numeral_numeral(int),N))) ) ) ).

% of_int_le_numeral_iff
tff(fact_3277_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,N: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),ring_1_of_int(A,Z)),aa(num,A,numeral_numeral(A),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),aa(num,int,numeral_numeral(int),N))) ) ) ).

% of_int_less_numeral_iff
tff(fact_3278_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(num,A,numeral_numeral(A),N)),ring_1_of_int(A,Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),N)),Z)) ) ) ).

% of_int_numeral_less_iff
tff(fact_3279_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,Z)),one_one(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),one_one(int))) ) ) ).

% of_int_le_1_iff
tff(fact_3280_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),ring_1_of_int(A,Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),Z)) ) ) ).

% of_int_1_le_iff
tff(fact_3281_zero__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X)) ) ) ).

% zero_le_floor
tff(fact_3282_of__int__1__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),ring_1_of_int(A,Z)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),Z)) ) ) ).

% of_int_1_less_iff
tff(fact_3283_of__int__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),ring_1_of_int(A,Z)),one_one(A)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),one_one(int))) ) ) ).

% of_int_less_1_iff
tff(fact_3284_floor__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),zero_zero(A))) ) ) ).

% floor_less_zero
tff(fact_3285_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),V)),X)) ) ) ).

% numeral_le_floor
tff(fact_3286_zero__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ).

% zero_less_floor
tff(fact_3287_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),zero_zero(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ).

% floor_le_zero
tff(fact_3288_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(num,A,numeral_numeral(A),V))) ) ) ).

% floor_less_numeral
tff(fact_3289_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ).

% one_le_floor
tff(fact_3290_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ).

% floor_less_one
tff(fact_3291_floor__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)) ) ).

% floor_neg_numeral
tff(fact_3292_floor__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] : archim6421214686448440834_floor(A,aa(A,A,minus_minus(A,X),aa(num,A,numeral_numeral(A),V))) = aa(int,int,minus_minus(int,archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V)) ) ).

% floor_diff_numeral
tff(fact_3293_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,B2: int,W: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,X)),aa(nat,A,power_power(A,ring_1_of_int(A,B2)),W)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),aa(nat,int,power_power(int,B2),W))) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_3294_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W: nat,X: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,ring_1_of_int(A,B2)),W)),ring_1_of_int(A,X)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,power_power(int,B2),W)),X)) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_3295_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: num,N: nat,Y: int] :
          ( ( aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),N) = ring_1_of_int(A,Y) )
        <=> ( aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N) = Y ) ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_3296_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,X: num,N: nat] :
          ( ( ring_1_of_int(A,Y) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),N) )
        <=> ( Y = aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N) ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_3297_floor__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archim6421214686448440834_floor(A,aa(A,A,minus_minus(A,X),one_one(A))) = aa(int,int,minus_minus(int,archim6421214686448440834_floor(A,X)),one_one(int)) ) ).

% floor_diff_one
tff(fact_3298_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W: nat,X: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,ring_1_of_int(A,B2)),W)),ring_1_of_int(A,X)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,power_power(int,B2),W)),X)) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_3299_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: int,B2: int,W: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),ring_1_of_int(A,X)),aa(nat,A,power_power(A,ring_1_of_int(A,B2)),W)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),aa(nat,int,power_power(int,B2),W))) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_3300_of__nat__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
         => ( aa(nat,A,semiring_1_of_nat(A),nat2(Z)) = ring_1_of_int(A,Z) ) ) ) ).

% of_nat_nat
tff(fact_3301_floor__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: num,N: nat] : archim6421214686448440834_floor(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),N)) = aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N) ) ).

% floor_numeral_power
tff(fact_3302_floor__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : 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_3303_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))),X)) ) ) ).

% numeral_less_floor
tff(fact_3304_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(num,int,numeral_numeral(int),V)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A)))) ) ) ).

% floor_le_numeral
tff(fact_3305_one__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),one_one(int)),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X)) ) ) ).

% one_less_floor
tff(fact_3306_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),one_one(int)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ) ).

% floor_le_one
tff(fact_3307_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),X)) ) ) ).

% neg_numeral_le_floor
tff(fact_3308_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)))) ) ) ).

% floor_less_neg_numeral
tff(fact_3309_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),N)),ring_1_of_int(A,A2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N)),A2)) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_3310_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,A2)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N))) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_3311_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),N)),ring_1_of_int(A,A2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N)),A2)) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_3312_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),ring_1_of_int(A,A2)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),X)),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),X)),N))) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_3313_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,X: num,N: nat] :
          ( ( ring_1_of_int(A,Y) = aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N) )
        <=> ( Y = aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N) ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_3314_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [X: num,N: nat,Y: int] :
          ( ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N) = ring_1_of_int(A,Y) )
        <=> ( aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N) = Y ) ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_3315_floor__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : 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_3316_floor__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : 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_3317_sin__int__2pin,axiom,
    ! [N: 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),aa(num,num,bit0,one2))),pi)),ring_1_of_int(real,N))) = zero_zero(real) ).

% sin_int_2pin
tff(fact_3318_cos__int__2pin,axiom,
    ! [N: int] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)),ring_1_of_int(real,N))) = one_one(real) ).

% cos_int_2pin
tff(fact_3319_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),X)) ) ) ).

% neg_numeral_less_floor
tff(fact_3320_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,V: num] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A)))) ) ) ).

% floor_le_neg_numeral
tff(fact_3321_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,A2)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N))) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_3322_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)),ring_1_of_int(A,A2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N)),A2)) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_3323_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: num,N: nat,A2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)),ring_1_of_int(A,A2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N)),A2)) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_3324_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,X: num,N: nat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),ring_1_of_int(A,A2)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X))),N)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),X))),N))) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_3325_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : 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_3326_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z2: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),ring_1_of_int(A,Z2))) ) ).

% ex_le_of_int
tff(fact_3327_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z2: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),ring_1_of_int(A,Z2)),X)) ) ).

% ex_of_int_less
tff(fact_3328_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z2: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),ring_1_of_int(A,Z2))) ) ).

% ex_less_of_int
tff(fact_3329_int__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),archim6421214686448440834_floor(A,X)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),ring_1_of_int(A,Z)),X)) ) ).

% int_add_floor
tff(fact_3330_floor__add__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),Z) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),ring_1_of_int(A,Z))) ) ).

% floor_add_int
tff(fact_3331_floor__divide__of__int__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [K: int,L: int] : archim6421214686448440834_floor(A,divide_divide(A,ring_1_of_int(A,K),ring_1_of_int(A,L))) = divide_divide(int,K,L) ) ).

% floor_divide_of_int_eq
tff(fact_3332_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,archim6421214686448440834_floor(A,X))),X)) ) ).

% of_int_floor_le
tff(fact_3333_floor__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),Z))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),ring_1_of_int(A,Z))) ) ) ).

% floor_less_iff
tff(fact_3334_le__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,Z)),X)) ) ) ).

% le_floor_iff
tff(fact_3335_floor__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,N: nat] :
          ( ( X = ring_1_of_int(A,archim6421214686448440834_floor(A,X)) )
         => ( archim6421214686448440834_floor(A,aa(nat,A,power_power(A,X),N)) = aa(nat,int,power_power(int,archim6421214686448440834_floor(A,X)),N) ) ) ) ).

% floor_power
tff(fact_3336_real__of__int__floor__add__one__gt,axiom,
    ! [R3: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),R3),aa(real,real,aa(real,fun(real,real),plus_plus(real),ring_1_of_int(real,archim6421214686448440834_floor(real,R3))),one_one(real)))) ).

% real_of_int_floor_add_one_gt
tff(fact_3337_floor__eq,axiom,
    ! [N: int,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),ring_1_of_int(real,N)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),plus_plus(real),ring_1_of_int(real,N)),one_one(real))))
       => ( archim6421214686448440834_floor(real,X) = N ) ) ) ).

% floor_eq
tff(fact_3338_real__of__int__floor__add__one__ge,axiom,
    ! [R3: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),R3),aa(real,real,aa(real,fun(real,real),plus_plus(real),ring_1_of_int(real,archim6421214686448440834_floor(real,R3))),one_one(real)))) ).

% real_of_int_floor_add_one_ge
tff(fact_3339_real__of__int__floor__gt__diff__one,axiom,
    ! [R3: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,minus_minus(real,R3),one_one(real))),ring_1_of_int(real,archim6421214686448440834_floor(real,R3)))) ).

% real_of_int_floor_gt_diff_one
tff(fact_3340_real__of__int__floor__ge__diff__one,axiom,
    ! [R3: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,minus_minus(real,R3),one_one(real))),ring_1_of_int(real,archim6421214686448440834_floor(real,R3)))) ).

% real_of_int_floor_ge_diff_one
tff(fact_3341_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,bool),T2: A] :
          ( pp(aa(int,bool,P,archim6421214686448440834_floor(A,T2)))
        <=> ! [I3: int] :
              ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,I3)),T2))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),T2),aa(A,A,aa(A,fun(A,A),plus_plus(A),ring_1_of_int(A,I3)),one_one(A)))) )
             => pp(aa(int,bool,P,I3)) ) ) ) ).

% floor_split
tff(fact_3342_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A2: int] :
          ( ( archim6421214686448440834_floor(A,X) = A2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,A2)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),ring_1_of_int(A,A2)),one_one(A)))) ) ) ) ).

% floor_eq_iff
tff(fact_3343_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,Z)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),ring_1_of_int(A,Z)),one_one(A))))
           => ( archim6421214686448440834_floor(A,X) = Z ) ) ) ) ).

% floor_unique
tff(fact_3344_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),archim6421214686448440834_floor(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),ring_1_of_int(A,Z)),one_one(A))),X)) ) ) ).

% less_floor_iff
tff(fact_3345_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),Z))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),ring_1_of_int(A,Z)),one_one(A)))) ) ) ).

% floor_le_iff
tff(fact_3346_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,archim6421214686448440834_floor(A,X))),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),ring_1_of_int(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int))))) ) ) ).

% floor_correct
tff(fact_3347_floor__eq2,axiom,
    ! [N: int,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),ring_1_of_int(real,N)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(real,real,aa(real,fun(real,real),plus_plus(real),ring_1_of_int(real,N)),one_one(real))))
       => ( archim6421214686448440834_floor(real,X) = N ) ) ) ).

% floor_eq2
tff(fact_3348_floor__divide__real__eq__div,axiom,
    ! [B2: int,A2: real] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),B2))
     => ( archim6421214686448440834_floor(real,divide_divide(real,A2,ring_1_of_int(real,B2))) = divide_divide(int,archim6421214686448440834_floor(real,A2),B2) ) ) ).

% floor_divide_real_eq_div
tff(fact_3349_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))) ) ) ).

% floor_mono
tff(fact_3350_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ).

% floor_less_cancel
tff(fact_3351_floor__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archimedean_ceiling(A,X))) ) ).

% floor_le_ceiling
tff(fact_3352_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),ring_1_of_int(A,archimedean_ceiling(A,X)))) ) ).

% le_of_int_ceiling
tff(fact_3353_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),ring_1_of_int(A,archim6421214686448440834_floor(A,divide_divide(A,P2,Q3)))),Q3)),P2)) ) ) ).

% floor_divide_lower
tff(fact_3354_take__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,K: int] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),ring_1_of_int(A,K)) = ring_1_of_int(A,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ) ).

% take_bit_of_int
tff(fact_3355_of__int__and__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : 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),ring_1_of_int(A,K)),ring_1_of_int(A,L)) ) ).

% of_int_and_eq
tff(fact_3356_of__int__not__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int] : ring_1_of_int(A,aa(int,int,bit_ri4277139882892585799ns_not(int),K)) = aa(A,A,bit_ri4277139882892585799ns_not(A),ring_1_of_int(A,K)) ) ).

% of_int_not_eq
tff(fact_3357_of__int__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : 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),ring_1_of_int(A,K)),ring_1_of_int(A,L)) ) ).

% of_int_xor_eq
tff(fact_3358_of__int__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : ring_1_of_int(A,bit_se2239418461657761734s_mask(int,N)) = bit_se2239418461657761734s_mask(A,N) ) ).

% of_int_mask_eq
tff(fact_3359_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),P2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),ring_1_of_int(A,archim6421214686448440834_floor(A,divide_divide(A,P2,Q3)))),one_one(A))),Q3))) ) ) ).

% floor_divide_upper
tff(fact_3360_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)))) ) ).

% le_floor_add
tff(fact_3361_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),ring_1_of_int(A,A2)))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),A2)) ) ) ).

% ceiling_le
tff(fact_3362_ceiling__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_ceiling(A,X)),Z))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),ring_1_of_int(A,Z))) ) ) ).

% ceiling_le_iff
tff(fact_3363_less__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),ring_1_of_int(A,Z)),X)) ) ) ).

% less_ceiling_iff
tff(fact_3364_real__of__int__div4,axiom,
    ! [N: int,X: int] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),ring_1_of_int(real,divide_divide(int,N,X))),divide_divide(real,ring_1_of_int(real,N),ring_1_of_int(real,X)))) ).

% real_of_int_div4
tff(fact_3365_real__of__int__div,axiom,
    ! [D2: int,N: int] :
      ( pp(dvd_dvd(int,D2,N))
     => ( ring_1_of_int(real,divide_divide(int,N,D2)) = divide_divide(real,ring_1_of_int(real,N),ring_1_of_int(real,D2)) ) ) ).

% real_of_int_div
tff(fact_3366_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),R3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),nat2(archim6421214686448440834_floor(A,R3)))),R3)) ) ) ).

% of_nat_floor
tff(fact_3367_one__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),one_one(int)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) ) ).

% one_add_floor
tff(fact_3368_le__mult__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(archim6421214686448440834_floor(A,A2))),nat2(archim6421214686448440834_floor(A,B2)))),nat2(archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))))) ) ).

% le_mult_nat_floor
tff(fact_3369_floor__divide__of__nat__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [M: nat,N: nat] : archim6421214686448440834_floor(A,divide_divide(A,aa(nat,A,semiring_1_of_nat(A),M),aa(nat,A,semiring_1_of_nat(A),N))) = aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,M,N)) ) ).

% floor_divide_of_nat_eq
tff(fact_3370_nat__floor__neg,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
     => ( nat2(archim6421214686448440834_floor(real,X)) = zero_zero(nat) ) ) ).

% nat_floor_neg
tff(fact_3371_floor__log__eq__powr__iff,axiom,
    ! [X: real,B2: real,K: int] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(B2),X)) = K )
        <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),powr(real,B2,ring_1_of_int(real,K))),X))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),powr(real,B2,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_3372_of__int__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Z))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),ring_1_of_int(A,Z))) ) ) ).

% of_int_nonneg
tff(fact_3373_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),ring_1_of_int(A,N))),X))
         => ( ( N = zero_zero(int) )
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ) ).

% of_int_leD
tff(fact_3374_of__int__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Z))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),ring_1_of_int(A,Z))) ) ) ).

% of_int_pos
tff(fact_3375_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),ring_1_of_int(A,N))),X))
         => ( ( N = zero_zero(int) )
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ) ).

% of_int_lessD
tff(fact_3376_floor__eq3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,semiring_1_of_nat(real),N)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
       => ( nat2(archim6421214686448440834_floor(real,X)) = N ) ) ) ).

% floor_eq3
tff(fact_3377_le__nat__floor,axiom,
    ! [X: nat,A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),X)),A2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X),nat2(archim6421214686448440834_floor(real,A2)))) ) ).

% le_nat_floor
tff(fact_3378_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [X3: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,X3)),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),ring_1_of_int(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),one_one(int)))))
          & ! [Y3: int] :
              ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,Y3)),X))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),ring_1_of_int(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),Y3),one_one(int))))) )
             => ( Y3 = X3 ) ) ) ) ).

% floor_exists1
tff(fact_3379_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
        ? [Z2: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,Z2)),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),ring_1_of_int(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),one_one(int))))) ) ) ).

% floor_exists
tff(fact_3380_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,archimedean_ceiling(A,R3))),aa(A,A,aa(A,fun(A,A),plus_plus(A),R3),one_one(A)))) ) ).

% of_int_ceiling_le_add_one
tff(fact_3381_of__int__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : 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_3382_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R3: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,minus_minus(A,ring_1_of_int(A,archimedean_ceiling(A,R3))),one_one(A))),R3)) ) ).

% of_int_ceiling_diff_one_le
tff(fact_3383_ceiling__diff__floor__le__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,minus_minus(int,archimedean_ceiling(A,X)),archim6421214686448440834_floor(A,X))),one_one(int))) ) ).

% ceiling_diff_floor_le_1
tff(fact_3384_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,X: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,semiring_1_of_nat(A),N)),ring_1_of_int(A,X)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),N)),X)) ) ) ).

% of_nat_less_of_int_iff
tff(fact_3385_int__le__real__less,axiom,
    ! [N: int,M: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),N),M))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),ring_1_of_int(real,N)),aa(real,real,aa(real,fun(real,real),plus_plus(real),ring_1_of_int(real,M)),one_one(real)))) ) ).

% int_le_real_less
tff(fact_3386_int__less__real__le,axiom,
    ! [N: int,M: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),N),M))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),ring_1_of_int(real,N)),one_one(real))),ring_1_of_int(real,M))) ) ).

% int_less_real_le
tff(fact_3387_of__int__not__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : 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_3388_ceiling__divide__eq__div,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: int,B2: int] : archimedean_ceiling(A,divide_divide(A,ring_1_of_int(A,A2),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_3389_real__of__int__div__aux,axiom,
    ! [X: int,D2: int] : divide_divide(real,ring_1_of_int(real,X),ring_1_of_int(real,D2)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),ring_1_of_int(real,divide_divide(int,X,D2))),divide_divide(real,ring_1_of_int(real,modulo_modulo(int,X,D2)),ring_1_of_int(real,D2))) ).

% real_of_int_div_aux
tff(fact_3390_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A2)),archim6421214686448440834_floor(A,B2))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))) ) ) ) ).

% le_mult_floor
tff(fact_3391_floor__eq4,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),N)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N))))
       => ( nat2(archim6421214686448440834_floor(real,X)) = N ) ) ) ).

% floor_eq4
tff(fact_3392_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,bool),T2: A] :
          ( pp(aa(int,bool,P,archimedean_ceiling(A,T2)))
        <=> ! [I3: int] :
              ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,minus_minus(A,ring_1_of_int(A,I3)),one_one(A))),T2))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),T2),ring_1_of_int(A,I3))) )
             => pp(aa(int,bool,P,I3)) ) ) ) ).

% ceiling_split
tff(fact_3393_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A2: int] :
          ( ( archimedean_ceiling(A,X) = A2 )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,minus_minus(A,ring_1_of_int(A,A2)),one_one(A))),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),ring_1_of_int(A,A2))) ) ) ) ).

% ceiling_eq_iff
tff(fact_3394_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,minus_minus(A,ring_1_of_int(A,Z)),one_one(A))),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),ring_1_of_int(A,Z)))
           => ( archimedean_ceiling(A,X) = Z ) ) ) ) ).

% ceiling_unique
tff(fact_3395_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,minus_minus(A,ring_1_of_int(A,archimedean_ceiling(A,X))),one_one(A))),X))
          & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),ring_1_of_int(A,archimedean_ceiling(A,X)))) ) ) ).

% ceiling_correct
tff(fact_3396_cot__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X2: A] : aa(A,A,cot(A),X2) = divide_divide(A,cos(A,X2),sin(A,X2)) ) ).

% cot_def
tff(fact_3397_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Z: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),archimedean_ceiling(A,X)),Z))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,minus_minus(A,ring_1_of_int(A,Z)),one_one(A)))) ) ) ).

% ceiling_less_iff
tff(fact_3398_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,X: A] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Z),archimedean_ceiling(A,X)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,minus_minus(A,ring_1_of_int(A,Z)),one_one(A))),X)) ) ) ).

% le_ceiling_iff
tff(fact_3399_real__of__int__div2,axiom,
    ! [N: int,X: int] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,minus_minus(real,divide_divide(real,ring_1_of_int(real,N),ring_1_of_int(real,X))),ring_1_of_int(real,divide_divide(int,N,X))))) ).

% real_of_int_div2
tff(fact_3400_real__of__int__div3,axiom,
    ! [N: int,X: int] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,minus_minus(real,divide_divide(real,ring_1_of_int(real,N),ring_1_of_int(real,X))),ring_1_of_int(real,divide_divide(int,N,X)))),one_one(real))) ).

% real_of_int_div3
tff(fact_3401_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P2),aa(A,A,aa(A,fun(A,A),times_times(A),ring_1_of_int(A,archimedean_ceiling(A,divide_divide(A,P2,Q3)))),Q3))) ) ) ).

% ceiling_divide_upper
tff(fact_3402_even__of__int__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),ring_1_of_int(A,K)))
        <=> pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)) ) ) ).

% even_of_int_iff
tff(fact_3403_of__int__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
           => ( ring_1_of_int(A,K) = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),nat2(aa(int,int,uminus_uminus(int),K)))) ) )
          & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
           => ( ring_1_of_int(A,K) = aa(nat,A,semiring_1_of_nat(A),nat2(K)) ) ) ) ) ).

% of_int_of_nat
tff(fact_3404_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q3: A,P2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),Q3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,ring_1_of_int(A,archimedean_ceiling(A,divide_divide(A,P2,Q3)))),one_one(A))),Q3)),P2)) ) ) ).

% ceiling_divide_lower
tff(fact_3405_ceiling__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: int,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),ring_1_of_int(A,N)),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),plus_plus(A),ring_1_of_int(A,N)),one_one(A))))
           => ( archimedean_ceiling(A,X) = aa(int,int,aa(int,fun(int,int),plus_plus(int),N),one_one(int)) ) ) ) ) ).

% ceiling_eq
tff(fact_3406_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( cos(real,X) = one_one(real) )
    <=> ? [X4: int] : X = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),ring_1_of_int(real,X4)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),pi) ) ).

% cos_one_2pi_int
tff(fact_3407_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] : aa(real,real,arccos,cos(real,Theta)) != aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Theta),aa(real,real,aa(real,fun(real,real),times_times(real),ring_1_of_int(real,K2)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_3408_cot__gt__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,cot(real),X))) ) ) ).

% cot_gt_zero
tff(fact_3409_floor__log2__div2,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))
     => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),N))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(nat,real,semiring_1_of_nat(real),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),one_one(int)) ) ) ).

% floor_log2_div2
tff(fact_3410_tan__cot_H,axiom,
    ! [X: real] : aa(real,real,tan(real),aa(real,real,minus_minus(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),X)) = aa(real,real,cot(real),X) ).

% tan_cot'
tff(fact_3411_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( cos(real,X) = zero_zero(real) )
    <=> ? [I3: int] :
          ( ~ pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),I3))
          & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),ring_1_of_int(real,I3)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_3412_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( sin(real,X) = zero_zero(real) )
    <=> ? [I3: int] :
          ( pp(dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),I3))
          & ( X = aa(real,real,aa(real,fun(real,real),times_times(real),ring_1_of_int(real,I3)),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_3413_powr__int,axiom,
    ! [X: real,I2: int] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I2))
         => ( powr(real,X,ring_1_of_int(real,I2)) = aa(nat,real,power_power(real,X),nat2(I2)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),I2))
         => ( powr(real,X,ring_1_of_int(real,I2)) = divide_divide(real,one_one(real),aa(nat,real,power_power(real,X),nat2(aa(int,int,uminus_uminus(int),I2)))) ) ) ) ) ).

% powr_int
tff(fact_3414_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,minus_minus(A,X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),ring_1_of_int(A,Y)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))))
           => ( archimedean_round(A,X) = Y ) ) ) ) ).

% round_unique
tff(fact_3415_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,N: int] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X),ring_1_of_int(A,N)))),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))
         => ( archimedean_round(A,X) = N ) ) ) ).

% round_unique'
tff(fact_3416_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,ring_1_of_int(A,archimedean_round(A,X))),X))),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% of_int_round_abs_le
tff(fact_3417_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,minus_minus(A,X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),ring_1_of_int(A,archimedean_round(A,X)))) ) ).

% of_int_round_gt
tff(fact_3418_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,minus_minus(A,X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))),ring_1_of_int(A,archimedean_round(A,X)))) ) ).

% of_int_round_ge
tff(fact_3419_round__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: num] : archimedean_round(A,aa(num,A,numeral_numeral(A),N)) = aa(num,int,numeral_numeral(int),N) ) ).

% round_numeral
tff(fact_3420_round__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).

% round_1
tff(fact_3421_round__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: num] : archimedean_round(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)) ) ).

% round_neg_numeral
tff(fact_3422_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_round(A,X)),archimedean_round(A,Y))) ) ) ).

% round_mono
tff(fact_3423_floor__le__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archim6421214686448440834_floor(A,X)),archimedean_round(A,X))) ) ).

% floor_le_round
tff(fact_3424_ceiling__ge__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),archimedean_round(A,X)),archimedean_ceiling(A,X))) ) ).

% ceiling_ge_round
tff(fact_3425_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: A,M: int] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Z),ring_1_of_int(A,archimedean_round(A,Z))))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Z),ring_1_of_int(A,M))))) ) ).

% round_diff_minimal
tff(fact_3426_round__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_round(A,X) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))))) ) ).

% round_def
tff(fact_3427_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,archimedean_round(A,X))),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))))) ) ).

% of_int_round_le
tff(fact_3428_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),archimedean_frac(A,X)))
           => ( archimedean_round(A,X) = archimedean_ceiling(A,X) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),archimedean_frac(A,X)))
           => ( archimedean_round(A,X) = archim6421214686448440834_floor(A,X) ) ) ) ) ).

% round_altdef
tff(fact_3429_powr__real__of__int,axiom,
    ! [X: real,N: int] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
         => ( powr(real,X,ring_1_of_int(real,N)) = aa(nat,real,power_power(real,X),nat2(N)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N))
         => ( powr(real,X,ring_1_of_int(real,N)) = aa(real,real,inverse_inverse(real),aa(nat,real,power_power(real,X),nat2(aa(int,int,uminus_uminus(int),N)))) ) ) ) ) ).

% powr_real_of_int
tff(fact_3430_cis__2pi,axiom,
    cis(aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)) = one_one(complex) ).

% cis_2pi
tff(fact_3431_i__even__power,axiom,
    ! [N: nat] : aa(nat,complex,power_power(complex,imaginary_unit),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(nat,complex,power_power(complex,aa(complex,complex,uminus_uminus(complex),one_one(complex))),N) ).

% i_even_power
tff(fact_3432_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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,minus_minus(A,A2),one_one(A))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_3433_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_3434_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_3435_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_3436_inverse__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% inverse_zero
tff(fact_3437_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_3438_inverse__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A] :
          ( ( aa(A,A,inverse_inverse(A),X) = one_one(A) )
        <=> ( X = one_one(A) ) ) ) ).

% inverse_eq_1_iff
tff(fact_3439_inverse__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).

% inverse_1
tff(fact_3440_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_3441_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_3442_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_3443_inverse__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),sgn_sgn(A,A2)) = sgn_sgn(A,A2) ) ).

% inverse_sgn
tff(fact_3444_sgn__inverse,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A] : sgn_sgn(A,aa(A,A,inverse_inverse(A),A2)) = aa(A,A,inverse_inverse(A),sgn_sgn(A,A2)) ) ).

% sgn_inverse
tff(fact_3445_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_3446_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2)) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_3447_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),zero_zero(A))) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_3448_inverse__positive__iff__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ).

% inverse_positive_iff_positive
tff(fact_3449_inverse__negative__iff__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ).

% inverse_negative_iff_negative
tff(fact_3450_inverse__less__iff__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_3451_inverse__less__iff__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ) ).

% inverse_less_iff_less
tff(fact_3452_gbinomial__0_I2_J,axiom,
    ! [B: $tType] :
      ( ( semiring_char_0(B)
        & semidom_divide(B) )
     => ! [K: nat] : aa(nat,B,gbinomial(B,zero_zero(B)),aa(nat,nat,suc,K)) = zero_zero(B) ) ).

% gbinomial_0(2)
tff(fact_3453_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_3454_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_3455_norm__ii,axiom,
    real_V7770717601297561774m_norm(complex,imaginary_unit) = one_one(real) ).

% norm_ii
tff(fact_3456_norm__cis,axiom,
    ! [A2: real] : real_V7770717601297561774m_norm(complex,cis(A2)) = one_one(real) ).

% norm_cis
tff(fact_3457_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_3458_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_3459_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_3460_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_3461_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_3462_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_3463_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),aa(num,num,bit0,one2))))) = one_one(complex) ).

% exp_two_pi_i'
tff(fact_3464_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),aa(num,num,bit0,one2))),real_Vector_of_real(complex,pi))),imaginary_unit)) = one_one(complex) ).

% exp_two_pi_i
tff(fact_3465_cis__pi__half,axiom,
    cis(divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = imaginary_unit ).

% cis_pi_half
tff(fact_3466_power2__i,axiom,
    aa(nat,complex,power_power(complex,imaginary_unit),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% power2_i
tff(fact_3467_cis__minus__pi__half,axiom,
    cis(aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) = aa(complex,complex,uminus_uminus(complex),imaginary_unit) ).

% cis_minus_pi_half
tff(fact_3468_mult__commute__imp__mult__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Y: A,X: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Y),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Y)),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),Y)) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_3469_power__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,N: nat] : aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),A2)),N) = aa(A,A,inverse_inverse(A),aa(nat,A,power_power(A,A2),N)) ) ).

% power_inverse
tff(fact_3470_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_3471_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_3472_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_3473_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_3474_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_3475_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_3476_norm__inverse__le__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [R3: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),R3),real_V7770717601297561774m_norm(A,X)))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),X))),aa(real,real,inverse_inverse(real),R3))) ) ) ) ).

% norm_inverse_le_norm
tff(fact_3477_positive__imp__inverse__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2))) ) ) ).

% positive_imp_inverse_positive
tff(fact_3478_negative__imp__inverse__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A))) ) ) ).

% negative_imp_inverse_negative
tff(fact_3479_inverse__positive__imp__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2)))
         => ( ( A2 != zero_zero(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2)) ) ) ) ).

% inverse_positive_imp_positive
tff(fact_3480_inverse__negative__imp__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A)))
         => ( ( A2 != zero_zero(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% inverse_negative_imp_negative
tff(fact_3481_less__imp__inverse__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2))) ) ) ) ).

% less_imp_inverse_less_neg
tff(fact_3482_inverse__less__imp__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ).

% inverse_less_imp_less_neg
tff(fact_3483_less__imp__inverse__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2))) ) ) ) ).

% less_imp_inverse_less
tff(fact_3484_inverse__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ) ).

% inverse_less_imp_less
tff(fact_3485_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_3486_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_3487_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_3488_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_3489_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_3490_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_3491_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_3492_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_3493_power__mult__inverse__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),M)),aa(A,A,inverse_inverse(A),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),X)),aa(nat,A,power_power(A,X),M)) ) ).

% power_mult_inverse_distrib
tff(fact_3494_power__mult__power__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),M)),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),X)),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),X)),N)),aa(nat,A,power_power(A,X),M)) ) ).

% power_mult_power_inverse_commute
tff(fact_3495_mult__inverse__of__nat__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xa: nat,X: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa))),X) = aa(A,A,aa(A,fun(A,A),times_times(A),X),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xa))) ) ).

% mult_inverse_of_nat_commute
tff(fact_3496_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_3497_divide__real__def,axiom,
    ! [X: real,Y: real] : divide_divide(real,X,Y) = aa(real,real,aa(real,fun(real,real),times_times(real),X),aa(real,real,inverse_inverse(real),Y)) ).

% divide_real_def
tff(fact_3498_frac__ge__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),archimedean_frac(A,X))) ) ).

% frac_ge_0
tff(fact_3499_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),archimedean_frac(A,X)),one_one(A))) ) ).

% frac_lt_1
tff(fact_3500_frac__1__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] : archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),one_one(A))) = archimedean_frac(A,X) ) ).

% frac_1_eq
tff(fact_3501_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),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_3502_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_3503_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2))) ) ) ) ).

% le_imp_inverse_le
tff(fact_3504_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% inverse_le_imp_le
tff(fact_3505_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),X)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),X)) ) ) ) ).

% inverse_le_1_iff
tff(fact_3506_one__less__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),A2))) ) ) ) ).

% one_less_inverse
tff(fact_3507_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(A,A,inverse_inverse(A),X)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ) ).

% one_less_inverse_iff
tff(fact_3508_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_3509_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_3510_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_3511_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,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,minus_minus(A,B2),A2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_diff
tff(fact_3512_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_3513_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_3514_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N)),K) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,nat,minus_minus(nat,N),K)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_3515_inverse__powr,axiom,
    ! [Y: real,A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),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_3516_Complex__eq__i,axiom,
    ! [X: real,Y: real] :
      ( ( complex2(X,Y) = imaginary_unit )
    <=> ( ( X = zero_zero(real) )
        & ( Y = one_one(real) ) ) ) ).

% Complex_eq_i
tff(fact_3517_imaginary__unit_Ocode,axiom,
    imaginary_unit = complex2(zero_zero(real),one_one(real)) ).

% imaginary_unit.code
tff(fact_3518_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ) ).

% inverse_le_iff
tff(fact_3519_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)))
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) )
            & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ) ) ).

% inverse_less_iff
tff(fact_3520_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),one_one(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),A2))) ) ) ) ).

% one_le_inverse
tff(fact_3521_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),X)),one_one(A)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),X)) ) ) ) ).

% inverse_less_1_iff
tff(fact_3522_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),X)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),one_one(A))) ) ) ) ).

% one_le_inverse_iff
tff(fact_3523_inverse__diff__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(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,minus_minus(A,A2),B2))),aa(A,A,inverse_inverse(A),B2))) ) ) ) ) ).

% inverse_diff_inverse
tff(fact_3524_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ? [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N2)))),X)) ) ) ).

% reals_Archimedean
tff(fact_3525_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,minus_minus(A,A2),one_one(A))),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),K)) ) ).

% gbinomial_addition_formula
tff(fact_3526_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,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,minus_minus(A,A2),one_one(A))),K)) ) ).

% gbinomial_absorb_comp
tff(fact_3527_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),K)),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(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_3528_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_3529_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_3530_forall__pos__mono__1,axiom,
    ! [P: fun(real,bool),E2: real] :
      ( ! [D3: real,E: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),D3),E))
         => ( pp(aa(real,bool,P,D3))
           => pp(aa(real,bool,P,E)) ) )
     => ( ! [N2: nat] : pp(aa(real,bool,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N2)))))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
         => pp(aa(real,bool,P,E2)) ) ) ) ).

% forall_pos_mono_1
tff(fact_3531_forall__pos__mono,axiom,
    ! [P: fun(real,bool),E2: real] :
      ( ! [D3: real,E: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),D3),E))
         => ( pp(aa(real,bool,P,D3))
           => pp(aa(real,bool,P,E)) ) )
     => ( ! [N2: nat] :
            ( ( N2 != zero_zero(nat) )
           => pp(aa(real,bool,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N2)))) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
         => pp(aa(real,bool,P,E2)) ) ) ) ).

% forall_pos_mono
tff(fact_3532_real__arch__inverse,axiom,
    ! [E2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
    <=> ? [N5: nat] :
          ( ( N5 != zero_zero(nat) )
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N5))))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N5))),E2)) ) ) ).

% real_arch_inverse
tff(fact_3533_sqrt__divide__self__eq,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( divide_divide(real,aa(real,real,sqrt,X),X) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X)) ) ) ).

% sqrt_divide_self_eq
tff(fact_3534_ln__inverse,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(real,real,ln_ln(real),aa(real,real,inverse_inverse(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,ln_ln(real),X)) ) ) ).

% ln_inverse
tff(fact_3535_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),X))
         => ? [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N2))),X)) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_3536_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [X: A,M: nat,N: nat] :
          ( ( X != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => ( aa(nat,A,power_power(A,X),aa(nat,nat,minus_minus(nat,N),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),N)),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),X)),M)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_3537_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_3538_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,minus_minus(A,A2),one_one(A))),K)) ) ).

% gbinomial_absorption
tff(fact_3539_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,M: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),M))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),M)),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),M)),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,nat,minus_minus(nat,M),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_3540_log__inverse,axiom,
    ! [A2: real,X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
     => ( ( A2 != one_one(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( aa(real,real,log(A2),aa(real,real,inverse_inverse(real),X)) = aa(real,real,uminus_uminus(real),aa(real,real,log(A2),X)) ) ) ) ) ).

% log_inverse
tff(fact_3541_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( archimedean_frac(A,X) = X )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),one_one(A))) ) ) ) ).

% frac_eq
tff(fact_3542_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)) ) ) ) ) ).

% frac_add
tff(fact_3543_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_3544_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_3545_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,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,aa(nat,A,semiring_1_of_nat(A),K)),A2)),one_one(A))),K)) ) ).

% gbinomial_negated_upper
tff(fact_3546_gbinomial__index__swap,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N))),one_one(A))),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),N)) ) ).

% gbinomial_index_swap
tff(fact_3547_exp__plus__inverse__exp,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),exp(real,X)),aa(real,real,inverse_inverse(real),exp(real,X))))) ).

% exp_plus_inverse_exp
tff(fact_3548_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,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_3549_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,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),K)) ) ).

% gbinomial_minus
tff(fact_3550_plus__inverse__ge__2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,inverse_inverse(real),X)))) ) ).

% plus_inverse_ge_2
tff(fact_3551_real__inv__sqrt__pow2,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( aa(nat,real,power_power(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(real,real,inverse_inverse(real),X) ) ) ).

% real_inv_sqrt_pow2
tff(fact_3552_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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,minus_minus(A,A2),one_one(A))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),K)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_3553_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,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_3554_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,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_3555_tan__cot,axiom,
    ! [X: real] : aa(real,real,tan(real),aa(real,real,minus_minus(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),X)) = aa(real,real,inverse_inverse(real),aa(real,real,tan(real),X)) ).

% tan_cot
tff(fact_3556_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,X)),archimedean_frac(A,Y))),one_one(A)))
           => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y))),one_one(int)) ) ) ) ) ).

% floor_add
tff(fact_3557_real__le__x__sinh,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),divide_divide(real,aa(real,real,minus_minus(real,exp(real,X)),aa(real,real,inverse_inverse(real),exp(real,X))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ) ).

% real_le_x_sinh
tff(fact_3558_real__le__abs__sinh,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),aa(real,real,abs_abs(real),divide_divide(real,aa(real,real,minus_minus(real,exp(real,X)),aa(real,real,inverse_inverse(real),exp(real,X))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))) ).

% real_le_abs_sinh
tff(fact_3559_tan__sec,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,power_power(A,aa(A,A,tan(A),X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),cos(A,X))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ) ) ) ).

% tan_sec
tff(fact_3560_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),aa(num,num,bit0,one2))) ).

% Arg_minus_ii
tff(fact_3561_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),aa(num,num,bit0,one2))))) ).

% csqrt_ii
tff(fact_3562_Arg__ii,axiom,
    arg(imaginary_unit) = divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ).

% Arg_ii
tff(fact_3563_sinh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( sinh(real,aa(real,real,ln_ln(real),X)) = divide_divide(real,aa(real,real,minus_minus(real,X),aa(real,real,inverse_inverse(real),X)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ).

% sinh_ln_real
tff(fact_3564_cis__multiple__2pi,axiom,
    ! [N: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),N),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),aa(num,num,bit0,one2))),pi)),N)) = one_one(complex) ) ) ).

% cis_multiple_2pi
tff(fact_3565_sinh__real__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),sinh(real,X)),sinh(real,Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ).

% sinh_real_less_iff
tff(fact_3566_sinh__real__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),sinh(real,X)),sinh(real,Y)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y)) ) ).

% sinh_real_le_iff
tff(fact_3567_sinh__real__neg__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),sinh(real,X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real))) ) ).

% sinh_real_neg_iff
tff(fact_3568_sinh__real__pos__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),sinh(real,X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X)) ) ).

% sinh_real_pos_iff
tff(fact_3569_sinh__real__nonneg__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),sinh(real,X)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X)) ) ).

% sinh_real_nonneg_iff
tff(fact_3570_sinh__real__nonpos__iff,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),sinh(real,X)),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) ) ).

% sinh_real_nonpos_iff
tff(fact_3571_floor__add2,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,Y: A] :
          ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
            | pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),ring_1_Ints(A))) )
         => ( archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,X)),archim6421214686448440834_floor(A,Y)) ) ) ) ).

% floor_add2
tff(fact_3572_frac__gt__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),archimedean_frac(A,X)))
        <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A))) ) ) ).

% frac_gt_0_iff
tff(fact_3573_power2__csqrt,axiom,
    ! [Z: complex] : aa(nat,complex,power_power(complex,csqrt(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = Z ).

% power2_csqrt
tff(fact_3574_Ints__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,power_power(A,A2),N)),ring_1_Ints(A))) ) ) ).

% Ints_power
tff(fact_3575_Ints__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: num] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(num,A,numeral_numeral(A),N)),ring_1_Ints(A))) ) ).

% Ints_numeral
tff(fact_3576_Ints__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),one_one(A)),ring_1_Ints(A))) ) ).

% Ints_1
tff(fact_3577_Ints__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),ring_1_Ints(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),ring_1_Ints(A))) ) ) ) ).

% Ints_add
tff(fact_3578_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),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_3579_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),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_3580_of__int__divide__in__Ints,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: int,A2: int] :
          ( pp(dvd_dvd(int,B2,A2))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),divide_divide(A,ring_1_of_int(A,A2),ring_1_of_int(A,B2))),ring_1_Ints(A))) ) ) ).

% of_int_divide_in_Ints
tff(fact_3581_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),ring_1_Ints(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),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)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),zero_zero(A))) ) ) ) ).

% Ints_odd_less_0
tff(fact_3582_Ints__nonzero__abs__ge1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
         => ( ( X != zero_zero(A) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(A,A,abs_abs(A),X))) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_3583_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),X)),one_one(A)))
           => ( X = zero_zero(A) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_3584_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),ring_1_Ints(A)))
           => ( ( X = Y )
            <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X),Y))),one_one(A))) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_3585_of__real__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( real_Vector_of_real(complex,aa(real,real,sqrt,X)) = csqrt(real_Vector_of_real(complex,X)) ) ) ).

% of_real_sqrt
tff(fact_3586_frac__neg,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A] :
          ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
           => ( archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = zero_zero(A) ) )
          & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),ring_1_Ints(A)))
           => ( archimedean_frac(A,aa(A,A,uminus_uminus(A),X)) = aa(A,A,minus_minus(A,one_one(A)),archimedean_frac(A,X)) ) ) ) ) ).

% frac_neg
tff(fact_3587_le__mult__floor__Ints,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & linordered_idom(A) )
     => ! [A2: B,B2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),A2))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),ring_1_Ints(B)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(B,A2)),archim6421214686448440834_floor(B,B2)))),ring_1_of_int(A,archim6421214686448440834_floor(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2))))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_3588_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [X: A,A2: A] :
          ( ( archimedean_frac(A,X) = A2 )
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,minus_minus(A,X),A2)),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),A2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),one_one(A))) ) ) ) ).

% frac_unique_iff
tff(fact_3589_mult__ceiling__le__Ints,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & linordered_idom(A) )
     => ! [A2: B,B2: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),A2))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),ring_1_Ints(B)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),ring_1_of_int(A,archimedean_ceiling(B,aa(B,B,aa(B,fun(B,B),times_times(B),A2),B2)))),ring_1_of_int(A,aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(B,A2)),archimedean_ceiling(B,B2))))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_3590_Arg__bounded,axiom,
    ! [Z: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),pi)),arg(Z)))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),arg(Z)),pi)) ) ).

% Arg_bounded
tff(fact_3591_sin__integer__2pi,axiom,
    ! [N: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),N),ring_1_Ints(real)))
     => ( sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)),N)) = zero_zero(real) ) ) ).

% sin_integer_2pi
tff(fact_3592_cos__integer__2pi,axiom,
    ! [N: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),N),ring_1_Ints(real)))
     => ( cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),pi)),N)) = one_one(real) ) ) ).

% cos_integer_2pi
tff(fact_3593_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,power_power(real,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,B2),aa(num,nat,numeral_numeral(nat),aa(num,num,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,power_power(real,A2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,B2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% complex_inverse
tff(fact_3594_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,minus_minus(A,exp(A,Z)),exp(A,aa(A,A,uminus_uminus(A),Z))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% sinh_field_def
tff(fact_3595_cis__Arg__unique,axiom,
    ! [Z: complex,X: real] :
      ( ( sgn_sgn(complex,Z) = cis(X) )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),pi)),X))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),pi))
         => ( arg(Z) = X ) ) ) ) ).

% cis_Arg_unique
tff(fact_3596_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( ( sgn_sgn(complex,Z) = cis(arg(Z)) )
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),pi)),arg(Z)))
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),arg(Z)),pi)) ) ) ).

% Arg_correct
tff(fact_3597_cosh__ln__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( cosh(real,aa(real,real,ln_ln(real),X)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),aa(real,real,inverse_inverse(real),X)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ).

% cosh_ln_real
tff(fact_3598_cosh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : cosh(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,cosh(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,power_power(A,sinh(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ).

% cosh_double
tff(fact_3599_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list(bool)] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),groups4207007520872428315er_sum(bool,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),Bs)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(list(bool),nat,size_size(list(bool)),Bs)))) ).

% horner_sum_of_bool_2_less
tff(fact_3600_cosh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cosh(A,X) = zero_zero(A) )
        <=> ( aa(nat,A,power_power(A,exp(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% cosh_zero_iff
tff(fact_3601_push__bit__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),N)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),N))) ) ).

% push_bit_numeral_minus_1
tff(fact_3602_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se4730199178511100633sh_bit(int,N),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% push_bit_nonnegative_int_iff
tff(fact_3603_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_se4730199178511100633sh_bit(int,N),K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% push_bit_negative_int_iff
tff(fact_3604_push__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A2: A] :
          ( ( aa(A,A,bit_se4730199178511100633sh_bit(A,N),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_3605_push__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).

% push_bit_of_0
tff(fact_3606_push__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,M),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A2)) = aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),A2) ) ).

% push_bit_push_bit
tff(fact_3607_push__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A2)),aa(A,A,bit_se4730199178511100633sh_bit(A,N),B2)) ) ).

% push_bit_and
tff(fact_3608_push__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A2)),aa(A,A,bit_se4730199178511100633sh_bit(A,N),B2)) ) ).

% push_bit_xor
tff(fact_3609_concat__bit__of__zero__1,axiom,
    ! [N: nat,L: int] : aa(int,int,bit_concat_bit(N,zero_zero(int)),L) = aa(int,int,bit_se4730199178511100633sh_bit(int,N),L) ).

% concat_bit_of_zero_1
tff(fact_3610_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_3611_push__bit__Suc__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,K: num] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),K)) = aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) ) ).

% push_bit_Suc_numeral
tff(fact_3612_push__bit__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,K: num] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K)))) ) ).

% push_bit_Suc_minus_numeral
tff(fact_3613_push__bit__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),K)) = aa(A,A,bit_se4730199178511100633sh_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) ) ).

% push_bit_numeral
tff(fact_3614_push__bit__minus__one__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)) ) ).

% push_bit_minus_one_eq_not_mask
tff(fact_3615_push__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N)),A2) = aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% push_bit_Suc
tff(fact_3616_push__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A)) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) ) ).

% push_bit_of_1
tff(fact_3617_push__bit__of__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,bit_se4730199178511100633sh_bit(nat,N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) ).

% push_bit_of_Suc_0
tff(fact_3618_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A2)))
        <=> ( ( N != zero_zero(nat) )
            | pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2)) ) ) ) ).

% even_push_bit_iff
tff(fact_3619_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),aa(num,num,bit0,K)))) ) ).

% push_bit_minus_numeral
tff(fact_3620_push__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,K: int] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),ring_1_of_int(A,K)) = ring_1_of_int(A,aa(int,int,bit_se4730199178511100633sh_bit(int,N),K)) ) ).

% push_bit_of_int
tff(fact_3621_of__nat__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,M),N)) = aa(A,A,bit_se4730199178511100633sh_bit(A,M),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_push_bit
tff(fact_3622_push__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,N),M)) ) ).

% push_bit_of_nat
tff(fact_3623_push__bit__nat__eq,axiom,
    ! [N: nat,K: int] : aa(nat,nat,bit_se4730199178511100633sh_bit(nat,N),nat2(K)) = nat2(aa(int,int,bit_se4730199178511100633sh_bit(int,N),K)) ).

% push_bit_nat_eq
tff(fact_3624_push__bit__minus,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A2)) ) ).

% push_bit_minus
tff(fact_3625_push__bit__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),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,N),A2)),aa(A,A,bit_se4730199178511100633sh_bit(A,N),B2)) ) ).

% push_bit_add
tff(fact_3626_cosh__real__pos,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),cosh(real,X))) ).

% cosh_real_pos
tff(fact_3627_cosh__real__nonpos__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),zero_zero(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),cosh(real,X)),cosh(real,Y)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),X)) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_3628_cosh__real__nonneg__le__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),cosh(real,X)),cosh(real,Y)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y)) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_3629_cosh__real__nonneg,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),cosh(real,X))) ).

% cosh_real_nonneg
tff(fact_3630_cosh__real__ge__1,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),cosh(real,X))) ).

% cosh_real_ge_1
tff(fact_3631_push__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),aa(A,A,bit_se4730199178511100633sh_bit(A,M),A2)) ) ).

% push_bit_take_bit
tff(fact_3632_take__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A2)) = aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,minus_minus(nat,M),N)),A2)) ) ).

% take_bit_push_bit
tff(fact_3633_sinh__less__cosh__real,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),sinh(real,X)),cosh(real,X))) ).

% sinh_less_cosh_real
tff(fact_3634_sinh__le__cosh__real,axiom,
    ! [X: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),sinh(real,X)),cosh(real,X))) ).

% sinh_le_cosh_real
tff(fact_3635_flip__bit__nat__def,axiom,
    ! [M: nat,N: nat] : bit_se8732182000553998342ip_bit(nat,M,N) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),N),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,M),one_one(nat))) ).

% flip_bit_nat_def
tff(fact_3636_cosh__real__strict__mono,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cosh(real,X)),cosh(real,Y))) ) ) ).

% cosh_real_strict_mono
tff(fact_3637_cosh__real__nonneg__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cosh(real,X)),cosh(real,Y)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_3638_cosh__real__nonpos__less__iff,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y),zero_zero(real)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),cosh(real,X)),cosh(real,Y)))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y),X)) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_3639_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,bit_se4730199178511100633sh_bit(int,M),K)),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),aa(nat,nat,minus_minus(nat,N),M))) ) ) ).

% bit_push_bit_iff_int
tff(fact_3640_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q3: nat,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,aa(nat,nat,bit_se4730199178511100633sh_bit(nat,M),Q3)),N))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
        & pp(aa(nat,bool,bit_se5641148757651400278ts_bit(nat,Q3),aa(nat,nat,minus_minus(nat,N),M))) ) ) ).

% bit_push_bit_iff_nat
tff(fact_3641_concat__bit__eq,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_concat_bit(N,K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(int,int,bit_se4730199178511100633sh_bit(int,N),L)) ).

% concat_bit_eq
tff(fact_3642_arcosh__cosh__real,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( aa(real,real,arcosh(real),cosh(real,X)) = X ) ) ).

% arcosh_cosh_real
tff(fact_3643_flip__bit__eq__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : bit_se8732182000553998342ip_bit(A,N,A2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A))) ) ).

% flip_bit_eq_xor
tff(fact_3644_cosh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : cosh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,X)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,X)),sinh(A,Y))) ) ).

% cosh_add
tff(fact_3645_sinh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sinh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,X)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,X)),sinh(A,Y))) ) ).

% sinh_add
tff(fact_3646_cosh__plus__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),cosh(A,X)),sinh(A,X)) = exp(A,X) ) ).

% cosh_plus_sinh
tff(fact_3647_sinh__plus__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),sinh(A,X)),cosh(A,X)) = exp(A,X) ) ).

% sinh_plus_cosh
tff(fact_3648_flip__bit__int__def,axiom,
    ! [N: nat,K: int] : bit_se8732182000553998342ip_bit(int,N,K) = aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),aa(int,int,bit_se4730199178511100633sh_bit(int,N),one_one(int))) ).

% flip_bit_int_def
tff(fact_3649_tanh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,tanh(A),X) = divide_divide(A,sinh(A,X),cosh(A,X)) ) ).

% tanh_def
tff(fact_3650_push__bit__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A2)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% push_bit_double
tff(fact_3651_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A))) != zero_zero(A) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_3652_push__bit__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] : aa(A,A,bit_se4730199178511100633sh_bit(A,M),bit_se2239418461657761734s_mask(A,N)) = 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),N),M))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,M))) ) ).

% push_bit_mask_eq
tff(fact_3653_unset__bit__eq__and__not,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),N),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,N),one_one(A)))) ) ).

% unset_bit_eq_and_not
tff(fact_3654_unset__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),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,N),one_one(int)))) ).

% unset_bit_int_def
tff(fact_3655_push__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_se4730199178511100633sh_bit(int,N),K) = aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ).

% push_bit_int_def
tff(fact_3656_push__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,bit_se4730199178511100633sh_bit(nat,N),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ).

% push_bit_nat_def
tff(fact_3657_push__bit__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) ) ).

% push_bit_eq_mult
tff(fact_3658_exp__dvdE,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( pp(dvd_dvd(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N),A2))
         => ~ ! [B5: A] : A2 != aa(A,A,bit_se4730199178511100633sh_bit(A,N),B5) ) ) ).

% exp_dvdE
tff(fact_3659_sinh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : sinh(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),X)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),sinh(A,X))),cosh(A,X)) ) ).

% sinh_double
tff(fact_3660_push__bit__minus__one,axiom,
    ! [N: nat] : aa(int,int,bit_se4730199178511100633sh_bit(int,N),aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ).

% push_bit_minus_one
tff(fact_3661_tanh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] :
          ( ( cosh(A,X) != zero_zero(A) )
         => ( ( cosh(A,Y) != zero_zero(A) )
           => ( aa(A,A,tanh(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tanh(A),X)),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),X)),aa(A,A,tanh(A),Y)))) ) ) ) ) ).

% tanh_add
tff(fact_3662_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),aa(num,num,bit0,one2))) ) ).

% cosh_field_def
tff(fact_3663_cosh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,power_power(A,cosh(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,power_power(A,sinh(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% cosh_square_eq
tff(fact_3664_sinh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(nat,A,power_power(A,sinh(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,cosh(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(A)) ) ).

% sinh_square_eq
tff(fact_3665_hyperbolic__pythagoras,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : aa(A,A,minus_minus(A,aa(nat,A,power_power(A,cosh(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,A,power_power(A,sinh(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(A) ) ).

% hyperbolic_pythagoras
tff(fact_3666_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs: list(bool),N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,groups4207007520872428315er_sum(bool,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),Bs)),N))
        <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(bool),nat,size_size(list(bool)),Bs)))
            & pp(aa(nat,bool,nth(bool,Bs),N)) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_3667_Cauchy__iff2,axiom,
    ! [X6: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X6)
    <=> ! [J3: nat] :
        ? [M8: nat] :
        ! [M7: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),M7))
         => ! [N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),N5))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(nat,real,X6,M7)),aa(nat,real,X6,N5)))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3))))) ) ) ) ).

% Cauchy_iff2
tff(fact_3668_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ vEBT_VEBT_membermima(X,Xa)
     => ( ! [Uu2: bool,Uv2: bool] : X != vEBT_Leaf(Uu2,Uv2)
       => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : X != vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
         => ( ! [Mi3: nat,Ma3: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = 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)),zero_zero(nat),Va3,Vb2)
               => ( ( Xa = Mi3 )
                  | ( Xa = Ma3 ) ) )
           => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList3: list(vEBT_VEBT)] :
                  ( ? [Vc: vEBT_VEBT] : X = 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)),aa(nat,nat,suc,V3),TreeList3,Vc)
                 => ( ( Xa = Mi3 )
                    | ( Xa = Ma3 )
                    | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3)))
                       => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3))) ) ) )
             => ~ ! [V3: nat,TreeList3: list(vEBT_VEBT)] :
                    ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList3,Vd)
                   => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3)))
                       => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_3669_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: bool] :
      ( ( vEBT_VEBT_membermima(X,Xa)
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => pp(Y) )
       => ( ( ? [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
           => pp(Y) )
         => ( ! [Mi3: nat,Ma3: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = 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)),zero_zero(nat),Va3,Vb2)
               => ( pp(Y)
                <=> ~ ( ( Xa = Mi3 )
                      | ( Xa = Ma3 ) ) ) )
           => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList3: list(vEBT_VEBT)] :
                  ( ? [Vc: vEBT_VEBT] : X = 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)),aa(nat,nat,suc,V3),TreeList3,Vc)
                 => ( pp(Y)
                  <=> ~ ( ( Xa = Mi3 )
                        | ( Xa = Ma3 )
                        | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3)))
                           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3))) ) ) ) )
             => ~ ! [V3: nat,TreeList3: list(vEBT_VEBT)] :
                    ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList3,Vd)
                   => ( pp(Y)
                    <=> ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3)))
                           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_3670_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] : re(csqrt(Z)) = 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),aa(num,num,bit0,one2)))) ).

% csqrt.simps(1)
tff(fact_3671_complex__Re__numeral,axiom,
    ! [V: num] : re(aa(num,complex,numeral_numeral(complex),V)) = aa(num,real,numeral_numeral(real),V) ).

% complex_Re_numeral
tff(fact_3672_Re__divide__numeral,axiom,
    ! [Z: complex,W: num] : re(divide_divide(complex,Z,aa(num,complex,numeral_numeral(complex),W))) = divide_divide(real,re(Z),aa(num,real,numeral_numeral(real),W)) ).

% Re_divide_numeral
tff(fact_3673_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_aa(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_3674_VEBT_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),F2: fun(bool,fun(bool,A)),VEBT: vEBT_VEBT] : aa(A,B,H,vEBT_case_VEBT(A,F1,F2,VEBT)) = vEBT_case_VEBT(B,aa(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)))),aTP_Lamp_ab(fun(A,B),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))))),H),F1),aa(fun(bool,fun(bool,A)),fun(bool,fun(bool,B)),aTP_Lamp_ac(fun(A,B),fun(fun(bool,fun(bool,A)),fun(bool,fun(bool,B))),H),F2),VEBT) ).

% VEBT.case_distrib
tff(fact_3675_less__set__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
    <=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A3)),aTP_Lamp_a(set(A),fun(A,bool),B3))) ) ).

% less_set_def
tff(fact_3676_less__eq__set__def,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
    <=> pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),A3)),aTP_Lamp_a(set(A),fun(A,bool),B3))) ) ).

% less_eq_set_def
tff(fact_3677_Collect__subset,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,bool)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ad(set(A),fun(fun(A,bool),fun(A,bool)),A3),P))),A3)) ).

% Collect_subset
tff(fact_3678_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aTP_Lamp_ae(A,fun(A,bool),A2))),collect(A,aTP_Lamp_ae(A,fun(A,bool),B2))))
        <=> pp(dvd_dvd(A,A2,B2)) ) ) ).

% subset_divisors_dvd
tff(fact_3679_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),collect(A,aTP_Lamp_ae(A,fun(A,bool),A2))),collect(A,aTP_Lamp_ae(A,fun(A,bool),B2))))
        <=> ( pp(dvd_dvd(A,A2,B2))
            & ~ pp(dvd_dvd(A,B2,A2)) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_3680_numeral__code_I2_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit0,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N)) ) ).

% numeral_code(2)
tff(fact_3681_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_af(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_3682_nat__less__as__int,axiom,
    ! [X2: nat,Xa2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X2),Xa2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(nat,int,semiring_1_of_nat(int),X2)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ) ).

% nat_less_as_int
tff(fact_3683_nat__leq__as__int,axiom,
    ! [X2: nat,Xa2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X2),Xa2))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),X2)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ) ).

% nat_leq_as_int
tff(fact_3684_numeral__code_I3_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)) = 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),N)),aa(num,A,numeral_numeral(A),N))),one_one(A)) ) ).

% numeral_code(3)
tff(fact_3685_power__numeral__even,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z: A,W: num] : aa(nat,A,power_power(A,Z),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,W))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Z),aa(num,nat,numeral_numeral(nat),W))),aa(nat,A,power_power(A,Z),aa(num,nat,numeral_numeral(nat),W))) ) ).

% power_numeral_even
tff(fact_3686_power__numeral__odd,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z: A,W: num] : aa(nat,A,power_power(A,Z),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,W))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),aa(nat,A,power_power(A,Z),aa(num,nat,numeral_numeral(nat),W)))),aa(nat,A,power_power(A,Z),aa(num,nat,numeral_numeral(nat),W))) ) ).

% power_numeral_odd
tff(fact_3687_nat__plus__as__int,axiom,
    ! [X2: nat,Xa2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X2),Xa2) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X2)),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_plus_as_int
tff(fact_3688_nat__div__as__int,axiom,
    ! [X2: nat,Xa2: nat] : divide_divide(nat,X2,Xa2) = nat2(divide_divide(int,aa(nat,int,semiring_1_of_nat(int),X2),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_div_as_int
tff(fact_3689_nat__mod__as__int,axiom,
    ! [X2: nat,Xa2: nat] : modulo_modulo(nat,X2,Xa2) = nat2(modulo_modulo(int,aa(nat,int,semiring_1_of_nat(int),X2),aa(nat,int,semiring_1_of_nat(int),Xa2))) ).

% nat_mod_as_int
tff(fact_3690_complex__Re__le__cmod,axiom,
    ! [X: complex] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),re(X)),real_V7770717601297561774m_norm(complex,X))) ).

% complex_Re_le_cmod
tff(fact_3691_one__complex_Osimps_I1_J,axiom,
    re(one_one(complex)) = one_one(real) ).

% one_complex.simps(1)
tff(fact_3692_set__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = collect(A,aTP_Lamp_ag(list(A),fun(A,bool),Xs)) ).

% set_conv_nth
tff(fact_3693_diff__nat__eq__if,axiom,
    ! [Z4: int,Z: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z4),zero_zero(int)))
       => ( aa(nat,nat,minus_minus(nat,nat2(Z)),nat2(Z4)) = nat2(Z) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z4),zero_zero(int)))
       => ( aa(nat,nat,minus_minus(nat,nat2(Z)),nat2(Z4)) = if(nat,aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,minus_minus(int,Z),Z4)),zero_zero(int)),zero_zero(nat),nat2(aa(int,int,minus_minus(int,Z),Z4))) ) ) ) ).

% diff_nat_eq_if
tff(fact_3694_abs__Re__le__cmod,axiom,
    ! [X: complex] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),re(X))),real_V7770717601297561774m_norm(complex,X))) ).

% abs_Re_le_cmod
tff(fact_3695_Re__csqrt,axiom,
    ! [Z: complex] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),re(csqrt(Z)))) ).

% Re_csqrt
tff(fact_3696_set__decode__def,axiom,
    ! [X: nat] : nat_set_decode(X) = collect(nat,aTP_Lamp_ah(nat,fun(nat,bool),X)) ).

% set_decode_def
tff(fact_3697_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = if(A,aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A2)),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A2)),aa(A,A,bit_se4730199178511100633sh_bit(A,aa(nat,nat,suc,N)),aa(A,A,uminus_uminus(A),one_one(A)))),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A2)) ) ).

% signed_take_bit_code
tff(fact_3698_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat,A2: A] :
          ( ( ( N = zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,A2,N) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( comm_s3205402744901411588hammer(A,A2,N) = set_fo6178422350223883121st_nat(A,aTP_Lamp_ai(A,fun(nat,fun(A,A)),A2),zero_zero(nat),aa(nat,nat,minus_minus(nat,N),one_one(nat)),one_one(A)) ) ) ) ) ).

% pochhammer_code
tff(fact_3699_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),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_3700_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( ( ( K = zero_zero(nat) )
           => ( aa(nat,A,gbinomial(A,A2),K) = one_one(A) ) )
          & ( ( K != zero_zero(nat) )
           => ( aa(nat,A,gbinomial(A,A2),K) = divide_divide(A,set_fo6178422350223883121st_nat(A,aTP_Lamp_aj(A,fun(nat,fun(A,A)),A2),zero_zero(nat),aa(nat,nat,minus_minus(nat,K),one_one(nat)),one_one(A)),semiring_char_0_fact(A,K)) ) ) ) ) ).

% gbinomial_code
tff(fact_3701_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option(product_prod(nat,nat)),V: nat,TreeList: list(vEBT_VEBT),S: vEBT_VEBT,X: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Node(Uy,aa(nat,nat,suc,V),TreeList,S),X)
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
         => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(X,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_3702_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),E2: real] :
          ( topolo3814608138187158403Cauchy(A,X6)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
           => ? [M9: nat] :
              ! [M2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),M2))
               => ! [N7: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),N7))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X6,M2)),aa(nat,A,X6,N7)))),E2)) ) ) ) ) ) ).

% CauchyD
tff(fact_3703_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
             => ? [M10: nat] :
                ! [M5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M10),M5))
                 => ! [N2: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M10),N2))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X6,M5)),aa(nat,A,X6,N2)))),E)) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% CauchyI
tff(fact_3704_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [E3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
             => ? [M8: nat] :
                ! [M7: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),M7))
                 => ! [N5: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),N5))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X6,M7)),aa(nat,A,X6,N5)))),E3)) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_3705_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list(vEBT_VEBT),Vd2: vEBT_VEBT,X: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V),TreeList,Vd2),X)
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
         => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(X,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_3706_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list(vEBT_VEBT),Vc2: vEBT_VEBT,X: 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,Mi),Ma)),aa(nat,nat,suc,V),TreeList,Vc2),X)
    <=> ( ( X = Mi )
        | ( X = Ma )
        | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)))
           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(X,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(X,divide_divide(nat,aa(nat,nat,suc,V),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_3707_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: bool] :
      ( ( vEBT_V5719532721284313246member(X,Xa)
      <=> pp(Y) )
     => ( ! [A5: bool,B5: bool] :
            ( ( X = vEBT_Leaf(A5,B5) )
           => ( pp(Y)
            <=> ~ ( ( ( Xa = zero_zero(nat) )
                   => pp(A5) )
                  & ( ( Xa != zero_zero(nat) )
                   => ( ( ( Xa = one_one(nat) )
                       => pp(B5) )
                      & ( Xa = one_one(nat) ) ) ) ) ) )
       => ( ( ? [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
           => pp(Y) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList3: list(vEBT_VEBT)] :
                ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList3,S3)
               => ( pp(Y)
                <=> ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3)))
                       => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_3708_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_V5719532721284313246member(X,Xa)
     => ( ! [A5: bool,B5: bool] :
            ( ( X = vEBT_Leaf(A5,B5) )
           => ~ ( ( ( Xa = zero_zero(nat) )
                 => pp(A5) )
                & ( ( Xa != zero_zero(nat) )
                 => ( ( ( Xa = one_one(nat) )
                     => pp(B5) )
                    & ( Xa = one_one(nat) ) ) ) ) )
       => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList3: list(vEBT_VEBT)] :
              ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList3,S3)
             => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3)))
                   => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                  & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3))) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_3709_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ vEBT_V5719532721284313246member(X,Xa)
     => ( ! [A5: bool,B5: bool] :
            ( ( X = vEBT_Leaf(A5,B5) )
           => ( ( ( Xa = zero_zero(nat) )
               => pp(A5) )
              & ( ( Xa != zero_zero(nat) )
               => ( ( ( Xa = one_one(nat) )
                   => pp(B5) )
                  & ( Xa = one_one(nat) ) ) ) ) )
       => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : X != vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList3: list(vEBT_VEBT)] :
                ( ? [S3: vEBT_VEBT] : X = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList3,S3)
               => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3)))
                   => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                  & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3))) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_3710_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_membermima(X,Xa)
     => ( ! [Mi3: nat,Ma3: nat] :
            ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : X = 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)),zero_zero(nat),Va3,Vb2)
           => ~ ( ( Xa = Mi3 )
                | ( Xa = Ma3 ) ) )
       => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList3: list(vEBT_VEBT)] :
              ( ? [Vc: vEBT_VEBT] : X = 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)),aa(nat,nat,suc,V3),TreeList3,Vc)
             => ~ ( ( Xa = Mi3 )
                  | ( Xa = Ma3 )
                  | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3)))
                     => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                    & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3))) ) ) )
         => ~ ! [V3: nat,TreeList3: list(vEBT_VEBT)] :
                ( ? [Vd: vEBT_VEBT] : X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList3,Vd)
               => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3)))
                     => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                    & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3))) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_3711_of__int__code__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          ( ( ( K = zero_zero(int) )
           => ( ring_1_of_int(A,K) = zero_zero(A) ) )
          & ( ( K != zero_zero(int) )
           => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
               => ( ring_1_of_int(A,K) = aa(A,A,uminus_uminus(A),ring_1_of_int(A,aa(int,int,uminus_uminus(int),K))) ) )
              & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
               => ( ring_1_of_int(A,K) = if(A,aa(int,bool,aa(int,fun(int,bool),fequal(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),zero_zero(int)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),ring_1_of_int(A,divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),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),aa(num,num,bit0,one2))),ring_1_of_int(A,divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))),one_one(A))) ) ) ) ) ) ) ).

% of_int_code_if
tff(fact_3712_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => topological_monoseq(real,aTP_Lamp_ak(real,fun(nat,real),X)) ) ).

% monoseq_arctan_series
tff(fact_3713_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),aa(num,num,bit0,one2)))),aa(real,real,aa(real,fun(real,real),times_times(real),if(real,aa(real,bool,aa(real,fun(real,bool),fequal(real),im(Z)),zero_zero(real)),one_one(real),sgn_sgn(real,im(Z)))),aa(real,real,sqrt,divide_divide(real,aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(complex,Z)),re(Z)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))) ).

% csqrt.code
tff(fact_3714_ln__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
       => ( aa(real,real,ln_ln(real),X) = suminf(real,aTP_Lamp_al(real,fun(nat,real),X)) ) ) ) ).

% ln_series
tff(fact_3715_predicate1I,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P,X3))
         => pp(aa(A,bool,Q,X3)) )
     => pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Q)) ) ).

% predicate1I
tff(fact_3716_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F3: fun(nat,A)] : suminf(A,aTP_Lamp_am(fun(nat,A),fun(nat,A),F3)) = aa(nat,A,F3,zero_zero(nat)) ) ).

% powser_zero
tff(fact_3717_Re__power__real,axiom,
    ! [X: complex,N: nat] :
      ( ( im(X) = zero_zero(real) )
     => ( re(aa(nat,complex,power_power(complex,X),N)) = aa(nat,real,power_power(real,re(X)),N) ) ) ).

% Re_power_real
tff(fact_3718_Im__divide__numeral,axiom,
    ! [Z: complex,W: num] : im(divide_divide(complex,Z,aa(num,complex,numeral_numeral(complex),W))) = divide_divide(real,im(Z),aa(num,real,numeral_numeral(real),W)) ).

% Im_divide_numeral
tff(fact_3719_csqrt__of__real__nonneg,axiom,
    ! [X: complex] :
      ( ( im(X) = zero_zero(real) )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),re(X)))
       => ( csqrt(X) = real_Vector_of_real(complex,aa(real,real,sqrt,re(X))) ) ) ) ).

% csqrt_of_real_nonneg
tff(fact_3720_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),im(X)),zero_zero(real)))
        | ( ( im(X) = zero_zero(real) )
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),re(X))) ) )
     => ( csqrt(aa(complex,complex,uminus_uminus(complex),X)) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),csqrt(X)) ) ) ).

% csqrt_minus
tff(fact_3721_csqrt__of__real__nonpos,axiom,
    ! [X: complex] :
      ( ( im(X) = zero_zero(real) )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),re(X)),zero_zero(real)))
       => ( csqrt(X) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,aa(real,real,sqrt,aa(real,real,abs_abs(real),re(X))))) ) ) ) ).

% csqrt_of_real_nonpos
tff(fact_3722_predicate1D,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),X: A] :
      ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Q))
     => ( pp(aa(A,bool,P,X))
       => pp(aa(A,bool,Q,X)) ) ) ).

% predicate1D
tff(fact_3723_rev__predicate1D,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Q: fun(A,bool)] :
      ( pp(aa(A,bool,P,X))
     => ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),P),Q))
       => pp(aa(A,bool,Q,X)) ) ) ).

% rev_predicate1D
tff(fact_3724_imaginary__unit_Osimps_I2_J,axiom,
    im(imaginary_unit) = one_one(real) ).

% imaginary_unit.simps(2)
tff(fact_3725_abs__Im__le__cmod,axiom,
    ! [X: complex] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),im(X))),real_V7770717601297561774m_norm(complex,X))) ).

% abs_Im_le_cmod
tff(fact_3726_cmod__Im__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( re(X) = re(Y) )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(complex,X)),real_V7770717601297561774m_norm(complex,Y)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),im(X))),aa(real,real,abs_abs(real),im(Y)))) ) ) ).

% cmod_Im_le_iff
tff(fact_3727_cmod__Re__le__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( im(X) = im(Y) )
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(complex,X)),real_V7770717601297561774m_norm(complex,Y)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),re(X))),aa(real,real,abs_abs(real),re(Y)))) ) ) ).

% cmod_Re_le_iff
tff(fact_3728_csqrt__principal,axiom,
    ! [Z: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),re(csqrt(Z))))
      | ( ( re(csqrt(Z)) = zero_zero(real) )
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),im(csqrt(Z)))) ) ) ).

% csqrt_principal
tff(fact_3729_cmod__le,axiom,
    ! [Z: complex] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(complex,Z)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,abs_abs(real),re(Z))),aa(real,real,abs_abs(real),im(Z))))) ).

% cmod_le
tff(fact_3730_monoseq__realpow,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => topological_monoseq(real,power_power(real,X)) ) ) ).

% monoseq_realpow
tff(fact_3731_cmod__power2,axiom,
    ! [Z: complex] : aa(nat,real,power_power(real,real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,im(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ).

% cmod_power2
tff(fact_3732_Im__power2,axiom,
    ! [X: complex] : im(aa(nat,complex,power_power(complex,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),re(X))),im(X)) ).

% Im_power2
tff(fact_3733_Re__power2,axiom,
    ! [X: complex] : re(aa(nat,complex,power_power(complex,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(real,real,minus_minus(real,aa(nat,real,power_power(real,re(X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,im(X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ).

% Re_power2
tff(fact_3734_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero(complex) )
    <=> ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,im(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = zero_zero(real) ) ) ).

% complex_eq_0
tff(fact_3735_norm__complex__def,axiom,
    ! [Z: complex] : real_V7770717601297561774m_norm(complex,Z) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,im(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% norm_complex_def
tff(fact_3736_inverse__complex_Osimps_I1_J,axiom,
    ! [X: complex] : re(aa(complex,complex,inverse_inverse(complex),X)) = divide_divide(real,re(X),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,im(X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% inverse_complex.simps(1)
tff(fact_3737_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,im(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% complex_neq_0
tff(fact_3738_Re__divide,axiom,
    ! [X: complex,Y: complex] : re(divide_divide(complex,X,Y)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),im(Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,im(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% Re_divide
tff(fact_3739_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa(nat,complex,power_power(complex,W),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) = Z )
     => ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),re(W)))
          | ( ( re(W) = zero_zero(real) )
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),im(W))) ) )
       => ( csqrt(Z) = W ) ) ) ).

% csqrt_unique
tff(fact_3740_csqrt__square,axiom,
    ! [B2: complex] :
      ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),re(B2)))
        | ( ( re(B2) = zero_zero(real) )
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),im(B2))) ) )
     => ( csqrt(aa(nat,complex,power_power(complex,B2),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = B2 ) ) ).

% csqrt_square
tff(fact_3741_inverse__complex_Osimps_I2_J,axiom,
    ! [X: complex] : im(aa(complex,complex,inverse_inverse(complex),X)) = divide_divide(real,aa(real,real,uminus_uminus(real),im(X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,im(X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% inverse_complex.simps(2)
tff(fact_3742_Im__divide,axiom,
    ! [X: complex,Y: complex] : im(divide_divide(complex,X,Y)) = divide_divide(real,aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),im(Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,im(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% Im_divide
tff(fact_3743_complex__abs__le__norm,axiom,
    ! [Z: complex] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,abs_abs(real),re(Z))),aa(real,real,abs_abs(real),im(Z)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),real_V7770717601297561774m_norm(complex,Z)))) ).

% complex_abs_le_norm
tff(fact_3744_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,divide_divide(real,re(Z),real_V7770717601297561774m_norm(complex,Z))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,divide_divide(real,im(Z),real_V7770717601297561774m_norm(complex,Z))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = one_one(real) ) ) ).

% complex_unit_circle
tff(fact_3745_inverse__complex_Ocode,axiom,
    ! [X: complex] : aa(complex,complex,inverse_inverse(complex),X) = complex2(divide_divide(real,re(X),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,im(X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),divide_divide(real,aa(real,real,uminus_uminus(real),im(X)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,im(X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% inverse_complex.code
tff(fact_3746_Complex__divide,axiom,
    ! [X: complex,Y: complex] : divide_divide(complex,X,Y) = complex2(divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),im(Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,im(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),divide_divide(real,aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),im(X)),re(Y))),aa(real,real,aa(real,fun(real,real),times_times(real),re(X)),im(Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,im(Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% Complex_divide
tff(fact_3747_pi__series,axiom,
    divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,aa(num,num,bit0,one2)))) = suminf(real,aTP_Lamp_an(nat,real)) ).

% pi_series
tff(fact_3748_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] : im(csqrt(Z)) = aa(real,real,aa(real,fun(real,real),times_times(real),if(real,aa(real,bool,aa(real,fun(real,bool),fequal(real),im(Z)),zero_zero(real)),one_one(real),sgn_sgn(real,im(Z)))),aa(real,real,sqrt,divide_divide(real,aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(complex,Z)),re(Z)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))) ).

% csqrt.simps(2)
tff(fact_3749_arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => ( aa(real,real,arctan,X) = suminf(real,aTP_Lamp_ao(real,fun(nat,real),X)) ) ) ).

% arctan_series
tff(fact_3750_suminf__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real)))
         => ( suminf(A,power_power(A,C2)) = divide_divide(A,one_one(A),aa(A,A,minus_minus(A,one_one(A)),C2)) ) ) ) ).

% suminf_geometric
tff(fact_3751_summable__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => summable(real,aTP_Lamp_ao(real,fun(nat,real),X)) ) ).

% summable_arctan_series
tff(fact_3752_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( ( ( X = zero_zero(nat) )
         => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
       => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != vEBT_Leaf(fFalse,fFalse) ) )
         => ~ ! [Va2: nat] :
                ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
               => ~ ( ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
                     => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) )
                    & ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
                     => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_3753_Im__Reals__divide,axiom,
    ! [R3: complex,Z: complex] :
      ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),R3),real_Vector_Reals(complex)))
     => ( im(divide_divide(complex,R3,Z)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),re(R3))),im(Z)),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% Im_Reals_divide
tff(fact_3754_intind,axiom,
    ! [A: $tType,I2: nat,N: nat,P: fun(A,bool),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
     => ( pp(aa(A,bool,P,X))
       => pp(aa(A,bool,P,aa(nat,A,nth(A,replicate(A,N,X)),I2))) ) ) ).

% intind
tff(fact_3755_replicate__eq__replicate,axiom,
    ! [A: $tType,M: nat,X: A,N: nat,Y: A] :
      ( ( replicate(A,M,X) = replicate(A,N,Y) )
    <=> ( ( M = N )
        & ( ( M != zero_zero(nat) )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_3756_length__replicate,axiom,
    ! [A: $tType,N: nat,X: A] : aa(list(A),nat,size_size(list(A)),replicate(A,N,X)) = N ).

% length_replicate
tff(fact_3757_summable__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),K: nat] :
          ( summable(A,aa(nat,fun(nat,A),aTP_Lamp_ap(fun(nat,A),fun(nat,fun(nat,A)),F3),K))
        <=> summable(A,F3) ) ) ).

% summable_iff_shift
tff(fact_3758_Ball__set__replicate,axiom,
    ! [A: $tType,N: nat,A2: A,P: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),replicate(A,N,A2))))
         => pp(aa(A,bool,P,X4)) )
    <=> ( pp(aa(A,bool,P,A2))
        | ( N = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_3759_Bex__set__replicate,axiom,
    ! [A: $tType,N: nat,A2: A,P: fun(A,bool)] :
      ( ? [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),replicate(A,N,A2))))
          & pp(aa(A,bool,P,X4)) )
    <=> ( pp(aa(A,bool,P,A2))
        & ( N != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_3760_in__set__replicate,axiom,
    ! [A: $tType,X: A,N: nat,Y: A] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),replicate(A,N,Y))))
    <=> ( ( X = Y )
        & ( N != zero_zero(nat) ) ) ) ).

% in_set_replicate
tff(fact_3761_nth__replicate,axiom,
    ! [A: $tType,I2: nat,N: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
     => ( aa(nat,A,nth(A,replicate(A,N,X)),I2) = X ) ) ).

% nth_replicate
tff(fact_3762_summable__divide__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(nat,A),C2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_aq(fun(nat,A),fun(A,fun(nat,A)),F3),C2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F3) ) ) ) ).

% summable_divide_iff
tff(fact_3763_summable__geometric__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( summable(A,power_power(A,C2))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real))) ) ) ).

% summable_geometric_iff
tff(fact_3764_summable__comparison__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),G: fun(nat,real)] :
          ( ? [N8: nat] :
            ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N2))),aa(nat,real,G,N2))) )
         => ( summable(real,G)
           => summable(A,F3) ) ) ) ).

% summable_comparison_test
tff(fact_3765_summable__comparison__test_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G: fun(nat,real),N3: nat,F3: fun(nat,A)] :
          ( summable(real,G)
         => ( ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N3),N2))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N2))),aa(nat,real,G,N2))) )
           => summable(A,F3) ) ) ) ).

% summable_comparison_test'
tff(fact_3766_suminf__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),G: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N2)),aa(nat,A,G,N2)))
         => ( summable(A,F3)
           => ( summable(A,G)
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),suminf(A,F3)),suminf(A,G))) ) ) ) ) ).

% suminf_le
tff(fact_3767_Reals__power,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),real_Vector_Reals(A)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,power_power(A,A2),N)),real_Vector_Reals(A))) ) ) ).

% Reals_power
tff(fact_3768_Reals__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),real_Vector_Reals(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),real_Vector_Reals(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),divide_divide(A,A2,B2)),real_Vector_Reals(A))) ) ) ) ).

% Reals_divide
tff(fact_3769_Reals__add,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),real_Vector_Reals(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),real_Vector_Reals(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),real_Vector_Reals(A))) ) ) ) ).

% Reals_add
tff(fact_3770_Reals__1,axiom,
    ! [B: $tType] :
      ( real_V2191834092415804123ebra_1(B)
     => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),one_one(B)),real_Vector_Reals(B))) ) ).

% Reals_1
tff(fact_3771_Reals__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W: num] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(num,A,numeral_numeral(A),W)),real_Vector_Reals(A))) ) ).

% Reals_numeral
tff(fact_3772_summable__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(nat,A),C2: A] :
          ( summable(A,F3)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_aq(fun(nat,A),fun(A,fun(nat,A)),F3),C2)) ) ) ).

% summable_divide
tff(fact_3773_summable__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),K: nat] :
          ( summable(A,F3)
         => summable(A,aa(nat,fun(nat,A),aTP_Lamp_ap(fun(nat,A),fun(nat,fun(nat,A)),F3),K)) ) ) ).

% summable_ignore_initial_segment
tff(fact_3774_summable__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F3: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F3)
         => ( summable(A,G)
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ar(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F3),G)) ) ) ) ).

% summable_add
tff(fact_3775_summable__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( summable(A,aTP_Lamp_as(fun(nat,A),fun(nat,A),F3))
        <=> summable(A,F3) ) ) ).

% summable_Suc_iff
tff(fact_3776_summable__zero__power,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,power_power(A,zero_zero(A))) ) ).

% summable_zero_power
tff(fact_3777_suminf__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F3: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F3)
         => ( summable(A,G)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),suminf(A,F3)),suminf(A,G)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ar(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F3),G)) ) ) ) ) ).

% suminf_add
tff(fact_3778_suminf__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(nat,A),C2: A] :
          ( summable(A,F3)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_aq(fun(nat,A),fun(A,fun(nat,A)),F3),C2)) = divide_divide(A,suminf(A,F3),C2) ) ) ) ).

% suminf_divide
tff(fact_3779_series__comparison__complex,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G: fun(nat,complex),N3: nat,F3: fun(nat,A)] :
          ( summable(complex,G)
         => ( ! [N2: nat] : pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),aa(nat,complex,G,N2)),real_Vector_Reals(complex)))
           => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),re(aa(nat,complex,G,N2))))
             => ( ! [N2: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N3),N2))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N2))),real_V7770717601297561774m_norm(complex,aa(nat,complex,G,N2)))) )
               => summable(A,F3) ) ) ) ) ) ).

% series_comparison_complex
tff(fact_3780_powser__insidea,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(nat,A),X: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_at(fun(nat,A),fun(A,fun(nat,A)),F3),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,X)))
           => summable(real,aa(A,fun(nat,real),aTP_Lamp_au(fun(nat,A),fun(A,fun(nat,real)),F3),Z)) ) ) ) ).

% powser_insidea
tff(fact_3781_suminf__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),suminf(A,F3))) ) ) ) ).

% suminf_nonneg
tff(fact_3782_suminf__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N2)))
           => ( ( suminf(A,F3) = zero_zero(A) )
            <=> ! [N5: nat] : aa(nat,A,F3,N5) = zero_zero(A) ) ) ) ) ).

% suminf_eq_zero_iff
tff(fact_3783_suminf__pos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F3,N2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),suminf(A,F3))) ) ) ) ).

% suminf_pos
tff(fact_3784_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(nat,A)] : summable(A,aTP_Lamp_av(fun(nat,A),fun(nat,A),F3)) ) ).

% summable_0_powser
tff(fact_3785_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F3: fun(nat,A)] : summable(A,aTP_Lamp_aw(fun(nat,A),fun(nat,A),F3)) ) ).

% summable_zero_power'
tff(fact_3786_summable__powser__split__head,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ax(fun(nat,A),fun(A,fun(nat,A)),F3),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_at(fun(nat,A),fun(A,fun(nat,A)),F3),Z)) ) ) ).

% summable_powser_split_head
tff(fact_3787_powser__split__head_I3_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ay(fun(nat,A),fun(A,fun(nat,A)),F3),Z))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_az(fun(nat,A),fun(A,fun(nat,A)),F3),Z)) ) ) ).

% powser_split_head(3)
tff(fact_3788_replicate__eqI,axiom,
    ! [A: $tType,Xs: list(A),N: nat,X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = N )
     => ( ! [Y4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),aa(list(A),set(A),set2(A),Xs)))
           => ( Y4 = X ) )
       => ( Xs = replicate(A,N,X) ) ) ) ).

% replicate_eqI
tff(fact_3789_replicate__length__same,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => ( X3 = X ) )
     => ( replicate(A,aa(list(A),nat,size_size(list(A)),Xs),X) = Xs ) ) ).

% replicate_length_same
tff(fact_3790_summable__powser__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(nat,A),M: nat,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_ba(fun(nat,A),fun(nat,fun(A,fun(nat,A))),F3),M),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_at(fun(nat,A),fun(A,fun(nat,A)),F3),Z)) ) ) ).

% summable_powser_ignore_initial_segment
tff(fact_3791_summable__norm__comparison__test,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),G: fun(nat,real)] :
          ( ? [N8: nat] :
            ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N2))),aa(nat,real,G,N2))) )
         => ( summable(real,G)
           => summable(real,aTP_Lamp_bb(fun(nat,A),fun(nat,real),F3)) ) ) ) ).

% summable_norm_comparison_test
tff(fact_3792_summable__rabs__comparison__test,axiom,
    ! [F3: fun(nat,real),G: fun(nat,real)] :
      ( ? [N8: nat] :
        ! [N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(nat,real,F3,N2))),aa(nat,real,G,N2))) )
     => ( summable(real,G)
       => summable(real,aTP_Lamp_bc(fun(nat,real),fun(nat,real),F3)) ) ) ).

% summable_rabs_comparison_test
tff(fact_3793_nonzero__Reals__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),real_Vector_Reals(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),real_Vector_Reals(A)))
           => ( ( B2 != zero_zero(A) )
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),divide_divide(A,A2,B2)),real_Vector_Reals(A))) ) ) ) ) ).

% nonzero_Reals_divide
tff(fact_3794_summable__rabs,axiom,
    ! [F3: fun(nat,real)] :
      ( summable(real,aTP_Lamp_bc(fun(nat,real),fun(nat,real),F3))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),suminf(real,F3))),suminf(real,aTP_Lamp_bc(fun(nat,real),fun(nat,real),F3)))) ) ).

% summable_rabs
tff(fact_3795_suminf__pos2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),I2: nat] :
          ( summable(A,F3)
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F3,I2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),suminf(A,F3))) ) ) ) ) ).

% suminf_pos2
tff(fact_3796_suminf__pos__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),suminf(A,F3)))
            <=> ? [I3: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F3,I3))) ) ) ) ) ).

% suminf_pos_iff
tff(fact_3797_summable__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real)))
         => summable(A,power_power(A,C2)) ) ) ).

% summable_geometric
tff(fact_3798_complete__algebra__summable__geometric,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),one_one(real)))
         => summable(A,power_power(A,X)) ) ) ).

% complete_algebra_summable_geometric
tff(fact_3799_suminf__split__head,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
         => ( suminf(A,aTP_Lamp_as(fun(nat,A),fun(nat,A),F3)) = aa(A,A,minus_minus(A,suminf(A,F3)),aa(nat,A,F3,zero_zero(nat))) ) ) ) ).

% suminf_split_head
tff(fact_3800_summable__norm,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A)] :
          ( summable(real,aTP_Lamp_bd(fun(nat,A),fun(nat,real),F3))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,suminf(A,F3))),suminf(real,aTP_Lamp_bd(fun(nat,A),fun(nat,real),F3)))) ) ) ).

% summable_norm
tff(fact_3801_powser__inside,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(nat,A),X: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ay(fun(nat,A),fun(A,fun(nat,A)),F3),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,X)))
           => summable(A,aa(A,fun(nat,A),aTP_Lamp_ay(fun(nat,A),fun(A,fun(nat,A)),F3),Z)) ) ) ) ).

% powser_inside
tff(fact_3802_summable__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] : summable(A,aTP_Lamp_be(A,fun(nat,A),X)) ) ).

% summable_exp
tff(fact_3803_powser__split__head_I1_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ay(fun(nat,A),fun(A,fun(nat,A)),F3),Z))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_ay(fun(nat,A),fun(A,fun(nat,A)),F3),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,F3,zero_zero(nat))),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_az(fun(nat,A),fun(A,fun(nat,A)),F3),Z))),Z)) ) ) ) ).

% powser_split_head(1)
tff(fact_3804_powser__split__head_I2_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ay(fun(nat,A),fun(A,fun(nat,A)),F3),Z))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_az(fun(nat,A),fun(A,fun(nat,A)),F3),Z))),Z) = aa(A,A,minus_minus(A,suminf(A,aa(A,fun(nat,A),aTP_Lamp_ay(fun(nat,A),fun(A,fun(nat,A)),F3),Z))),aa(nat,A,F3,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_3805_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R3: real,F3: fun(nat,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
         => ( summable(A,F3)
           => ? [N9: nat] :
              ! [N7: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N9),N7))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,suminf(A,aa(nat,fun(nat,A),aTP_Lamp_ap(fun(nat,A),fun(nat,fun(nat,A)),F3),N7)))),R3)) ) ) ) ) ).

% suminf_exist_split
tff(fact_3806_summable__power__series,axiom,
    ! [F3: fun(nat,real),Z: real] :
      ( ! [I4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F3,I4)),one_one(real)))
     => ( ! [I4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,F3,I4)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Z))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z),one_one(real)))
           => summable(real,aa(real,fun(nat,real),aTP_Lamp_bf(fun(nat,real),fun(real,fun(nat,real)),F3),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_3807_Abel__lemma,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R3: real,R0: real,A2: fun(nat,A),M6: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),R3))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),R3),R0))
           => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,A2,N2))),aa(nat,real,power_power(real,R0),N2))),M6))
             => summable(real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_bg(real,fun(fun(nat,A),fun(nat,real)),R3),A2)) ) ) ) ) ).

% Abel_lemma
tff(fact_3808_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [C2: real,N3: nat,F3: fun(nat,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),one_one(real)))
         => ( ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N3),N2))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,aa(nat,nat,suc,N2)))),aa(real,real,aa(real,fun(real,real),times_times(real),C2),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N2))))) )
           => summable(A,F3) ) ) ) ).

% summable_ratio_test
tff(fact_3809_Re__Reals__divide,axiom,
    ! [R3: complex,Z: complex] :
      ( pp(aa(set(complex),bool,aa(complex,fun(set(complex),bool),member(complex),R3),real_Vector_Reals(complex)))
     => ( re(divide_divide(complex,R3,Z)) = divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),re(R3)),re(Z)),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ).

% Re_Reals_divide
tff(fact_3810_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(nat,nat,suc,aa(nat,nat,suc,Va))))
       => ( vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) )
      & ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(nat,nat,suc,aa(nat,nat,suc,Va))))
       => ( vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ).

% vebt_buildup.simps(3)
tff(fact_3811_sin__paired,axiom,
    ! [X: real] : sums(real,aTP_Lamp_bh(real,fun(nat,real),X),sin(real,X)) ).

% sin_paired
tff(fact_3812_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: bool] :
      ( ( vEBT_VEBT_membermima(X,Xa)
      <=> pp(Y) )
     => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa)))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ~ pp(Y)
               => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,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)),Xa))) ) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ( ~ pp(Y)
                 => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,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)),Xa))) ) )
           => ( ! [Mi3: nat,Ma3: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( X = 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)),zero_zero(nat),Va3,Vb2) )
                 => ( ( pp(Y)
                    <=> ( ( Xa = Mi3 )
                        | ( Xa = Ma3 ) ) )
                   => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,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,Mi3),Ma3)),zero_zero(nat),Va3,Vb2)),Xa))) ) )
             => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList3: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( X = 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)),aa(nat,nat,suc,V3),TreeList3,Vc) )
                   => ( ( pp(Y)
                      <=> ( ( Xa = Mi3 )
                          | ( Xa = Ma3 )
                          | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3)))
                             => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3))) ) ) )
                     => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,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,Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList3,Vc)),Xa))) ) )
               => ~ ! [V3: nat,TreeList3: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList3,Vd) )
                     => ( ( pp(Y)
                        <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3)))
                             => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                            & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3))) ) )
                       => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,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,V3),TreeList3,Vd)),Xa))) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_3813_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ vEBT_VEBT_membermima(X,Xa)
     => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa)))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,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)),Xa))) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( X = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,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)),Xa))) )
           => ( ! [Mi3: nat,Ma3: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( X = 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)),zero_zero(nat),Va3,Vb2) )
                 => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,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,Mi3),Ma3)),zero_zero(nat),Va3,Vb2)),Xa)))
                   => ( ( Xa = Mi3 )
                      | ( Xa = Ma3 ) ) ) )
             => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList3: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( X = 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)),aa(nat,nat,suc,V3),TreeList3,Vc) )
                   => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,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,Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList3,Vc)),Xa)))
                     => ( ( Xa = Mi3 )
                        | ( Xa = Ma3 )
                        | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3)))
                           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3))) ) ) ) )
               => ~ ! [V3: nat,TreeList3: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                      ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList3,Vd) )
                     => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,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,V3),TreeList3,Vd)),Xa)))
                       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3)))
                           => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3))) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_3814_geometric__deriv__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z)),one_one(real)))
         => sums(A,aTP_Lamp_bi(A,fun(nat,A),Z),divide_divide(A,one_one(A),aa(nat,A,power_power(A,aa(A,A,minus_minus(A,one_one(A)),Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ) ).

% geometric_deriv_sums
tff(fact_3815_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A),X: A] :
          ( sums(A,aTP_Lamp_av(fun(nat,A),fun(nat,A),A2),X)
        <=> ( aa(nat,A,A2,zero_zero(nat)) = X ) ) ) ).

% powser_sums_zero_iff
tff(fact_3816_sums__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),G: fun(nat,A),S: A,T2: A] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N2)),aa(nat,A,G,N2)))
         => ( sums(A,F3,S)
           => ( sums(A,G,T2)
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),S),T2)) ) ) ) ) ).

% sums_le
tff(fact_3817_sums__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(nat,A),A2: A,C2: A] :
          ( sums(A,F3,A2)
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_aq(fun(nat,A),fun(A,fun(nat,A)),F3),C2),divide_divide(A,A2,C2)) ) ) ).

% sums_divide
tff(fact_3818_sums__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F3: fun(nat,A),A2: A,G: fun(nat,A),B2: A] :
          ( sums(A,F3,A2)
         => ( sums(A,G,B2)
           => sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ar(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F3),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% sums_add
tff(fact_3819_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F3: fun(nat,A),A2: A] :
          ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bj(A,fun(fun(nat,A),fun(nat,A)),C2),F3),A2)
         => ( ( C2 != zero_zero(A) )
           => sums(A,F3,divide_divide(A,A2,C2)) ) ) ) ).

% sums_mult_D
tff(fact_3820_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),S: A] :
          ( ( aa(nat,A,F3,zero_zero(nat)) = zero_zero(A) )
         => ( sums(A,aTP_Lamp_as(fun(nat,A),fun(nat,A),F3),S)
           => sums(A,F3,S) ) ) ) ).

% sums_Suc_imp
tff(fact_3821_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),S: A] :
          ( sums(A,aTP_Lamp_as(fun(nat,A),fun(nat,A),F3),S)
        <=> sums(A,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),S),aa(nat,A,F3,zero_zero(nat)))) ) ) ).

% sums_Suc_iff
tff(fact_3822_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F3: fun(nat,A),L: A] :
          ( sums(A,aTP_Lamp_bk(fun(nat,A),fun(nat,A),F3),L)
         => sums(A,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),L),aa(nat,A,F3,zero_zero(nat)))) ) ) ).

% sums_Suc
tff(fact_3823_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [N: nat,F3: fun(nat,A),S: A] :
          ( ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N))
             => ( aa(nat,A,F3,I4) = zero_zero(A) ) )
         => ( sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bl(nat,fun(fun(nat,A),fun(nat,A)),N),F3),S)
          <=> sums(A,F3,S) ) ) ) ).

% sums_zero_iff_shift
tff(fact_3824_powser__sums__if,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [M: nat,Z: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_bm(nat,fun(A,fun(nat,A)),M),Z),aa(nat,A,power_power(A,Z),M)) ) ).

% powser_sums_if
tff(fact_3825_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A)] : sums(A,aTP_Lamp_av(fun(nat,A),fun(nat,A),A2),aa(nat,A,A2,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_3826_geometric__sums,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,C2)),one_one(real)))
         => sums(A,power_power(A,C2),divide_divide(A,one_one(A),aa(A,A,minus_minus(A,one_one(A)),C2))) ) ) ).

% geometric_sums
tff(fact_3827_power__half__series,axiom,
    sums(real,aTP_Lamp_bn(nat,real),one_one(real)) ).

% power_half_series
tff(fact_3828_sums__if_H,axiom,
    ! [G: fun(nat,real),X: real] :
      ( sums(real,G,X)
     => sums(real,aTP_Lamp_bo(fun(nat,real),fun(nat,real),G),X) ) ).

% sums_if'
tff(fact_3829_sums__if,axiom,
    ! [G: fun(nat,real),X: real,F3: fun(nat,real),Y: real] :
      ( sums(real,G,X)
     => ( sums(real,F3,Y)
       => sums(real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_bp(fun(nat,real),fun(fun(nat,real),fun(nat,real)),G),F3),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)) ) ) ).

% sums_if
tff(fact_3830_cos__paired,axiom,
    ! [X: real] : sums(real,aTP_Lamp_bq(real,fun(nat,real),X),cos(real,X)) ).

% cos_paired
tff(fact_3831_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_membermima(X,Xa)
     => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa)))
       => ( ! [Mi3: nat,Ma3: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
              ( ( X = 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)),zero_zero(nat),Va3,Vb2) )
             => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,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,Mi3),Ma3)),zero_zero(nat),Va3,Vb2)),Xa)))
               => ~ ( ( Xa = Mi3 )
                    | ( Xa = Ma3 ) ) ) )
         => ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList3: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                ( ( X = 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)),aa(nat,nat,suc,V3),TreeList3,Vc) )
               => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,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,Mi3),Ma3)),aa(nat,nat,suc,V3),TreeList3,Vc)),Xa)))
                 => ~ ( ( Xa = Mi3 )
                      | ( Xa = Ma3 )
                      | ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3)))
                         => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3))) ) ) ) )
           => ~ ! [V3: nat,TreeList3: list(vEBT_VEBT),Vd: vEBT_VEBT] :
                  ( ( X = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList3,Vd) )
                 => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,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,V3),TreeList3,Vd)),Xa)))
                   => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3)))
                         => vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3))) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_3832_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ vEBT_V5719532721284313246member(X,Xa)
     => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa)))
       => ( ! [A5: bool,B5: bool] :
              ( ( X = vEBT_Leaf(A5,B5) )
             => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(A5,B5)),Xa)))
               => ( ( ( Xa = zero_zero(nat) )
                   => pp(A5) )
                  & ( ( Xa != zero_zero(nat) )
                   => ( ( ( Xa = one_one(nat) )
                       => pp(B5) )
                      & ( Xa = one_one(nat) ) ) ) ) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,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)),Xa))) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList3: list(vEBT_VEBT),S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList3,S3) )
                 => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,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,V3),TreeList3,S3)),Xa)))
                   => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3)))
                       => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3))) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_3833_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_V5719532721284313246member(X,Xa)
     => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa)))
       => ( ! [A5: bool,B5: bool] :
              ( ( X = vEBT_Leaf(A5,B5) )
             => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(A5,B5)),Xa)))
               => ~ ( ( ( Xa = zero_zero(nat) )
                     => pp(A5) )
                    & ( ( Xa != zero_zero(nat) )
                     => ( ( ( Xa = one_one(nat) )
                         => pp(B5) )
                        & ( Xa = one_one(nat) ) ) ) ) ) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList3: list(vEBT_VEBT),S3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList3,S3) )
               => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,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,V3),TreeList3,S3)),Xa)))
                 => ~ ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3)))
                       => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_3834_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: bool] :
      ( ( vEBT_V5719532721284313246member(X,Xa)
      <=> pp(Y) )
     => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa)))
       => ( ! [A5: bool,B5: bool] :
              ( ( X = vEBT_Leaf(A5,B5) )
             => ( ( pp(Y)
                <=> ( ( ( Xa = zero_zero(nat) )
                     => pp(A5) )
                    & ( ( Xa != zero_zero(nat) )
                     => ( ( ( Xa = one_one(nat) )
                         => pp(B5) )
                        & ( Xa = one_one(nat) ) ) ) ) )
               => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,vEBT_Leaf(A5,B5)),Xa))) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( X = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ( ~ pp(Y)
                 => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,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)),Xa))) ) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList3: list(vEBT_VEBT),S3: vEBT_VEBT] :
                  ( ( X = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList3,S3) )
                 => ( ( pp(Y)
                    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3)))
                         => vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_VEBT_low(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),vEBT_VEBT_high(Xa,divide_divide(nat,aa(nat,nat,suc,V3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3))) ) )
                   => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,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,V3),TreeList3,S3)),Xa))) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_3835_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(A,fun(nat,A)),C2),X))
         => sums(A,aa(A,fun(nat,A),aTP_Lamp_bs(fun(nat,A),fun(A,fun(nat,A)),C2),X),suminf(A,aa(A,fun(nat,A),aTP_Lamp_br(fun(nat,A),fun(A,fun(nat,A)),C2),X))) ) ) ).

% diffs_equiv
tff(fact_3836_diffs__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X2: nat] : aa(nat,A,diffs(A,C2),X2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,X2))),aa(nat,A,C2,aa(nat,nat,suc,X2))) ) ).

% diffs_def
tff(fact_3837_termdiff__converges__all,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [X3: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(A,fun(nat,A)),C2),X3))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_bu(fun(nat,A),fun(A,fun(nat,A)),C2),X)) ) ) ).

% termdiff_converges_all
tff(fact_3838_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,K5: real,C2: fun(nat,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),K5))
         => ( ! [X3: A] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X3)),K5))
               => summable(A,aa(A,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(A,fun(nat,A)),C2),X3)) )
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bv(A,fun(fun(nat,A),fun(nat,A)),X),C2)) ) ) ) ).

% termdiff_converges
tff(fact_3839_exp__first__two__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : exp(A,X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),X)),suminf(A,aTP_Lamp_bw(A,fun(nat,A),X))) ) ).

% exp_first_two_terms
tff(fact_3840_monoseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( topological_monoseq(A,X6)
        <=> ( ! [M7: nat,N5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),N5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,M7)),aa(nat,A,X6,N5))) )
            | ! [M7: nat,N5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),N5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N5)),aa(nat,A,X6,M7))) ) ) ) ) ).

% monoseq_def
tff(fact_3841_monoI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [M5: nat,N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M5),N2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N2)),aa(nat,A,X6,M5))) )
         => topological_monoseq(A,X6) ) ) ).

% monoI2
tff(fact_3842_monoI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [M5: nat,N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M5),N2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,M5)),aa(nat,A,X6,N2))) )
         => topological_monoseq(A,X6) ) ) ).

% monoI1
tff(fact_3843_scaleR__one,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : aa(A,A,real_V8093663219630862766scaleR(A,one_one(real)),X) = X ) ).

% scaleR_one
tff(fact_3844_scaleR__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: A,U: real,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,U),A2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,U),B2)) )
        <=> ( ( A2 = B2 )
            | ( U = one_one(real) ) ) ) ) ).

% scaleR_eq_iff
tff(fact_3845_scaleR__power,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: real,Y: A,N: nat] : aa(nat,A,power_power(A,aa(A,A,real_V8093663219630862766scaleR(A,X),Y)),N) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,power_power(real,X),N)),aa(nat,A,power_power(A,Y),N)) ) ).

% scaleR_power
tff(fact_3846_scaleR__minus1__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),one_one(real))),X) = aa(A,A,uminus_uminus(A),X) ) ).

% scaleR_minus1_left
tff(fact_3847_scaleR__collapse,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,one_one(real)),U)),A2)),aa(A,A,real_V8093663219630862766scaleR(A,U),A2)) = A2 ) ).

% scaleR_collapse
tff(fact_3848_scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,W: num,A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(num,real,numeral_numeral(real),U)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A2)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),W))),A2) ) ).

% scaleR_times
tff(fact_3849_inverse__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [V: num,W: num,A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A2)) = aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,aa(num,real,numeral_numeral(real),W),aa(num,real,numeral_numeral(real),V))),A2) ) ).

% inverse_scaleR_times
tff(fact_3850_fraction__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,V: num,W: num,A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,aa(num,real,numeral_numeral(real),U),aa(num,real,numeral_numeral(real),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A2)) = aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),W)),aa(num,real,numeral_numeral(real),V))),A2) ) ).

% fraction_scaleR_times
tff(fact_3851_scaleR__half__double,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: A] : aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)) = A2 ) ).

% scaleR_half_double
tff(fact_3852_scaleR__right__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,X: A,Y: A] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ).

% scaleR_right_distrib
tff(fact_3853_scaleR__left__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,X: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,B2),X)) ) ).

% scaleR_left_distrib
tff(fact_3854_scaleR__left_Oadd,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: real,Y: real,Xa: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),Y)),Xa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,X),Xa)),aa(A,A,real_V8093663219630862766scaleR(A,Y),Xa)) ) ).

% scaleR_left.add
tff(fact_3855_of__real__def,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R3: real] : real_Vector_of_real(A,R3) = aa(A,A,real_V8093663219630862766scaleR(A,R3),one_one(A)) ) ).

% of_real_def
tff(fact_3856_scaleR__right__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,B2),X))) ) ) ) ).

% scaleR_right_mono
tff(fact_3857_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: real,A2: real,C2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B2),C2))) ) ) ) ).

% scaleR_right_mono_neg
tff(fact_3858_scaleR__le__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2)))
        <=> ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) )
            & ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_3859_scaleR__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_3860_scaleR__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_3861_scaleR__left__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [X: A,Y: A,A2: real] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y))) ) ) ) ).

% scaleR_left_mono
tff(fact_3862_scaleR__left__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: A,A2: A,C2: real] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),C2),zero_zero(real)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))) ) ) ) ).

% scaleR_left_mono_neg
tff(fact_3863_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E2: A,C2: A,B2: real,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E2)),D2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,B2),A2)),E2)),D2))) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_3864_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E2: A,C2: A,B2: real,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E2)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E2)),D2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,A2),B2)),E2)),C2)),D2)) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_3865_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)))
        <=> ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),zero_zero(real)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% zero_le_scaleR_iff
tff(fact_3866_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)),zero_zero(A)))
        <=> ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) )
            | ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),zero_zero(real)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% scaleR_le_0_iff
tff(fact_3867_scaleR__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2))) ) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_3868_scaleR__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),zero_zero(A))) ) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_3869_scaleR__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),zero_zero(A))) ) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_3870_scaleR__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),X))) ) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_3871_split__scaleR__pos__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),B2)) )
            | ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),zero_zero(real)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),zero_zero(A))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2))) ) ) ).

% split_scaleR_pos_le
tff(fact_3872_split__scaleR__neg__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,X: A] :
          ( ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),zero_zero(A))) )
            | ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),zero_zero(real)))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),zero_zero(A))) ) ) ).

% split_scaleR_neg_le
tff(fact_3873_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,C2: A,D2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),A2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),C2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B2),D2))) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_3874_scaleR__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,X: A,Y: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),B2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Y))) ) ) ) ) ) ).

% scaleR_mono
tff(fact_3875_scaleR__left__le__one__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [X: A,A2: real] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),one_one(real)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),X)) ) ) ) ).

% scaleR_left_le_one_le
tff(fact_3876_scaleR__2,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),X) ) ).

% scaleR_2
tff(fact_3877_real__vector__affinity__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [M: real,X: A,C2: A,Y: A] :
          ( ( M != zero_zero(real) )
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,M),X)),C2) = Y )
          <=> ( X = aa(A,A,minus_minus(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),Y)),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),C2)) ) ) ) ) ).

% real_vector_affinity_eq
tff(fact_3878_real__vector__eq__affinity,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [M: real,Y: A,X: A,C2: A] :
          ( ( M != zero_zero(real) )
         => ( ( Y = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,M),X)),C2) )
          <=> ( aa(A,A,minus_minus(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),Y)),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),C2)) = X ) ) ) ) ).

% real_vector_eq_affinity
tff(fact_3879_pos__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% pos_divideR_le_eq
tff(fact_3880_pos__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2)) ) ) ) ).

% pos_le_divideR_eq
tff(fact_3881_neg__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2)) ) ) ) ).

% neg_divideR_le_eq
tff(fact_3882_neg__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% neg_le_divideR_eq
tff(fact_3883_neg__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% neg_less_divideR_eq
tff(fact_3884_neg__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2)) ) ) ) ).

% neg_divideR_less_eq
tff(fact_3885_pos__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2)) ) ) ) ).

% pos_less_divideR_eq
tff(fact_3886_pos__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% pos_divideR_less_eq
tff(fact_3887_summable__exp__generic,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(A,aTP_Lamp_bx(A,fun(nat,A),X)) ) ).

% summable_exp_generic
tff(fact_3888_sin__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sums(A,aTP_Lamp_by(A,fun(nat,A),X),sin(A,X)) ) ).

% sin_converges
tff(fact_3889_sin__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X2: A] : sin(A,X2) = suminf(A,aTP_Lamp_by(A,fun(nat,A),X2)) ) ).

% sin_def
tff(fact_3890_cos__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sums(A,aTP_Lamp_bz(A,fun(nat,A),X),cos(A,X)) ) ).

% cos_converges
tff(fact_3891_cos__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X2: A] : cos(A,X2) = suminf(A,aTP_Lamp_bz(A,fun(nat,A),X2)) ) ).

% cos_def
tff(fact_3892_summable__norm__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(real,aTP_Lamp_ca(A,fun(nat,real),X)) ) ).

% summable_norm_sin
tff(fact_3893_summable__norm__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(real,aTP_Lamp_cb(A,fun(nat,real),X)) ) ).

% summable_norm_cos
tff(fact_3894_pos__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_3895_pos__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_3896_neg__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_3897_neg__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_3898_neg__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_3899_neg__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_3900_pos__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2))) ) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_3901_pos__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2))) ) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_3902_exp__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sums(A,aTP_Lamp_bx(A,fun(nat,A),X),exp(A,X)) ) ).

% exp_converges
tff(fact_3903_exp__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X2: A] : exp(A,X2) = suminf(A,aTP_Lamp_bx(A,fun(nat,A),X2)) ) ).

% exp_def
tff(fact_3904_summable__norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : summable(real,aTP_Lamp_cc(A,fun(nat,real),X)) ) ).

% summable_norm_exp
tff(fact_3905_sin__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sums(A,aTP_Lamp_cd(A,fun(nat,A),X),sin(A,X)) ) ).

% sin_minus_converges
tff(fact_3906_cos__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sums(A,aTP_Lamp_ce(A,fun(nat,A),X),cos(A,X)) ) ).

% cos_minus_converges
tff(fact_3907_cosh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : cosh(A,X) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),exp(A,X)),exp(A,aa(A,A,uminus_uminus(A),X)))) ) ).

% cosh_def
tff(fact_3908_sinh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sinh(A,X) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(A,A,minus_minus(A,exp(A,X)),exp(A,aa(A,A,uminus_uminus(A),X)))) ) ).

% sinh_def
tff(fact_3909_exp__first__term,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : exp(A,X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),suminf(A,aTP_Lamp_cf(A,fun(nat,A),X))) ) ).

% exp_first_term
tff(fact_3910_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sums(A,aTP_Lamp_cg(A,fun(nat,A),X),cosh(A,X)) ) ).

% cosh_converges
tff(fact_3911_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A] : sums(A,aTP_Lamp_ch(A,fun(nat,A),X),sinh(A,X)) ) ).

% sinh_converges
tff(fact_3912_mono__SucI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N2)),aa(nat,A,X6,aa(nat,nat,suc,N2))))
         => topological_monoseq(A,X6) ) ) ).

% mono_SucI1
tff(fact_3913_mono__SucI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,aa(nat,nat,suc,N2))),aa(nat,A,X6,N2)))
         => topological_monoseq(A,X6) ) ) ).

% mono_SucI2
tff(fact_3914_monoseq__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( topological_monoseq(A,X6)
        <=> ( ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N5)),aa(nat,A,X6,aa(nat,nat,suc,N5))))
            | ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,aa(nat,nat,suc,N5))),aa(nat,A,X6,N5))) ) ) ) ).

% monoseq_Suc
tff(fact_3915_of__nat__code,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),N) = semiri8178284476397505188at_aux(A,aTP_Lamp_ci(A,A),N,zero_zero(A)) ) ).

% of_nat_code
tff(fact_3916_Arg__def,axiom,
    ! [Z: complex] :
      ( ( ( Z = zero_zero(complex) )
       => ( arg(Z) = zero_zero(real) ) )
      & ( ( Z != zero_zero(complex) )
       => ( arg(Z) = fChoice(real,aTP_Lamp_cj(complex,fun(real,bool),Z)) ) ) ) ).

% Arg_def
tff(fact_3917_set__vebt__def,axiom,
    ! [T2: vEBT_VEBT] : vEBT_set_vebt(T2) = collect(nat,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,T2)) ).

% set_vebt_def
tff(fact_3918_sin__x__sin__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_cl(A,fun(A,fun(nat,A)),X),Y),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,X)),sin(A,Y))) ) ).

% sin_x_sin_y
tff(fact_3919_atMost__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,set(A),set_ord_atMost(A),X) = aa(A,set(A),set_ord_atMost(A),Y) )
        <=> ( X = Y ) ) ) ).

% atMost_eq_iff
tff(fact_3920_atMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),aa(A,set(A),set_ord_atMost(A),K)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),K)) ) ) ).

% atMost_iff
tff(fact_3921_atMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),X)),aa(A,set(A),set_ord_atMost(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% atMost_subset_iff
tff(fact_3922_sum_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),aa(nat,set(nat),set_ord_atMost(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% sum.atMost_Suc
tff(fact_3923_norm__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(B,A),A3: set(B)] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A3))),aa(set(B),real,groups7311177749621191930dd_sum(B,real,aTP_Lamp_cm(fun(B,A),fun(B,real),F3)),A3))) ) ).

% norm_sum
tff(fact_3924_sum__norm__le,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [S2: set(B),F3: fun(B,A),G: fun(B,real)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S2))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(B,A,F3,X3))),aa(B,real,G,X3))) )
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),S2))),aa(set(B),real,groups7311177749621191930dd_sum(B,real,G),S2))) ) ) ).

% sum_norm_le
tff(fact_3925_sum__choose__upper,axiom,
    ! [M: nat,N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_cn(nat,fun(nat,nat),M)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,M)) ).

% sum_choose_upper
tff(fact_3926_mod__sum__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F3: fun(B,A),A2: A,A3: set(B)] : modulo_modulo(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_co(fun(B,A),fun(A,fun(B,A)),F3),A2)),A3),A2) = modulo_modulo(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A3),A2) ) ).

% mod_sum_eq
tff(fact_3927_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cp(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ).

% sum.atMost_Suc_shift
tff(fact_3928_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),I2: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cq(fun(nat,A),fun(nat,A),F3)),aa(nat,set(nat),set_ord_atMost(nat),I2)) = aa(A,A,minus_minus(A,aa(nat,A,F3,zero_zero(nat))),aa(nat,A,F3,aa(nat,nat,suc,I2))) ) ).

% sum_telescope
tff(fact_3929_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat,D2: fun(nat,A)] :
          ( ! [X4: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_cr(fun(nat,A),fun(A,fun(nat,A)),C2),X4)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_cr(fun(nat,A),fun(A,fun(nat,A)),D2),X4)),aa(nat,set(nat),set_ord_atMost(nat),N))
        <=> ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),N))
             => ( aa(nat,A,C2,I3) = aa(nat,A,D2,I3) ) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_3930_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linord2810124833399127020strict(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: fun(nat,A),B3: A] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,A2,N2)))
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,A2),aa(nat,set(nat),set_ord_atMost(nat),N2))),B3))
           => summable(A,A2) ) ) ) ).

% bounded_imp_summable
tff(fact_3931_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_atMost(A),U) = collect(A,aTP_Lamp_cs(A,fun(A,bool),U)) ) ).

% atMost_def
tff(fact_3932_sum__choose__lower,axiom,
    ! [R3: nat,N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_ct(nat,fun(nat,nat),R3)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),R3),N))),N) ).

% sum_choose_lower
tff(fact_3933_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_cu(nat,fun(nat,nat),N)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),one_one(nat))),M) ).

% choose_rising_sum(2)
tff(fact_3934_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_cu(nat,fun(nat,nat),N)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ).

% choose_rising_sum(1)
tff(fact_3935_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat] :
          ( ! [X4: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_cr(fun(nat,A),fun(A,fun(nat,A)),C2),X4)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A)
        <=> ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),N))
             => ( aa(nat,A,C2,I3) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_3936_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C2: fun(nat,A),N: nat,K: nat] :
          ( ! [W2: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_cv(fun(nat,A),fun(A,fun(nat,A)),C2),W2)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
           => ( aa(nat,A,C2,K) = zero_zero(A) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_3937_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cw(A,fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),N)) = 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),N))),one_one(A))),N) ) ).

% gbinomial_parallel_sum
tff(fact_3938_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_cx(nat,fun(nat,fun(nat,nat)),M),N)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),M) ) ) ).

% sum_choose_diagonal
tff(fact_3939_vandermonde,axiom,
    ! [M: nat,N: nat,R3: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_cy(nat,fun(nat,fun(nat,fun(nat,nat))),M),N),R3)),aa(nat,set(nat),set_ord_atMost(nat),R3)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),R3) ).

% vandermonde
tff(fact_3940_sum__gp__basic,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),X)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,X),aa(nat,nat,suc,N))) ) ).

% sum_gp_basic
tff(fact_3941_choose__row__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,binomial(N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N) ).

% choose_row_sum
tff(fact_3942_binomial,axiom,
    ! [A2: nat,B2: nat,N: nat] : aa(nat,nat,power_power(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),N) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_cz(nat,fun(nat,fun(nat,fun(nat,nat))),A2),B2),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ).

% binomial
tff(fact_3943_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),aa(nat,set(nat),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),aa(num,num,bit0,one2))),N)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_da(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% sum.in_pairs_0
tff(fact_3944_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [M: nat,A2: fun(nat,A),N: nat,B2: fun(nat,A),X: A] :
          ( ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),I4))
             => ( aa(nat,A,A2,I4) = zero_zero(A) ) )
         => ( ! [J2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),J2))
               => ( aa(nat,A,B2,J2) = zero_zero(A) ) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),A2),X)),aa(nat,set(nat),set_ord_atMost(nat),M))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),B2),X)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(fun(nat,A),fun(A,fun(nat,A)),aTP_Lamp_dd(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),A2),B2),X)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ) ) ) ).

% polynomial_product
tff(fact_3945_gbinomial__sum__lower__neg,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_de(A,fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),M)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),M)) ) ).

% gbinomial_sum_lower_neg
tff(fact_3946_binomial__ring,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,B2: A,N: nat] : aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),N) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_df(A,fun(A,fun(nat,fun(nat,A))),A2),B2),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% binomial_ring
tff(fact_3947_polynomial__product__nat,axiom,
    ! [M: nat,A2: fun(nat,nat),N: nat,B2: fun(nat,nat),X: nat] :
      ( ! [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),I4))
         => ( aa(nat,nat,A2,I4) = zero_zero(nat) ) )
     => ( ! [J2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),J2))
           => ( aa(nat,nat,B2,J2) = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_dg(fun(nat,nat),fun(nat,fun(nat,nat)),A2),X)),aa(nat,set(nat),set_ord_atMost(nat),M))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_dg(fun(nat,nat),fun(nat,fun(nat,nat)),B2),X)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_di(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),A2),B2),X)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ) ) ).

% polynomial_product_nat
tff(fact_3948_choose__square__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_dj(nat,fun(nat,nat),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),N) ).

% choose_square_sum
tff(fact_3949_pochhammer__binomial__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A2: A,B2: A,N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),N) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_dk(A,fun(A,fun(nat,fun(nat,A))),A2),B2),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% pochhammer_binomial_sum
tff(fact_3950_of__nat__aux_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),N: nat,I2: A] : semiri8178284476397505188at_aux(A,Inc,aa(nat,nat,suc,N),I2) = semiri8178284476397505188at_aux(A,Inc,N,aa(A,A,Inc,I2)) ) ).

% of_nat_aux.simps(2)
tff(fact_3951_of__nat__aux_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),I2: A] : semiri8178284476397505188at_aux(A,Inc,zero_zero(nat),I2) = I2 ) ).

% of_nat_aux.simps(1)
tff(fact_3952_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,M: nat,I5: set(nat)] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_dl(A,fun(nat,fun(nat,A)),X),M)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),I5)) ) ).

% sum_power_add
tff(fact_3953_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),P2))
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_dm(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),P2)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_dn(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,minus_minus(nat,P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_3954_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A2: A,X: A,Y: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_do(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_dp(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) ) ).

% gbinomial_partial_sum_poly
tff(fact_3955_exp__series__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A,Y: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),X),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),X) )
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,N))),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_dq(A,fun(A,fun(nat,fun(nat,A))),X),Y),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ) ) ).

% exp_series_add_commuting
tff(fact_3956_root__polyfun,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,Z: A,A2: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N))
         => ( ( aa(nat,A,power_power(A,Z),N) = A2 )
          <=> ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_dr(nat,fun(A,fun(A,fun(nat,A))),N),Z),A2)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_3957_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] :
          ( ( ( X = one_one(A) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) )
          & ( ( X != one_one(A) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),aa(nat,set(nat),set_ord_atMost(nat),N)) = divide_divide(A,aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,X),aa(nat,nat,suc,N))),aa(A,A,minus_minus(A,one_one(A)),X)) ) ) ) ) ).

% sum_gp0
tff(fact_3958_choose__alternating__linear__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( ( N != one_one(nat) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ds(nat,fun(nat,A),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_3959_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_dt(nat,fun(nat,A),M)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M) ) ).

% gbinomial_sum_nat_pow2
tff(fact_3960_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A2: A,X: A,Y: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_do(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_du(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),X),Y)),aa(nat,set(nat),set_ord_atMost(nat),M)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_3961_binomial__r__part__sum,axiom,
    ! [M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)),one_one(nat)))),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)) ).

% binomial_r_part_sum
tff(fact_3962_choose__linear__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_dv(nat,fun(nat,nat),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,minus_minus(nat,N),one_one(nat)))) ).

% choose_linear_sum
tff(fact_3963_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_dw(nat,fun(nat,A),N)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_3964_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E2: real,C2: fun(nat,A),N: nat] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
         => ? [M9: real] :
            ! [Z3: A] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),M9),real_V7770717601297561774m_norm(A,Z3)))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_at(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)),aa(nat,set(nat),set_ord_atMost(nat),N)))),aa(real,real,aa(real,fun(real,real),times_times(real),E2),aa(nat,real,power_power(real,real_V7770717601297561774m_norm(A,Z3)),aa(nat,nat,suc,N))))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_3965_gbinomial__r__part__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,semiring_1_of_nat(A),M))),one_one(A)))),aa(nat,set(nat),set_ord_atMost(nat),M)) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M)) ) ).

% gbinomial_r_part_sum
tff(fact_3966_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_dx(nat,fun(nat,A),N)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) ) ) ) ).

% choose_odd_sum
tff(fact_3967_choose__even__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_dy(nat,fun(nat,A),N)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N) ) ) ) ).

% choose_even_sum
tff(fact_3968_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_dz(A,fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),M)) = 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),M)),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,gbinomial(A,A2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),one_one(nat)))) ) ).

% gbinomial_partial_row_sum
tff(fact_3969_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] : aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ea(nat,fun(nat,bool)),N))) ) ).

% mask_eq_sum_exp
tff(fact_3970_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),aa(nat,nat,suc,zero_zero(nat))) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ea(nat,fun(nat,bool)),N))) ).

% mask_eq_sum_exp_nat
tff(fact_3971_cos__x__cos__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_ec(A,fun(A,fun(nat,A)),X),Y),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,X)),cos(A,Y))) ) ).

% cos_x_cos_y
tff(fact_3972_sums__cos__x__plus__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A,Y: A] : sums(A,aa(A,fun(nat,A),aTP_Lamp_ee(A,fun(A,fun(nat,A)),X),Y),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y))) ) ).

% sums_cos_x_plus_y
tff(fact_3973_sum__abs__ge__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F3: fun(A,B),A3: set(A)] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,aTP_Lamp_ef(fun(A,B),fun(A,B),F3)),A3))) ) ).

% sum_abs_ge_zero
tff(fact_3974_sum__abs,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [F3: fun(A,B),A3: set(A)] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),A3))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,aTP_Lamp_ef(fun(A,B),fun(A,B),F3)),A3))) ) ).

% sum_abs
tff(fact_3975_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),X: fun(A,B),A2: fun(A,B),B2: B,Delta: B] :
          ( ! [I4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I4),I5))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,X,I4))) )
         => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,X),I5) = one_one(B) )
           => ( ! [I4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I4),I5))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,minus_minus(B,aa(A,B,A2,I4)),B2))),Delta)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,minus_minus(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_eg(fun(A,B),fun(fun(A,B),fun(A,B)),X),A2)),I5)),B2))),Delta)) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_3976_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
       => ? [T4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),T4))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),zero_zero(real)))
            & ( cos(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_eh(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi))),semiring_char_0_fact(real,N))),aa(nat,real,power_power(real,X),N))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_3977_lessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,set(A),set_ord_lessThan(A),X) = aa(A,set(A),set_ord_lessThan(A),Y) )
        <=> ( X = Y ) ) ) ).

% lessThan_eq_iff
tff(fact_3978_of__nat__id,axiom,
    ! [N: nat] : aa(nat,nat,semiring_1_of_nat(nat),N) = N ).

% of_nat_id
tff(fact_3979_lessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),aa(A,set(A),set_ord_lessThan(A),K)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),K)) ) ) ).

% lessThan_iff
tff(fact_3980_lessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_lessThan(A),X)),aa(A,set(A),set_ord_lessThan(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% lessThan_subset_iff
tff(fact_3981_sum_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(nat,A,G,N)) ) ).

% sum.lessThan_Suc
tff(fact_3982_sumr__cos__zero__one,axiom,
    ! [N: nat] : aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ei(nat,real)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = one_one(real) ).

% sumr_cos_zero_one
tff(fact_3983_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P: fun(A,nat),N: A] :
          ( ! [X3: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Q,X3)),aa(A,nat,P,X3)))
         => ( aa(nat,nat,minus_minus(nat,aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,P),aa(A,set(A),set_ord_lessThan(A),N))),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,Q),aa(A,set(A),set_ord_lessThan(A),N))) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_ej(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),aa(A,set(A),set_ord_lessThan(A),N)) ) ) ) ).

% sum_diff_distrib
tff(fact_3984_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : aa(A,set(A),set_ord_lessThan(A),U) = collect(A,aTP_Lamp_ek(A,fun(A,bool),U)) ) ).

% lessThan_def
tff(fact_3985_sum__subtractf__nat,axiom,
    ! [A: $tType,A3: set(A),G: fun(A,nat),F3: fun(A,nat)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,G,X3)),aa(A,nat,F3,X3))) )
     => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_el(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G),F3)),A3) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),A3)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,G),A3)) ) ) ).

% sum_subtractf_nat
tff(fact_3986_sum__SucD,axiom,
    ! [A: $tType,F3: fun(A,nat),A3: set(A),N: nat] :
      ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),A3) = aa(nat,nat,suc,N) )
     => ? [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,F3,X3))) ) ) ).

% sum_SucD
tff(fact_3987_lessThan__strict__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,N: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(A,set(A),set_ord_lessThan(A),M)),aa(A,set(A),set_ord_lessThan(A),N)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),N)) ) ) ).

% lessThan_strict_subset_iff
tff(fact_3988_lessThan__Suc__atMost,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K)) = aa(nat,set(nat),set_ord_atMost(nat),K) ).

% lessThan_Suc_atMost
tff(fact_3989_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),A2)),aa(A,set(A),set_ord_lessThan(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% Iic_subset_Iio_iff
tff(fact_3990_sum_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_em(fun(nat,A),fun(nat,fun(nat,A)),G),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.nat_diff_reindex
tff(fact_3991_suminf__le__const,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),X: A] :
          ( summable(A,F3)
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),aa(nat,set(nat),set_ord_lessThan(nat),N2))),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),suminf(A,F3)),X)) ) ) ) ).

% suminf_le_const
tff(fact_3992_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cp(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% sum.lessThan_Suc_shift
tff(fact_3993_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cq(fun(nat,A),fun(nat,A),F3)),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(A,A,minus_minus(A,aa(nat,A,F3,zero_zero(nat))),aa(nat,A,F3,M)) ) ).

% sum_lessThan_telescope'
tff(fact_3994_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F3: fun(nat,A),M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_en(fun(nat,A),fun(nat,A),F3)),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(A,A,minus_minus(A,aa(nat,A,F3,M)),aa(nat,A,F3,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_3995_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),X: A] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N2)))
         => ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),aa(nat,set(nat),set_ord_lessThan(nat),N2))),X))
           => summable(A,F3) ) ) ) ).

% summableI_nonneg_bounded
tff(fact_3996_sums__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),N: nat,S: A] :
          ( sums(A,aa(nat,fun(nat,A),aTP_Lamp_ap(fun(nat,A),fun(nat,fun(nat,A)),F3),N),S)
        <=> sums(A,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),S),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),aa(nat,set(nat),set_ord_lessThan(nat),N)))) ) ) ).

% sums_iff_shift
tff(fact_3997_sums__iff__shift_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),N: nat,S: A] :
          ( sums(A,aa(nat,fun(nat,A),aTP_Lamp_ap(fun(nat,A),fun(nat,fun(nat,A)),F3),N),aa(A,A,minus_minus(A,S),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),aa(nat,set(nat),set_ord_lessThan(nat),N))))
        <=> sums(A,F3,S) ) ) ).

% sums_iff_shift'
tff(fact_3998_sums__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),S: A,N: nat] :
          ( sums(A,F3,S)
         => sums(A,aa(nat,fun(nat,A),aTP_Lamp_ap(fun(nat,A),fun(nat,fun(nat,A)),F3),N),aa(A,A,minus_minus(A,S),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),aa(nat,set(nat),set_ord_lessThan(nat),N)))) ) ) ).

% sums_split_initial_segment
tff(fact_3999_sum__nth__roots,axiom,
    ! [N: nat,C2: complex] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N))
     => ( aa(set(complex),complex,groups7311177749621191930dd_sum(complex,complex,aTP_Lamp_eo(complex,complex)),collect(complex,aa(complex,fun(complex,bool),aTP_Lamp_ep(nat,fun(complex,fun(complex,bool)),N),C2))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_4000_power__diff__1__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),N)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),one_one(A))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% power_diff_1_eq
tff(fact_4001_one__diff__power__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,X),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),X)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% one_diff_power_eq
tff(fact_4002_geometric__sum,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [X: A,N: nat] :
          ( ( X != one_one(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = divide_divide(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),N)),one_one(A)),aa(A,A,minus_minus(A,X),one_one(A))) ) ) ) ).

% geometric_sum
tff(fact_4003_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cp(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% sum.atMost_shift
tff(fact_4004_suminf__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),K: nat] :
          ( summable(A,F3)
         => ( suminf(A,F3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_ap(fun(nat,A),fun(nat,fun(nat,A)),F3),K))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),aa(nat,set(nat),set_ord_lessThan(nat),K))) ) ) ) ).

% suminf_split_initial_segment
tff(fact_4005_suminf__minus__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),K: nat] :
          ( summable(A,F3)
         => ( suminf(A,aa(nat,fun(nat,A),aTP_Lamp_ap(fun(nat,A),fun(nat,fun(nat,A)),F3),K)) = aa(A,A,minus_minus(A,suminf(A,F3)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),aa(nat,set(nat),set_ord_lessThan(nat),K))) ) ) ) ).

% suminf_minus_initial_segment
tff(fact_4006_sum__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N))
     => ( aa(set(complex),complex,groups7311177749621191930dd_sum(complex,complex,aTP_Lamp_eo(complex,complex)),collect(complex,aTP_Lamp_eq(nat,fun(complex,bool),N))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_4007_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),N: nat] :
          ( summable(A,F3)
         => ( ! [M5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F3,M5))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),aa(nat,set(nat),set_ord_lessThan(nat),N))),suminf(A,F3))) ) ) ) ).

% sum_less_suminf
tff(fact_4008_sum__gp__strict,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] :
          ( ( ( X = one_one(A) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) )
          & ( ( X != one_one(A) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = divide_divide(A,aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,X),N)),aa(A,A,minus_minus(A,one_one(A)),X)) ) ) ) ) ).

% sum_gp_strict
tff(fact_4009_lemma__termdiff1,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Z: A,H: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_er(A,fun(A,fun(nat,fun(nat,A))),Z),H),M)),aa(nat,set(nat),set_ord_lessThan(nat),M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_es(A,fun(A,fun(nat,fun(nat,A))),Z),H),M)),aa(nat,set(nat),set_ord_lessThan(nat),M)) ) ).

% lemma_termdiff1
tff(fact_4010_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat,Y: A] : aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),aa(nat,nat,suc,N))),aa(nat,A,power_power(A,Y),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),Y)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_et(A,fun(nat,fun(A,fun(nat,A))),X),N),Y)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N)))) ) ).

% diff_power_eq_sum
tff(fact_4011_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat,Y: A] : aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),N)),aa(nat,A,power_power(A,Y),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),Y)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_eu(A,fun(nat,fun(A,fun(nat,A))),X),N),Y)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% power_diff_sumr2
tff(fact_4012_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),A2: A,N: nat] :
          ( ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),aa(nat,set(nat),set_ord_atMost(nat),N)) = zero_zero(A) )
         => ~ ! [B5: fun(nat,A)] :
                ~ ! [Z3: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Z3),A2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),B5),Z3)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ) ).

% polyfun_linear_factor_root
tff(fact_4013_polyfun__linear__factor,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),N: nat,A2: A] :
        ? [B5: fun(nat,A)] :
        ! [Z3: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Z3),A2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),B5),Z3)),aa(nat,set(nat),set_ord_lessThan(nat),N)))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ).

% polyfun_linear_factor
tff(fact_4014_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,F3: fun(nat,A),K5: A,K: nat] :
          ( ! [P4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),P4),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,P4)),K5)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),K5))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,minus_minus(nat,N),K)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),K5))) ) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_4015_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),N: nat,I2: nat] :
          ( summable(A,F3)
         => ( ! [M5: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,M5))) )
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),I2))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(nat,A,F3,I2)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),aa(nat,set(nat),set_ord_lessThan(nat),N))),suminf(A,F3))) ) ) ) ) ) ).

% sum_less_suminf2
tff(fact_4016_one__diff__power__eq_H,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [X: A,N: nat] : aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,X),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),X)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ev(A,fun(nat,fun(nat,A)),X),N)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% one_diff_power_eq'
tff(fact_4017_Maclaurin__zero,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [X: real,N: nat,Diff: fun(nat,fun(A,real))] :
          ( ( X = zero_zero(real) )
         => ( ( N != zero_zero(nat) )
           => ( aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_ew(real,fun(fun(nat,fun(A,real)),fun(nat,real)),X),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(A,real,aa(nat,fun(A,real),Diff,zero_zero(nat)),zero_zero(A)) ) ) ) ) ).

% Maclaurin_zero
tff(fact_4018_Maclaurin__lemma,axiom,
    ! [H: real,F3: fun(real,real),J: fun(nat,real),N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
     => ? [B7: real] : aa(real,real,F3,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_ex(real,fun(fun(nat,real),fun(nat,real)),H),J)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),B7),divide_divide(real,aa(nat,real,power_power(real,H),N),semiring_char_0_fact(real,N)))) ) ).

% Maclaurin_lemma
tff(fact_4019_sum__split__even__odd,axiom,
    ! [F3: fun(nat,real),G: fun(nat,real),N: nat] : aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_ey(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F3),G)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N))) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ez(fun(nat,real),fun(nat,real),F3)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fa(fun(nat,real),fun(nat,real),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ).

% sum_split_even_odd
tff(fact_4020_sum__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [K5: set(B),F3: fun(B,A),G: fun(B,A)] :
          ( ! [I4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),K5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I4)),aa(B,A,G,I4))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),K5)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),K5))) ) ) ).

% sum_mono
tff(fact_4021_sum_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),H: fun(B,A),A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fb(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),A3)) ) ).

% sum.distrib
tff(fact_4022_sum__divide__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F3: fun(B,A),A3: set(B),R3: A] : divide_divide(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A3),R3) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_fc(fun(B,A),fun(A,fun(B,A)),F3),R3)),A3) ) ).

% sum_divide_distrib
tff(fact_4023_Maclaurin__exp__le,axiom,
    ! [X: real,N: nat] :
    ? [T4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X)))
      & ( exp(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fd(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,exp(real,T4),semiring_char_0_fact(real,N))),aa(nat,real,power_power(real,X),N))) ) ) ).

% Maclaurin_exp_le
tff(fact_4024_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,A2: fun(nat,A),X: A,Y: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N))
         => ( aa(A,A,minus_minus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),A2),X)),aa(nat,set(nat),set_ord_atMost(nat),N))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),Y)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_ff(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),N),A2),X),Y)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ) ) ).

% polyfun_diff_alt
tff(fact_4025_exp__first__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X: A,K: nat] : exp(A,X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_bx(A,fun(nat,A),X)),aa(nat,set(nat),set_ord_lessThan(nat),K))),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_fg(A,fun(nat,fun(nat,A)),X),K))) ) ).

% exp_first_terms
tff(fact_4026_Maclaurin__sin__bound,axiom,
    ! [X: real,N: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,sin(real,X)),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fh(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,N))),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),X)),N)))) ).

% Maclaurin_sin_bound
tff(fact_4027_sum__pos__lt__pair,axiom,
    ! [F3: fun(nat,real),K: nat] :
      ( summable(real,F3)
     => ( ! [D3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D3)))),aa(nat,real,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D3)),one_one(nat)))))))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,F3),aa(nat,set(nat),set_ord_lessThan(nat),K))),suminf(real,F3))) ) ) ).

% sum_pos_lt_pair
tff(fact_4028_sum__nonpos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),zero_zero(A))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A3)),zero_zero(A))) ) ) ).

% sum_nonpos
tff(fact_4029_sum__nonneg,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A3))) ) ) ).

% sum_nonneg
tff(fact_4030_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(nat),F3: fun(nat,A),G: fun(nat,A)] :
          ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3))
         => ( ! [X3: nat] :
                ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,suc,X3)),A3))
               => ( aa(nat,A,F3,aa(nat,nat,suc,X3)) = aa(nat,A,G,aa(nat,nat,suc,X3)) ) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),A3) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),A3) ) ) ) ) ).

% sum_cong_Suc
tff(fact_4031_Maclaurin__exp__lt,axiom,
    ! [X: real,N: nat] :
      ( ( X != zero_zero(real) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ? [T4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T4)))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X)))
            & ( exp(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fd(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,exp(real,T4),semiring_char_0_fact(real,N))),aa(nat,real,power_power(real,X),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_4032_lemma__termdiff2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [H: A,Z: A,N: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,divide_divide(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H)),N)),aa(nat,A,power_power(A,Z),N)),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,power_power(A,Z),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),times_times(A),H),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fj(A,fun(A,fun(nat,fun(nat,A))),H),Z),N)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))) ) ) ) ).

% lemma_termdiff2
tff(fact_4033_Maclaurin__sin__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T4: real] : sin(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fh(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),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),T4),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),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi))),semiring_char_0_fact(real,N))),aa(nat,real,power_power(real,X),N))) ).

% Maclaurin_sin_expansion
tff(fact_4034_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N: nat] :
    ? [T4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X)))
      & ( sin(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fh(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),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),T4),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),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi))),semiring_char_0_fact(real,N))),aa(nat,real,power_power(real,X),N))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_4035_Maclaurin__cos__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T4: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X)))
      & ( cos(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_eh(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi))),semiring_char_0_fact(real,N))),aa(nat,real,power_power(real,X),N))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_4036_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ? [T4: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T4))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),X))
          & ( sin(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fh(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),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),T4),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),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi))),semiring_char_0_fact(real,N))),aa(nat,real,power_power(real,X),N))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_4037_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ? [T4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T4))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),X))
            & ( sin(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_fh(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),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),T4),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),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi))),semiring_char_0_fact(real,N))),aa(nat,real,power_power(real,X),N))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_4038_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ? [T4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T4))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),X))
            & ( cos(real,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_eh(real,fun(nat,real),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi))),semiring_char_0_fact(real,N))),aa(nat,real,power_power(real,X),N))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_4039_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => bij_betw(nat,complex,aTP_Lamp_fk(nat,fun(nat,complex),N),aa(nat,set(nat),set_ord_lessThan(nat),N),collect(complex,aTP_Lamp_eq(nat,fun(complex,bool),N))) ) ).

% bij_betw_roots_unity
tff(fact_4040_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [N: nat,M: nat,X: A] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( ( ( X = one_one(A) )
               => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),set_or1337092689740270186AtMost(nat,M,N)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),M)) ) )
              & ( ( X != one_one(A) )
               => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),set_or1337092689740270186AtMost(nat,M,N)) = divide_divide(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),M)),aa(nat,A,power_power(A,X),aa(nat,nat,suc,N))),aa(A,A,minus_minus(A,one_one(A)),X)) ) ) ) ) ) ) ).

% sum_gp
tff(fact_4041_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R3: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_dz(A,fun(nat,A),R3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),M)) = 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,M)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,gbinomial(A,R3),aa(nat,nat,suc,M))) ) ).

% gchoose_row_sum_weighted
tff(fact_4042_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% gauss_sum_from_Suc_0
tff(fact_4043_atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),set_or1337092689740270186AtMost(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),I2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),U)) ) ) ) ).

% atLeastAtMost_iff
tff(fact_4044_Icc__eq__Icc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,H: A,L3: A,H2: A] :
          ( ( set_or1337092689740270186AtMost(A,L,H) = set_or1337092689740270186AtMost(A,L3,H2) )
        <=> ( ( ( L = L3 )
              & ( H = H2 ) )
            | ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
              & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L3),H2)) ) ) ) ) ).

% Icc_eq_Icc
tff(fact_4045_atLeastatMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2)))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_4046_Icc__subset__Iic__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,H2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),aa(A,set(A),set_ord_atMost(A),H2)))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),H),H2)) ) ) ) ).

% Icc_subset_Iic_iff
tff(fact_4047_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ) ) ) ).

% sum.cl_ivl_Suc
tff(fact_4048_not__Iic__eq__Icc,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H2: A,L: A,H: A] : aa(A,set(A),set_ord_atMost(A),H2) != set_or1337092689740270186AtMost(A,L,H) ) ).

% not_Iic_eq_Icc
tff(fact_4049_not__Iic__le__Icc,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L3: A,H2: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),H)),set_or1337092689740270186AtMost(A,L3,H2))) ) ).

% not_Iic_le_Icc
tff(fact_4050_all__nat__less,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ! [M7: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),N))
         => pp(aa(nat,bool,P,M7)) )
    <=> ! [X4: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
         => pp(aa(nat,bool,P,X4)) ) ) ).

% all_nat_less
tff(fact_4051_ex__nat__less,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ? [M7: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),N))
          & pp(aa(nat,bool,P,M7)) )
    <=> ? [X4: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
          & pp(aa(nat,bool,P,X4)) ) ) ).

% ex_nat_less
tff(fact_4052_atMost__atLeast0,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_atMost(nat),N) = set_or1337092689740270186AtMost(nat,zero_zero(nat),N) ).

% atMost_atLeast0
tff(fact_4053_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cp(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_4054_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fl(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_4055_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2)))
        <=> ( ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
              | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2))
                & ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A2))
                  | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D2)) ) ) )
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),D2)) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_4056_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,N,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_fm(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,N,M)) ) ).

% sum.atLeastAtMost_rev
tff(fact_4057_sum__shift__lb__Suc0__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F3,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or1337092689740270186AtMost(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_4058_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% sum.atLeast0_atMost_Suc
tff(fact_4059_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_4060_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,aa(nat,nat,suc,N))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_4061_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cp(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_4062_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_en(fun(nat,A),fun(nat,A),F3)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,minus_minus(A,aa(nat,A,F3,aa(nat,nat,suc,N))),aa(nat,A,F3,M)) ) ) ) ).

% sum_Suc_diff
tff(fact_4063_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cp(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_4064_sum__bounds__lt__plus1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),Mm: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cp(fun(nat,A),fun(nat,A),F3)),aa(nat,set(nat),set_ord_lessThan(nat),Mm)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ).

% sum_bounds_lt_plus1
tff(fact_4065_sum_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fn(fun(nat,fun(nat,A)),fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fp(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.nested_swap'
tff(fact_4066_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),A2: nat,B2: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_fq(fun(nat,A),fun(nat,fun(A,A)),F3),A2,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_4067_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A),P2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_4068_sum__up__index__split,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),aa(nat,set(nat),set_ord_atMost(nat),M))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)))) ) ).

% sum_up_index_split
tff(fact_4069_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F3: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fr(fun(nat,A),fun(nat,A),F3)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,minus_minus(A,aa(nat,A,F3,M)),aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fr(fun(nat,A),fun(nat,A),F3)),set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) ) ) ) ).

% sum_natinterval_diff
tff(fact_4070_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_fs(fun(nat,A),fun(nat,A),F3)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) = aa(A,A,minus_minus(A,aa(nat,A,F3,N)),aa(nat,A,F3,M)) ) ) ) ).

% sum_telescope''
tff(fact_4071_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,N: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,minus_minus(nat,N),M)))) ) ) ) ).

% sum_power_shift
tff(fact_4072_summable__partial__sum__bound,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),E2: real] :
          ( summable(A,F3)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
           => ~ ! [N9: nat] :
                  ~ ! [M2: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N9),M2))
                     => ! [N7: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or1337092689740270186AtMost(nat,M2,N7)))),E2)) ) ) ) ) ).

% summable_partial_sum_bound
tff(fact_4073_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,N: nat,X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),X)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),set_or1337092689740270186AtMost(nat,M,N))) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,X),M)),aa(nat,A,power_power(A,X),aa(nat,nat,suc,N))) ) ) ) ).

% sum_gp_multiplied
tff(fact_4074_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_da(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.in_pairs
tff(fact_4075_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat,K: A] :
          ( ! [X4: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_cr(fun(nat,A),fun(A,fun(nat,A)),C2),X4)),aa(nat,set(nat),set_ord_atMost(nat),N)) = K
        <=> ( ( aa(nat,A,C2,zero_zero(nat)) = K )
            & ! [X4: nat] :
                ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),set_or1337092689740270186AtMost(nat,one_one(nat),N)))
               => ( aa(nat,A,C2,X4) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_4076_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ft(nat,fun(nat,A),K)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_4077_gauss__sum__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fu(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% gauss_sum_nat
tff(fact_4078_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,D2: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fv(A,fun(A,fun(nat,A)),A2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = 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),N)),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),aa(num,num,bit0,one2))),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D2))) ) ).

% double_arith_series
tff(fact_4079_double__gauss__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).

% double_gauss_sum
tff(fact_4080_arith__series__nat,axiom,
    ! [A2: nat,D2: nat,N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_fw(nat,fun(nat,fun(nat,nat)),A2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = divide_divide(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),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),aa(num,num,bit0,one2))),A2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),D2))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% arith_series_nat
tff(fact_4081_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fu(nat,nat)),set_or1337092689740270186AtMost(nat,M,N)) = divide_divide(nat,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,minus_minus(nat,M),one_one(nat)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Sum_Icc_nat
tff(fact_4082_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,D2: A,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_fx(A,fun(A,fun(nat,A)),A2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = 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),N)),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),aa(num,num,bit0,one2))),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D2))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% arith_series
tff(fact_4083_gauss__sum,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% gauss_sum
tff(fact_4084_double__gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_4085_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [X: A,M: nat,N: nat] :
          ( ( ( X = one_one(A) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)) ) )
          & ( ( X != one_one(A) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,power_power(A,X)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),M)),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,power_power(A,X),aa(nat,nat,suc,N)))),aa(A,A,minus_minus(A,one_one(A)),X)) ) ) ) ) ).

% sum_gp_offset
tff(fact_4086_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,A2: fun(nat,A),X: A,Y: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N))
         => ( aa(A,A,minus_minus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),A2),X)),aa(nat,set(nat),set_ord_atMost(nat),N))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,X),Y)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_fz(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),N),A2),X),Y)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ) ) ).

% polyfun_diff
tff(fact_4087_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,N))),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),aa(num,num,bit0,one2)))),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_ga(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),aa(num,num,bit0,one2))),N)),one_one(nat)))) ) ).

% pochhammer_times_pochhammer_half
tff(fact_4088_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(X) = Y )
     => ( pp(aa(nat,bool,accp(nat,vEBT_v4011308405150292612up_rel),X))
       => ( ( ( X = zero_zero(nat) )
           => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
             => ~ pp(aa(nat,bool,accp(nat,vEBT_v4011308405150292612up_rel),zero_zero(nat))) ) )
         => ( ( ( X = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = vEBT_Leaf(fFalse,fFalse) )
               => ~ pp(aa(nat,bool,accp(nat,vEBT_v4011308405150292612up_rel),aa(nat,nat,suc,zero_zero(nat)))) ) )
           => ~ ! [Va2: nat] :
                  ( ( X = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                 => ( ( ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
                       => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) )
                      & ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2))))
                       => ( Y = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),vEBT_vebt_buildup(aa(nat,nat,suc,divide_divide(nat,aa(nat,nat,suc,aa(nat,nat,suc,Va2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) )
                   => ~ pp(aa(nat,bool,accp(nat,vEBT_v4011308405150292612up_rel),aa(nat,nat,suc,aa(nat,nat,suc,Va2)))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_4089_divmod__algorithm__code_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit0,N)) = aa(product_prod(A,A),product_prod(A,A),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_gb(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N)) ) ).

% divmod_algorithm_code(6)
tff(fact_4090_arctan__def,axiom,
    ! [Y: real] : aa(real,real,arctan,Y) = the(real,aTP_Lamp_gc(real,fun(real,bool),Y)) ).

% arctan_def
tff(fact_4091_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_gd(B,A)),A3) = one_one(A) ) ).

% prod.neutral_const
tff(fact_4092_prod_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = 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),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(nat,A,G,N)) ) ).

% prod.lessThan_Suc
tff(fact_4093_prod_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = 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),aa(nat,set(nat),set_ord_atMost(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% prod.atMost_Suc
tff(fact_4094_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,N)),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = 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,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ) ) ) ).

% prod.cl_ivl_Suc
tff(fact_4095_divmod__algorithm__code_I5_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,aa(num,num,bit0,M),aa(num,num,bit0,N)) = aa(product_prod(A,A),product_prod(A,A),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_ge(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N)) ) ).

% divmod_algorithm_code(5)
tff(fact_4096_mod__prod__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F3: fun(B,A),A2: A,A3: set(B)] : modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(A,fun(B,A),aTP_Lamp_co(fun(B,A),fun(A,fun(B,A)),F3),A2)),A3),A2) = modulo_modulo(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A3),A2) ) ).

% mod_prod_eq
tff(fact_4097_norm__prod__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [F3: fun(B,A),A3: set(B)] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A3))),aa(set(B),real,aa(fun(B,real),fun(set(B),real),groups7121269368397514597t_prod(B,real),aTP_Lamp_gf(fun(B,A),fun(B,real),F3)),A3))) ) ).

% norm_prod_le
tff(fact_4098_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) )
         => ~ ! [A5: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),A3))
               => ( aa(B,A,G,A5) = one_one(A) ) ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_4099_prod_Oneutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),G: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => ( aa(B,A,G,X3) = one_one(A) ) )
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3) = one_one(A) ) ) ) ).

% prod.neutral
tff(fact_4100_prod__dividef,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F3: fun(B,A),G: 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),aTP_Lamp_gg(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G)),A3) = divide_divide(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A3),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3)) ) ).

% prod_dividef
tff(fact_4101_prod__power__distrib,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [F3: fun(A,B),A3: set(A),N: nat] : aa(nat,B,power_power(B,aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A3)),N) = aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),aa(nat,fun(A,B),aTP_Lamp_gh(fun(A,B),fun(nat,fun(A,B)),F3),N)),A3) ) ).

% prod_power_distrib
tff(fact_4102_prod__nonneg,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A3: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A3))) ) ) ).

% prod_nonneg
tff(fact_4103_prod__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A3: set(B),F3: fun(B,A),G: fun(B,A)] :
          ( ! [I4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I4)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I4)),aa(B,A,G,I4))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3))) ) ) ).

% prod_mono
tff(fact_4104_prod__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A3: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F3,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A3))) ) ) ).

% prod_pos
tff(fact_4105_prod__ge__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(B,A,F3,X3))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A3))) ) ) ).

% prod_ge_1
tff(fact_4106_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: 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,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gi(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_4107_power__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C2: A,F3: fun(B,nat),A3: set(B)] : aa(nat,A,power_power(A,C2),aa(set(B),nat,groups7311177749621191930dd_sum(B,nat,F3),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,nat),fun(B,A),aTP_Lamp_gj(A,fun(fun(B,nat),fun(B,A)),C2),F3)),A3) ) ).

% power_sum
tff(fact_4108_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: 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),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gk(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_4109_prod__le__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: set(B),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,X3)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),one_one(A))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A3)),one_one(A))) ) ) ).

% prod_le_1
tff(fact_4110_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gl(fun(nat,A),fun(nat,fun(nat,A)),G),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.nat_diff_reindex
tff(fact_4111_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_gm(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,N,M)) ) ).

% prod.atLeastAtMost_rev
tff(fact_4112_ln__neg__is__const,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real)))
     => ( aa(real,real,ln_ln(real),X) = the(real,aTP_Lamp_gn(real,bool)) ) ) ).

% ln_neg_is_const
tff(fact_4113_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: 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,N))) = 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),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_4114_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_4115_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,aa(nat,nat,suc,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_4116_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N))) = 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_gi(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% prod.lessThan_Suc_shift
tff(fact_4117_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( 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,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gi(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_4118_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N))) = 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_gi(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N))) ) ).

% prod.atMost_Suc_shift
tff(fact_4119_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: 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)),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gi(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_4120_fact__prod,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = 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_fu(nat,nat)),set_or1337092689740270186AtMost(nat,one_one(nat),N))) ) ).

% fact_prod
tff(fact_4121_prod_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_go(fun(nat,fun(nat,A)),fun(nat,A),A2)),aa(nat,set(nat),set_ord_atMost(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gq(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),N)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.nested_swap'
tff(fact_4122_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F3: fun(nat,A),A2: nat,B2: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),F3),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_gr(fun(nat,A),fun(nat,fun(A,A)),F3),A2,B2,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_4123_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A),P2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2))) = 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,M,N))),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),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_4124_periodic__finite__ex,axiom,
    ! [D2: int,P: fun(int,bool)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D2))
     => ( ! [X3: int,K2: int] :
            ( pp(aa(int,bool,P,X3))
          <=> pp(aa(int,bool,P,aa(int,int,minus_minus(int,X3),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D2)))) )
       => ( ? [X_1: int] : pp(aa(int,bool,P,X_1))
        <=> ? [X4: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X4),set_or1337092689740270186AtMost(int,one_one(int),D2)))
              & pp(aa(int,bool,P,X4)) ) ) ) ) ).

% periodic_finite_ex
tff(fact_4125_bset_I3_J,axiom,
    ! [D4: int,T2: int,B3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(int,int,minus_minus(int,T2),one_one(int))),B3))
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B3))
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
           => ( ( X2 = T2 )
             => ( aa(int,int,minus_minus(int,X2),D4) = T2 ) ) ) ) ) ).

% bset(3)
tff(fact_4126_bset_I4_J,axiom,
    ! [D4: int,T2: int,B3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),T2),B3))
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B3))
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
           => ( ( X2 != T2 )
             => ( aa(int,int,minus_minus(int,X2),D4) != T2 ) ) ) ) ) ).

% bset(4)
tff(fact_4127_bset_I5_J,axiom,
    ! [D4: int,B3: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B3))
                 => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X2),T2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,minus_minus(int,X2),D4)),T2)) ) ) ) ).

% bset(5)
tff(fact_4128_bset_I7_J,axiom,
    ! [D4: int,T2: int,B3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),T2),B3))
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B3))
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),X2))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),aa(int,int,minus_minus(int,X2),D4))) ) ) ) ) ).

% bset(7)
tff(fact_4129_aset_I3_J,axiom,
    ! [D4: int,T2: int,A3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),T2),one_one(int))),A3))
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                   => ( X2 != aa(int,int,minus_minus(int,Xb2),Xa3) ) ) )
           => ( ( X2 = T2 )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D4) = T2 ) ) ) ) ) ).

% aset(3)
tff(fact_4130_aset_I4_J,axiom,
    ! [D4: int,T2: int,A3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),T2),A3))
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                   => ( X2 != aa(int,int,minus_minus(int,Xb2),Xa3) ) ) )
           => ( ( X2 != T2 )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D4) != T2 ) ) ) ) ) ).

% aset(4)
tff(fact_4131_aset_I5_J,axiom,
    ! [D4: int,T2: int,A3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),T2),A3))
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                   => ( X2 != aa(int,int,minus_minus(int,Xb2),Xa3) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X2),T2))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D4)),T2)) ) ) ) ) ).

% aset(5)
tff(fact_4132_aset_I7_J,axiom,
    ! [D4: int,A3: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                 => ( X2 != aa(int,int,minus_minus(int,Xb2),Xa3) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),X2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),T2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D4))) ) ) ) ).

% aset(7)
tff(fact_4133_norm__prod__diff,axiom,
    ! [A: $tType,I6: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [I5: set(I6),Z: fun(I6,A),W: fun(I6,A)] :
          ( ! [I4: I6] :
              ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I4),I5))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(I6,A,Z,I4))),one_one(real))) )
         => ( ! [I4: I6] :
                ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I4),I5))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(I6,A,W,I4))),one_one(real))) )
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(set(I6),A,aa(fun(I6,A),fun(set(I6),A),groups7121269368397514597t_prod(I6,A),Z),I5)),aa(set(I6),A,aa(fun(I6,A),fun(set(I6),A),groups7121269368397514597t_prod(I6,A),W),I5)))),aa(set(I6),real,groups7311177749621191930dd_sum(I6,real,aa(fun(I6,A),fun(I6,real),aTP_Lamp_gs(fun(I6,A),fun(fun(I6,A),fun(I6,real)),Z),W)),I5))) ) ) ) ).

% norm_prod_diff
tff(fact_4134_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_atMost(nat),N)) = 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_gi(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N))) ) ).

% prod.atMost_shift
tff(fact_4135_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( semiring_char_0_fact(nat,M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,N)),aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_fu(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M))) ) ) ).

% fact_eq_fact_times
tff(fact_4136_bset_I6_J,axiom,
    ! [D4: int,B3: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B3))
                 => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X2),T2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,minus_minus(int,X2),D4)),T2)) ) ) ) ).

% bset(6)
tff(fact_4137_bset_I8_J,axiom,
    ! [D4: int,T2: int,B3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(int,int,minus_minus(int,T2),one_one(int))),B3))
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),B3))
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa3) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T2),X2))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T2),aa(int,int,minus_minus(int,X2),D4))) ) ) ) ) ).

% bset(8)
tff(fact_4138_aset_I6_J,axiom,
    ! [D4: int,T2: int,A3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),T2),one_one(int))),A3))
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
               => ! [Xb2: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                   => ( X2 != aa(int,int,minus_minus(int,Xb2),Xa3) ) ) )
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X2),T2))
             => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D4)),T2)) ) ) ) ) ).

% aset(6)
tff(fact_4139_aset_I8_J,axiom,
    ! [D4: int,A3: set(int),T2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa3),set_or1337092689740270186AtMost(int,one_one(int),D4)))
             => ! [Xb2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb2),A3))
                 => ( X2 != aa(int,int,minus_minus(int,Xb2),Xa3) ) ) )
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T2),X2))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),T2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D4))) ) ) ) ).

% aset(8)
tff(fact_4140_cppi,axiom,
    ! [D4: int,P: fun(int,bool),P3: fun(int,bool),A3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Z3),X3))
           => ( pp(aa(int,bool,P,X3))
            <=> pp(aa(int,bool,P3,X3)) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                 => ! [Xb3: int] :
                      ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),A3))
                     => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa2) ) ) )
             => ( pp(aa(int,bool,P,X3))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D4))) ) )
         => ( ! [X3: int,K2: int] :
                ( pp(aa(int,bool,P3,X3))
              <=> pp(aa(int,bool,P3,aa(int,int,minus_minus(int,X3),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D4)))) )
           => ( ? [X_1: int] : pp(aa(int,bool,P,X_1))
            <=> ( ? [X4: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X4),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                    & pp(aa(int,bool,P3,X4)) )
                | ? [X4: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X4),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                    & ? [Xa4: int] :
                        ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),A3))
                        & pp(aa(int,bool,P,aa(int,int,minus_minus(int,Xa4),X4))) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_4141_cpmi,axiom,
    ! [D4: int,P: fun(int,bool),P3: fun(int,bool),B3: set(int)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),D4))
     => ( ? [Z3: int] :
          ! [X3: int] :
            ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X3),Z3))
           => ( pp(aa(int,bool,P,X3))
            <=> pp(aa(int,bool,P3,X3)) ) )
       => ( ! [X3: int] :
              ( ! [Xa2: int] :
                  ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa2),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                 => ! [Xb3: int] :
                      ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xb3),B3))
                     => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa2) ) ) )
             => ( pp(aa(int,bool,P,X3))
               => pp(aa(int,bool,P,aa(int,int,minus_minus(int,X3),D4))) ) )
         => ( ! [X3: int,K2: int] :
                ( pp(aa(int,bool,P3,X3))
              <=> pp(aa(int,bool,P3,aa(int,int,minus_minus(int,X3),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D4)))) )
           => ( ? [X_1: int] : pp(aa(int,bool,P,X_1))
            <=> ( ? [X4: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X4),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                    & pp(aa(int,bool,P3,X4)) )
                | ? [X4: int] :
                    ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X4),set_or1337092689740270186AtMost(int,one_one(int),D4)))
                    & ? [Xa4: int] :
                        ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa4),B3))
                        & pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa4),X4))) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_4142_arccos__def,axiom,
    ! [Y: real] : aa(real,real,arccos,Y) = the(real,aTP_Lamp_gt(real,fun(real,bool),Y)) ).

% arccos_def
tff(fact_4143_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gu(A,fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod
tff(fact_4144_pochhammer__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,N) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gv(A,fun(nat,fun(nat,A)),A2),N)),set_or1337092689740270186AtMost(nat,one_one(nat),N)) ) ).

% pochhammer_prod_rev
tff(fact_4145_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( divide_divide(nat,semiring_char_0_fact(nat,M),semiring_char_0_fact(nat,N)) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_fu(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),M)) ) ) ).

% fact_div_fact
tff(fact_4146_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: 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),aa(num,num,bit0,one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gw(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.in_pairs
tff(fact_4147_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),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),aa(num,num,bit0,one2))),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gw(fun(nat,A),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% prod.in_pairs_0
tff(fact_4148_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gv(A,fun(nat,fun(nat,A)),A2),N)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_4149_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P2: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),P2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),P2))
           => ( 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_gx(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),P2)) = 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_gy(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,minus_minus(nat,P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_4150_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_gz(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_4151_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_ha(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_4152_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_hb(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_4153_pi__half,axiom,
    divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) = the(real,aTP_Lamp_hc(real,bool)) ).

% pi_half
tff(fact_4154_pi__def,axiom,
    pi = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),the(real,aTP_Lamp_hc(real,bool))) ).

% pi_def
tff(fact_4155_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M),N))
     => ( aa(set(int),int,groups7311177749621191930dd_sum(int,int,aTP_Lamp_hd(int,int)),set_or1337092689740270186AtMost(int,M,N)) = divide_divide(int,aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),N),one_one(int)))),aa(int,int,aa(int,fun(int,int),times_times(int),M),aa(int,int,minus_minus(int,M),one_one(int)))),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))) ) ) ).

% Sum_Icc_int
tff(fact_4156_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_he(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_4157_arcsin__def,axiom,
    ! [Y: real] : aa(real,real,arcsin,Y) = the(real,aTP_Lamp_hf(real,fun(real,bool),Y)) ).

% arcsin_def
tff(fact_4158_divmod__nat__if,axiom,
    ! [N: nat,M: nat] :
      ( ( ( ( N = zero_zero(nat) )
          | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) )
       => ( divmod_nat(M,N) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,zero_zero(nat)),M) ) )
      & ( ~ ( ( N = zero_zero(nat) )
            | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) )
       => ( divmod_nat(M,N) = aa(product_prod(nat,nat),product_prod(nat,nat),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_hg(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,minus_minus(nat,M),N),N)) ) ) ) ).

% divmod_nat_if
tff(fact_4159_set__n__lists,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,N,Xs)) = collect(list(A),aa(list(A),fun(list(A),bool),aTP_Lamp_hh(nat,fun(list(A),fun(list(A),bool)),N),Xs)) ).

% set_n_lists
tff(fact_4160_pred__subset__eq,axiom,
    ! [A: $tType,R: set(A),S2: set(A)] :
      ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),aTP_Lamp_a(set(A),fun(A,bool),R)),aTP_Lamp_a(set(A),fun(A,bool),S2)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),R),S2)) ) ).

% pred_subset_eq
tff(fact_4161_complex__mult__cnj,axiom,
    ! [Z: complex] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z)) = real_Vector_of_real(complex,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,power_power(real,re(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,power_power(real,im(Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% complex_mult_cnj
tff(fact_4162_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),N: 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,bool,aTP_Lamp_hi(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hk(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% prod.triangle_reindex_eq
tff(fact_4163_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),N: 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,bool,aTP_Lamp_hl(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_hk(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% prod.triangle_reindex
tff(fact_4164_prod__int__eq,axiom,
    ! [I2: 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,I2,J)) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_hd(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I2),aa(nat,int,semiring_1_of_nat(int),J))) ).

% prod_int_eq
tff(fact_4165_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,groups7311177749621191930dd_sum(product_prod(nat,nat),A,product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_hi(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_hn(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ).

% sum.triangle_reindex_eq
tff(fact_4166_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,groups7311177749621191930dd_sum(product_prod(nat,nat),A,product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_hl(nat,fun(nat,fun(nat,bool)),N)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_hn(fun(nat,fun(nat,A)),fun(nat,A),G)),aa(nat,set(nat),set_ord_lessThan(nat),N)) ) ).

% sum.triangle_reindex
tff(fact_4167_prod__int__plus__eq,axiom,
    ! [I2: 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,I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J))) = aa(set(int),int,aa(fun(int,int),fun(set(int),int),groups7121269368397514597t_prod(int,int),aTP_Lamp_hd(int,int)),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I2),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J)))) ).

% prod_int_plus_eq
tff(fact_4168_subrelI,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( ! [X3: A,Y4: B] :
          ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,X3),Y4)),R3))
         => pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,X3),Y4)),S)) )
     => pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R3),S)) ) ).

% subrelI
tff(fact_4169_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),aTP_Lamp_ho(set(product_prod(A,B)),fun(A,fun(B,bool)),R)),aTP_Lamp_ho(set(product_prod(A,B)),fun(A,fun(B,bool)),S2)))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R),S2)) ) ).

% pred_subset_eq2
tff(fact_4170_length__n__lists__elem,axiom,
    ! [A: $tType,Ys: list(A),N: nat,Xs: list(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Ys),aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,N,Xs))))
     => ( aa(list(A),nat,size_size(list(A)),Ys) = N ) ) ).

% length_n_lists_elem
tff(fact_4171_Re__complex__div__gt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),re(divide_divide(complex,A2,B2))))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2))))) ) ).

% Re_complex_div_gt_0
tff(fact_4172_Re__complex__div__lt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),re(divide_divide(complex,A2,B2))),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))),zero_zero(real))) ) ).

% Re_complex_div_lt_0
tff(fact_4173_Re__complex__div__ge__0,axiom,
    ! [A2: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),re(divide_divide(complex,A2,B2))))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2))))) ) ).

% Re_complex_div_ge_0
tff(fact_4174_Re__complex__div__le__0,axiom,
    ! [A2: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),re(divide_divide(complex,A2,B2))),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))),zero_zero(real))) ) ).

% Re_complex_div_le_0
tff(fact_4175_Im__complex__div__gt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),im(divide_divide(complex,A2,B2))))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2))))) ) ).

% Im_complex_div_gt_0
tff(fact_4176_Im__complex__div__lt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),im(divide_divide(complex,A2,B2))),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))),zero_zero(real))) ) ).

% Im_complex_div_lt_0
tff(fact_4177_Im__complex__div__ge__0,axiom,
    ! [A2: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),im(divide_divide(complex,A2,B2))))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2))))) ) ).

% Im_complex_div_ge_0
tff(fact_4178_Im__complex__div__le__0,axiom,
    ! [A2: complex,B2: complex] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),im(divide_divide(complex,A2,B2))),zero_zero(real)))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)))),zero_zero(real))) ) ).

% Im_complex_div_le_0
tff(fact_4179_complex__mod__mult__cnj,axiom,
    ! [Z: complex] : real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z))) = aa(nat,real,power_power(real,real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% complex_mod_mult_cnj
tff(fact_4180_complex__div__gt__0,axiom,
    ! [A2: complex,B2: complex] :
      ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),re(divide_divide(complex,A2,B2))))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2))))) )
      & ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),im(divide_divide(complex,A2,B2))))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),im(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2))))) ) ) ).

% complex_div_gt_0
tff(fact_4181_complex__norm__square,axiom,
    ! [Z: complex] : real_Vector_of_real(complex,aa(nat,real,power_power(real,real_V7770717601297561774m_norm(complex,Z)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(Z)) ).

% complex_norm_square
tff(fact_4182_divmod__nat__def,axiom,
    ! [M: nat,N: nat] : divmod_nat(M,N) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,divide_divide(nat,M,N)),modulo_modulo(nat,M,N)) ).

% divmod_nat_def
tff(fact_4183_complex__add__cnj,axiom,
    ! [Z: complex] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),Z),cnj(Z)) = real_Vector_of_real(complex,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),re(Z))) ).

% complex_add_cnj
tff(fact_4184_complex__diff__cnj,axiom,
    ! [Z: complex] : aa(complex,complex,minus_minus(complex,Z),cnj(Z)) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),im(Z)))),imaginary_unit) ).

% complex_diff_cnj
tff(fact_4185_complex__div__cnj,axiom,
    ! [A2: complex,B2: complex] : divide_divide(complex,A2,B2) = divide_divide(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),A2),cnj(B2)),real_Vector_of_real(complex,aa(nat,real,power_power(real,real_V7770717601297561774m_norm(complex,B2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% complex_div_cnj
tff(fact_4186_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(W))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),cnj(Z)),W)) = real_Vector_of_real(complex,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),re(aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Z),cnj(W))))) ).

% cnj_add_mult_eq_Re
tff(fact_4187_of__nat__code__if,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(nat,A,semiring_1_of_nat(A),N) = zero_zero(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,A,semiring_1_of_nat(A),N) = aa(product_prod(nat,nat),A,product_case_prod(nat,nat,A,aTP_Lamp_hp(nat,fun(nat,A))),divmod_nat(N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ) ) ) ) ).

% of_nat_code_if
tff(fact_4188_length__n__lists,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),n_lists(A,N,Xs)) = aa(nat,nat,power_power(nat,aa(list(A),nat,size_size(list(A)),Xs)),N) ).

% length_n_lists
tff(fact_4189_set__encode__def,axiom,
    nat_set_encode = groups7311177749621191930dd_sum(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) ).

% set_encode_def
tff(fact_4190_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),aa(list(vEBT_VEBT),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_4191_predicate2I,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),Q: fun(A,fun(B,bool))] :
      ( ! [X3: A,Y4: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),P,X3),Y4))
         => pp(aa(B,bool,aa(A,fun(B,bool),Q,X3),Y4)) )
     => pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),P),Q)) ) ).

% predicate2I
tff(fact_4192_rev__predicate2D,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),X: A,Y: B,Q: fun(A,fun(B,bool))] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y))
     => ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),P),Q))
       => pp(aa(B,bool,aa(A,fun(B,bool),Q,X),Y)) ) ) ).

% rev_predicate2D
tff(fact_4193_predicate2D,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Q: fun(A,fun(B,bool)),X: A,Y: B] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),P),Q))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y))
       => pp(aa(B,bool,aa(A,fun(B,bool),Q,X),Y)) ) ) ).

% predicate2D
tff(fact_4194_size__list__estimation,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: nat,F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),aa(A,nat,F3,X)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y),aa(list(A),nat,size_list(A,F3),Xs))) ) ) ).

% size_list_estimation
tff(fact_4195_size__list__estimation_H,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: nat,F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),aa(A,nat,F3,X)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y),aa(list(A),nat,size_list(A,F3),Xs))) ) ) ).

% size_list_estimation'
tff(fact_4196_size__list__pointwise,axiom,
    ! [A: $tType,Xs: list(A),F3: fun(A,nat),G: fun(A,nat)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,X3)),aa(A,nat,G,X3))) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_list(A,F3),Xs)),aa(list(A),nat,size_list(A,G),Xs))) ) ).

% size_list_pointwise
tff(fact_4197_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),aa(list(vEBT_VEBT),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_4198_accp__subset,axiom,
    ! [A: $tType,R1: fun(A,fun(A,bool)),R22: fun(A,fun(A,bool))] :
      ( pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),R1),R22))
     => pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),accp(A,R22)),accp(A,R1))) ) ).

% accp_subset
tff(fact_4199_Collect__case__prod__mono,axiom,
    ! [B: $tType,A: $tType,A3: fun(A,fun(B,bool)),B3: fun(A,fun(B,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),A3),B3))
     => pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),collect(product_prod(A,B),product_case_prod(A,B,bool,A3))),collect(product_prod(A,B),product_case_prod(A,B,bool,B3)))) ) ).

% Collect_case_prod_mono
tff(fact_4200_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fu(nat,nat)),set_or7035219750837199246ssThan(nat,M,N)) = divide_divide(nat,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,minus_minus(nat,N),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,minus_minus(nat,M),one_one(nat)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% Sum_Ico_nat
tff(fact_4201_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),set_or7035219750837199246ssThan(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),I2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),U)) ) ) ) ).

% atLeastLessThan_iff
tff(fact_4202_ivl__subset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I2: A,J: A,M: A,N: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,I2,J)),set_or7035219750837199246ssThan(A,M,N)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),J),I2))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),I2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),J),N)) ) ) ) ) ).

% ivl_subset
tff(fact_4203_ivl__diff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I2: A,N: A,M: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),N))
         => ( aa(set(A),set(A),minus_minus(set(A),set_or7035219750837199246ssThan(A,I2,M)),set_or7035219750837199246ssThan(A,I2,N)) = set_or7035219750837199246ssThan(A,N,M) ) ) ) ).

% ivl_diff
tff(fact_4204_lessThan__minus__lessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [N: A,M: A] : aa(set(A),set(A),minus_minus(set(A),aa(A,set(A),set_ord_lessThan(A),N)),aa(A,set(A),set_ord_lessThan(A),M)) = set_or7035219750837199246ssThan(A,M,N) ) ).

% lessThan_minus_lessThan
tff(fact_4205_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).

% sum.op_ivl_Suc
tff(fact_4206_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = 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,M,N))),aa(nat,A,G,N)) ) ) ) ) ).

% prod.op_ivl_Suc
tff(fact_4207_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
           => ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
            <=> ( ( A2 = C2 )
                & ( B2 = D2 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_4208_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
             => ( A2 = C2 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_4209_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),D2))
             => ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_4210_atLeastLessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2)))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_4211_ex__nat__less__eq,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ? [M7: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M7),N))
          & pp(aa(nat,bool,P,M7)) )
    <=> ? [X4: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
          & pp(aa(nat,bool,P,X4)) ) ) ).

% ex_nat_less_eq
tff(fact_4212_all__nat__less__eq,axiom,
    ! [N: nat,P: fun(nat,bool)] :
      ( ! [M7: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M7),N))
         => pp(aa(nat,bool,P,M7)) )
    <=> ! [X4: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
         => pp(aa(nat,bool,P,X4)) ) ) ).

% all_nat_less_eq
tff(fact_4213_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_4214_lessThan__atLeast0,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_lessThan(nat),N) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ).

% lessThan_atLeast0
tff(fact_4215_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_cp(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_4216_sum_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fl(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_4217_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: 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,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gi(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_4218_prod_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: 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),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gk(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_4219_sum_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & comm_monoid_add(A) )
     => ! [A2: B,C2: B,B2: B,D2: B,G: fun(B,A),H: fun(B,A)] :
          ( ( A2 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),X3))
                 => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),D2))
                   => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) ) )
             => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),set_or7035219750837199246ssThan(B,A2,B2)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),set_or7035219750837199246ssThan(B,C2,D2)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_4220_prod_Oivl__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & comm_monoid_mult(A) )
     => ! [A2: B,C2: B,B2: B,D2: B,G: fun(B,A),H: fun(B,A)] :
          ( ( A2 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),X3))
                 => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),X3),D2))
                   => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),set_or7035219750837199246ssThan(B,A2,B2)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),set_or7035219750837199246ssThan(B,C2,D2)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_4221_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,P2: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P2))
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,N,P2))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,P2)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_4222_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,P2: nat,F3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P2))
           => ( aa(A,A,minus_minus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or7035219750837199246ssThan(nat,M,P2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or7035219750837199246ssThan(nat,M,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or7035219750837199246ssThan(nat,N,P2)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_4223_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,P2: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),P2))
           => ( 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,M,N))),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,N,P2))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,P2)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_4224_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D2)) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_4225_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_4226_sum__shift__lb__Suc0__0__upt,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F3,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_4227_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,A,G,N)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_4228_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_4229_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: nat,B2: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),A2),B2))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,A2,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G,B2)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_4230_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: 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,N))) = 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),N))),aa(nat,A,G,N)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_4231_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_4232_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: nat,B2: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_4233_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,N)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N))) ) ) ) ).

% sum.last_plus
tff(fact_4234_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,N)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N))) ) ) ) ).

% prod.last_plus
tff(fact_4235_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,F3: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_en(fun(nat,A),fun(nat,A),F3)),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,minus_minus(A,aa(nat,A,F3,N)),aa(nat,A,F3,M)) ) ) ) ).

% sum_Suc_diff'
tff(fact_4236_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_hq(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or7035219750837199246ssThan(nat,N,M)) ) ).

% sum.atLeastLessThan_rev
tff(fact_4237_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_hr(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fp(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% sum.nested_swap
tff(fact_4238_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_hs(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or7035219750837199246ssThan(nat,N,M)) ) ).

% prod.atLeastLessThan_rev
tff(fact_4239_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),N: nat] : 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),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_gq(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% prod.nested_swap
tff(fact_4240_sum_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),K: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hu(fun(nat,A),fun(nat,fun(nat,A)),G),K)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ).

% sum.nat_group
tff(fact_4241_prod_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),K: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hv(fun(nat,A),fun(nat,fun(nat,A)),G),K)),aa(nat,set(nat),set_ord_lessThan(nat),N)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ).

% prod.nat_group
tff(fact_4242_prod__Suc__fact,axiom,
    ! [N: nat] : aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) = semiring_char_0_fact(nat,N) ).

% prod_Suc_fact
tff(fact_4243_prod__Suc__Suc__fact,axiom,
    ! [N: 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)),N)) = semiring_char_0_fact(nat,N) ).

% prod_Suc_Suc_fact
tff(fact_4244_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N)) = zero_zero(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N)) ) ) ) ) ).

% sum.head_if
tff(fact_4245_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [N: nat,M: nat,G: fun(nat,A)] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = one_one(A) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
           => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = 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,M,N))),aa(nat,A,G,N)) ) ) ) ) ).

% prod.head_if
tff(fact_4246_fact__prod__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = 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),N))) ) ).

% fact_prod_Suc
tff(fact_4247_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_fm(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_4248_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_gm(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_4249_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,N: nat] : comm_s3205402744901411588hammer(A,A2,N) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aTP_Lamp_gu(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% pochhammer_prod
tff(fact_4250_summable__Cauchy,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A)] :
          ( summable(A,F3)
        <=> ! [E3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
             => ? [N6: nat] :
                ! [M7: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),M7))
                 => ! [N5: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),set_or7035219750837199246ssThan(nat,M7,N5)))),E3)) ) ) ) ) ).

% summable_Cauchy
tff(fact_4251_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: bool,X222: bool] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Leaf(X21,X222)) = zero_zero(nat) ).

% VEBT.size_gen(2)
tff(fact_4252_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F3: fun(nat,A),S: A,K: nat] :
          ( sums(A,F3,S)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
           => sums(A,aa(nat,fun(nat,A),aTP_Lamp_hw(fun(nat,A),fun(nat,fun(nat,A)),F3),K),S) ) ) ) ).

% sums_group
tff(fact_4253_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_hx(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% take_bit_sum
tff(fact_4254_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( semiring_char_0_fact(A,N) = 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,minus_minus(nat,N),K),N)))),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,N),K))) ) ) ) ).

% fact_split
tff(fact_4255_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K)) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(nat,fun(nat,A),aTP_Lamp_hy(nat,fun(nat,fun(nat,A)),K),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_4256_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_hz(A,fun(nat,fun(nat,A)),A2),K)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_4257_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_gz(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_4258_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F3: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F3,A2,Xs) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_ia(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F3),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_4259_accp__subset__induct,axiom,
    ! [A: $tType,D4: fun(A,bool),R: fun(A,fun(A,bool)),X: A,P: fun(A,bool)] :
      ( pp(aa(fun(A,bool),bool,aa(fun(A,bool),fun(fun(A,bool),bool),ord_less_eq(fun(A,bool)),D4),accp(A,R)))
     => ( ! [X3: A,Z2: A] :
            ( pp(aa(A,bool,D4,X3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),R,Z2),X3))
             => pp(aa(A,bool,D4,Z2)) ) )
       => ( pp(aa(A,bool,D4,X))
         => ( ! [X3: A] :
                ( pp(aa(A,bool,D4,X3))
               => ( ! [Z3: A] :
                      ( pp(aa(A,bool,aa(A,fun(A,bool),R,Z3),X3))
                     => pp(aa(A,bool,P,Z3)) )
                 => pp(aa(A,bool,P,X3)) ) )
           => pp(aa(A,bool,P,X)) ) ) ) ) ).

% accp_subset_induct
tff(fact_4260_sum__power2,axiom,
    ! [K: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K)),one_one(nat)) ).

% sum_power2
tff(fact_4261_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A2: fun(nat,A),B2: fun(nat,A)] :
          ( ! [I4: nat,J2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),J2))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),N))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A2,I4)),aa(nat,A,A2,J2))) ) )
         => ( ! [I4: nat,J2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),J2))
               => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),N))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,B2,J2)),aa(nat,A,B2,I4))) ) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ib(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))))) ) ) ) ).

% Chebyshev_sum_upper
tff(fact_4262_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A2: fun(nat,nat),B2: fun(nat,nat)] :
      ( ! [I4: nat,J2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),J2))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),N))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,A2,I4)),aa(nat,nat,A2,J2))) ) )
     => ( ! [I4: nat,J2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I4),J2))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J2),N))
             => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,B2,J2)),aa(nat,nat,B2,I4))) ) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_ic(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_4263_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [A: $tType,F3: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc2: A] :
      ( pp(aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool,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),A2),aa(A,product_prod(nat,A),product_Pair(nat,A,B2),Acc2)))))
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A2))
         => ( set_fo6178422350223883121st_nat(A,F3,A2,B2,Acc2) = Acc2 ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B2),A2))
         => ( set_fo6178422350223883121st_nat(A,F3,A2,B2,Acc2) = set_fo6178422350223883121st_nat(A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F3,A2),Acc2)) ) ) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_4264_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [A: $tType,X: fun(nat,fun(A,A)),Xa: nat,Xb: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,X,Xa,Xb,Xc) = Y )
     => ( pp(aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool,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)),X),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),Xa),aa(A,product_prod(nat,A),product_Pair(nat,A,Xb),Xc)))))
       => ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb),Xa))
               => ( Y = Xc ) )
              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Xb),Xa))
               => ( Y = set_fo6178422350223883121st_nat(A,X,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xa),one_one(nat)),Xb,aa(A,A,aa(nat,fun(A,A),X,Xa),Xc)) ) ) )
           => ~ pp(aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool,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)),X),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),Xa),aa(A,product_prod(nat,A),product_Pair(nat,A,Xb),Xc))))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_4265_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_4266_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,bool))))] :
      ( pp(aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool,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)))))
     => ( ! [F4: fun(nat,fun(A,A)),A5: nat,B5: nat,Acc: A] :
            ( pp(aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),bool,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)),F4),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),product_Pair(nat,product_prod(nat,A),A5),aa(A,product_prod(nat,A),product_Pair(nat,A,B5),Acc)))))
           => ( ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),B5),A5))
               => pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P,F4),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A5),one_one(nat))),B5),aa(A,A,aa(nat,fun(A,A),F4,A5),Acc))) )
             => pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P,F4),A5),B5),Acc)) ) )
       => pp(aa(A,bool,aa(nat,fun(A,bool),aa(nat,fun(nat,fun(A,bool)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,bool))),P,A0),A1),A22),A32)) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_4267_in__measure,axiom,
    ! [A: $tType,X: A,Y: A,F3: fun(A,nat)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y)),measure(A,F3)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y))) ) ).

% in_measure
tff(fact_4268_int__ge__less__than2__def,axiom,
    ! [D2: int] : int_ge_less_than2(D2) = collect(product_prod(int,int),product_case_prod(int,int,bool,aTP_Lamp_id(int,fun(int,fun(int,bool)),D2))) ).

% int_ge_less_than2_def
tff(fact_4269_int__ge__less__than__def,axiom,
    ! [D2: int] : int_ge_less_than(D2) = collect(product_prod(int,int),product_case_prod(int,int,bool,aTP_Lamp_ie(int,fun(int,fun(int,bool)),D2))) ).

% int_ge_less_than_def
tff(fact_4270_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,bool))] :
      ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),product_Pair(int,int,A0),A1)))
     => ( ! [I4: int,J2: int] :
            ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),product_Pair(int,int,I4),J2)))
           => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I4),J2))
               => pp(aa(int,bool,aa(int,fun(int,bool),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I4),one_one(int))),J2)) )
             => pp(aa(int,bool,aa(int,fun(int,bool),P,I4),J2)) ) )
       => pp(aa(int,bool,aa(int,fun(int,bool),P,A0),A1)) ) ) ).

% upto.pinduct
tff(fact_4271_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_if(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_4272_case__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F3: fun(nat,A),V: num,N: nat] : case_nat(A,A2,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),N)) = aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),N)) ).

% case_nat_add_eq_if
tff(fact_4273_rec__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F3: fun(nat,fun(A,A)),V: num,N: nat] : aa(nat,A,rec_nat(A,A2,F3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),N)) = aa(A,A,aa(nat,fun(A,A),F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),N)),aa(nat,A,rec_nat(A,A2,F3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),N))) ).

% rec_nat_add_eq_if
tff(fact_4274_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(bool,int,zero_neq_one_of_bool(int),fdisj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ).

% or_int_rec
tff(fact_4275_or_Oright__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) ) ).

% or.right_idem
tff(fact_4276_or_Oleft__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) ) ).

% or.left_idem
tff(fact_4277_or_Oidem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),A2) = A2 ) ).

% or.idem
tff(fact_4278_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_4279_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_4280_take__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),B2)) ) ).

% take_bit_or
tff(fact_4281_push__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se4730199178511100633sh_bit(A,N),A2)),aa(A,A,bit_se4730199178511100633sh_bit(A,N),B2)) ) ).

% push_bit_or
tff(fact_4282_old_Onat_Osimps_I7_J,axiom,
    ! [T: $tType,F1: T,F2: fun(nat,fun(T,T)),Nat: nat] : aa(nat,T,rec_nat(T,F1,F2),aa(nat,nat,suc,Nat)) = aa(T,T,aa(nat,fun(T,T),F2,Nat),aa(nat,T,rec_nat(T,F1,F2),Nat)) ).

% old.nat.simps(7)
tff(fact_4283_old_Onat_Osimps_I6_J,axiom,
    ! [T: $tType,F1: T,F2: fun(nat,fun(T,T))] : aa(nat,T,rec_nat(T,F1,F2),zero_zero(nat)) = F1 ).

% old.nat.simps(6)
tff(fact_4284_bit_Odisj__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,uminus_uminus(A),one_one(A))),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_left
tff(fact_4285_bit_Odisj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_right
tff(fact_4286_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L)) ) ) ).

% or_nonnegative_int_iff
tff(fact_4287_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),zero_zero(int)))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
        | pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),L),zero_zero(int))) ) ) ).

% or_negative_int_iff
tff(fact_4288_bit_Ode__Morgan__disj,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) ) ).

% bit.de_Morgan_disj
tff(fact_4289_bit_Ode__Morgan__conj,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) ) ).

% bit.de_Morgan_conj
tff(fact_4290_case__nat__numeral,axiom,
    ! [A: $tType,A2: A,F3: fun(nat,A),V: num] : case_nat(A,A2,F3,aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,F3,pred_numeral(V)) ).

% case_nat_numeral
tff(fact_4291_rec__nat__numeral,axiom,
    ! [A: $tType,A2: A,F3: fun(nat,fun(A,A)),V: num] : aa(nat,A,rec_nat(A,A2,F3),aa(num,nat,numeral_numeral(nat),V)) = aa(A,A,aa(nat,fun(A,A),F3,pred_numeral(V)),aa(nat,A,rec_nat(A,A2,F3),pred_numeral(V))) ).

% rec_nat_numeral
tff(fact_4292_or__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).

% or_numerals(8)
tff(fact_4293_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_4294_bit_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),X)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_right
tff(fact_4295_bit_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),X) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_left
tff(fact_4296_or__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X))),aa(num,A,numeral_numeral(A),aa(num,num,bit0,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y))) ) ).

% or_numerals(3)
tff(fact_4297_or__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,X)) ) ).

% or_numerals(5)
tff(fact_4298_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),aa(num,num,bit0,Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).

% or_numerals(1)
tff(fact_4299_or__minus__numerals_I6_J,axiom,
    ! [N: 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,N)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N))) ).

% or_minus_numerals(6)
tff(fact_4300_or__minus__numerals_I2_J,axiom,
    ! [N: 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,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N))) ).

% or_minus_numerals(2)
tff(fact_4301_or__minus__minus__numerals,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),M)),one_one(int))),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),N)),one_one(int)))) ).

% or_minus_minus_numerals
tff(fact_4302_and__minus__minus__numerals,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),M)),one_one(int))),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),N)),one_one(int)))) ).

% and_minus_minus_numerals
tff(fact_4303_or__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),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),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(7)
tff(fact_4304_or__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,X))),aa(num,A,numeral_numeral(A),aa(num,num,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),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(6)
tff(fact_4305_or__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [X: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,X))),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),aa(num,num,bit0,one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),X)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(4)
tff(fact_4306_nat_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F1: A,F2: fun(nat,A),Nat: nat] : aa(A,B,H,case_nat(A,F1,F2,Nat)) = case_nat(B,aa(A,B,H,F1),aa(fun(nat,A),fun(nat,B),aTP_Lamp_ig(fun(A,B),fun(fun(nat,A),fun(nat,B)),H),F2),Nat) ).

% nat.case_distrib
tff(fact_4307_divmod__integer_H__def,axiom,
    ! [M: num,N: num] : unique8689654367752047608divmod(code_integer,M,N) = aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,divide_divide(code_integer,aa(num,code_integer,numeral_numeral(code_integer),M),aa(num,code_integer,numeral_numeral(code_integer),N))),modulo_modulo(code_integer,aa(num,code_integer,numeral_numeral(code_integer),M),aa(num,code_integer,numeral_numeral(code_integer),N))) ).

% divmod_integer'_def
tff(fact_4308_of__int__or__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : 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),ring_1_of_int(A,K)),ring_1_of_int(A,L)) ) ).

% of_int_or_eq
tff(fact_4309_bit__or__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),N))
    <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,K),N))
        | pp(aa(nat,bool,bit_se5641148757651400278ts_bit(int,L),N)) ) ) ).

% bit_or_int_iff
tff(fact_4310_or_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),C2)) ) ).

% or.left_commute
tff(fact_4311_or_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),A2) ) ).

% or.commute
tff(fact_4312_or_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B2),C2)) ) ).

% or.assoc
tff(fact_4313_bit__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)),N))
        <=> ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
            | pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N)) ) ) ) ).

% bit_or_iff
tff(fact_4314_bit_Oconj__disj__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Z)) ) ).

% bit.conj_disj_distrib
tff(fact_4315_bit_Odisj__conj__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Z)) ) ).

% bit.disj_conj_distrib
tff(fact_4316_bit_Oconj__disj__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,X: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y),Z)),X) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),X)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Z),X)) ) ).

% bit.conj_disj_distrib2
tff(fact_4317_bit_Odisj__conj__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),Z)),X) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y),X)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Z),X)) ) ).

% bit.disj_conj_distrib2
tff(fact_4318_bit_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),zero_zero(A)) = X ) ).

% bit.disj_zero_right
tff(fact_4319_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_4320_sgn__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_zero(code_integer) )
       => ( sgn_sgn(code_integer,K) = zero_zero(code_integer) ) )
      & ( ( K != zero_zero(code_integer) )
       => ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
           => ( sgn_sgn(code_integer,K) = aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)) ) )
          & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
           => ( sgn_sgn(code_integer,K) = one_one(code_integer) ) ) ) ) ) ).

% sgn_integer_code
tff(fact_4321_of__nat__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_or_eq
tff(fact_4322_less__eq__integer__code_I1_J,axiom,
    pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),zero_zero(code_integer)),zero_zero(code_integer))) ).

% less_eq_integer_code(1)
tff(fact_4323_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F1: A,F2: fun(nat,A),X22: nat] : case_nat(A,F1,F2,aa(nat,nat,suc,X22)) = aa(nat,A,F2,X22) ).

% old.nat.simps(5)
tff(fact_4324_old_Onat_Osimps_I4_J,axiom,
    ! [A: $tType,F1: A,F2: fun(nat,A)] : case_nat(A,F1,F2,zero_zero(nat)) = F1 ).

% old.nat.simps(4)
tff(fact_4325_OR__lower,axiom,
    ! [X: int,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
       => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y))) ) ) ).

% OR_lower
tff(fact_4326_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),L))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),K),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L))) ) ).

% or_greater_eq
tff(fact_4327_disjunctive__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( ! [N2: nat] :
              ( ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N2))
              | ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,B2),N2)) )
         => ( 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_4328_plus__and__or,axiom,
    ! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Y)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),X),Y) ).

% plus_and_or
tff(fact_4329_or__eq__not__not__and,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),aa(A,A,bit_ri4277139882892585799ns_not(A),B2))) ) ).

% or_eq_not_not_and
tff(fact_4330_and__eq__not__not__or,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),aa(A,A,bit_ri4277139882892585799ns_not(A),B2))) ) ).

% and_eq_not_not_or
tff(fact_4331_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> pp(case_nat(bool,fTrue,aTP_Lamp_ih(nat,bool),Nat)) ) ).

% nat.disc_eq_case(1)
tff(fact_4332_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> pp(case_nat(bool,fFalse,aTP_Lamp_ii(nat,bool),Nat)) ) ).

% nat.disc_eq_case(2)
tff(fact_4333_or__int__def,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),aa(int,int,bit_ri4277139882892585799ns_not(int),L))) ).

% or_int_def
tff(fact_4334_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_4335_bit_Oxor__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),aa(A,A,bit_ri4277139882892585799ns_not(A),Y))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),Y)) ) ).

% bit.xor_def
tff(fact_4336_bit_Oxor__def2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),X),Y) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),X)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y))) ) ).

% bit.xor_def2
tff(fact_4337_set__bit__eq__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),N),A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A))) ) ).

% set_bit_eq_or
tff(fact_4338_xor__int__def,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),aa(int,int,bit_ri4277139882892585799ns_not(int),L))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),L)) ).

% xor_int_def
tff(fact_4339_concat__bit__def,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_concat_bit(N,K),L) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(int,int,bit_se4730199178511100633sh_bit(int,N),L)) ).

% concat_bit_def
tff(fact_4340_set__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),aa(int,int,bit_se4730199178511100633sh_bit(int,N),one_one(int))) ).

% set_bit_int_def
tff(fact_4341_one__natural_Orsp,axiom,
    one_one(nat) = one_one(nat) ).

% one_natural.rsp
tff(fact_4342_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,M)),N))
    <=> pp(case_nat(bool,fFalse,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% less_eq_nat.simps(2)
tff(fact_4343_even__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)))
        <=> ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))
            & pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),B2)) ) ) ) ).

% even_or_iff
tff(fact_4344_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),X) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),X) = 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)) )
               => ( X = Y ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_4345_or__not__numerals_I4_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int)) ).

% or_not_numerals(4)
tff(fact_4346_or__not__numerals_I2_J,axiom,
    ! [N: 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,bit0,N)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N))) ).

% or_not_numerals(2)
tff(fact_4347_diff__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,N)) = case_nat(nat,zero_zero(nat),aTP_Lamp_fu(nat,nat),aa(nat,nat,minus_minus(nat,M),N)) ).

% diff_Suc
tff(fact_4348_bit_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),X),Y) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),X),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( aa(A,A,bit_ri4277139882892585799ns_not(A),X) = Y ) ) ) ) ).

% bit.compl_unique
tff(fact_4349_or__not__numerals_I3_J,axiom,
    ! [N: 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,N)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N))) ).

% or_not_numerals(3)
tff(fact_4350_or__not__numerals_I7_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) ).

% or_not_numerals(7)
tff(fact_4351_bit__numeral__rec_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,W))),N))
        <=> pp(case_nat(bool,fFalse,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W)),N)) ) ) ).

% bit_numeral_rec(1)
tff(fact_4352_bit__numeral__rec_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),aa(num,num,bit1,W))),N))
        <=> pp(case_nat(bool,fTrue,bit_se5641148757651400278ts_bit(A,aa(num,A,numeral_numeral(A),W)),N)) ) ) ).

% bit_numeral_rec(2)
tff(fact_4353_Nitpick_Ocase__nat__unfold,axiom,
    ! [A: $tType,N: nat,X: A,F3: fun(nat,A)] :
      ( ( ( N = zero_zero(nat) )
       => ( case_nat(A,X,F3,N) = X ) )
      & ( ( N != zero_zero(nat) )
       => ( case_nat(A,X,F3,N) = aa(nat,A,F3,aa(nat,nat,minus_minus(nat,N),one_one(nat))) ) ) ) ).

% Nitpick.case_nat_unfold
tff(fact_4354_signed__take__bit__eq__if__negative,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
         => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N))) ) ) ) ).

% signed_take_bit_eq_if_negative
tff(fact_4355_mask__Suc__exp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),bit_se2239418461657761734s_mask(A,N)) ) ).

% mask_Suc_exp
tff(fact_4356_or__not__numerals_I6_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% or_not_numerals(6)
tff(fact_4357_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(bool,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))) ) ).

% one_or_eq
tff(fact_4358_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(bool,A,zero_neq_one_of_bool(A),dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),A2))) ) ).

% or_one_eq
tff(fact_4359_mask__Suc__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,N)) = 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),aa(num,num,bit0,one2))),bit_se2239418461657761734s_mask(A,N))) ) ).

% mask_Suc_double
tff(fact_4360_OR__upper,axiom,
    ! [X: int,N: nat,Y: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Y),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)))
         => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),X),Y)),aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N))) ) ) ) ).

% OR_upper
tff(fact_4361_or__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = 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),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(5)
tff(fact_4362_signed__take__bit__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)))) ) ).

% signed_take_bit_def
tff(fact_4363_or__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = 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),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(9)
tff(fact_4364_or__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = 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),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(8)
tff(fact_4365_old_Orec__nat__def,axiom,
    ! [T: $tType,X2: T,Xa2: fun(nat,fun(T,T)),Xb3: nat] : aa(nat,T,rec_nat(T,X2,Xa2),Xb3) = the(T,rec_set_nat(T,X2,Xa2,Xb3)) ).

% old.rec_nat_def
tff(fact_4366_integer__of__int__code,axiom,
    ! [K: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
       => ( code_integer_of_int(K) = aa(code_integer,code_integer,uminus_uminus(code_integer),code_integer_of_int(aa(int,int,uminus_uminus(int),K))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int)))
       => ( ( ( K = zero_zero(int) )
           => ( code_integer_of_int(K) = zero_zero(code_integer) ) )
          & ( ( K != zero_zero(int) )
           => ( code_integer_of_int(K) = if(code_integer,aa(int,bool,aa(int,fun(int,bool),fequal(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),zero_zero(int)),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),aa(num,num,bit0,one2))),code_integer_of_int(divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),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),aa(num,num,bit0,one2))),code_integer_of_int(divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))),one_one(code_integer))) ) ) ) ) ) ).

% integer_of_int_code
tff(fact_4367_or__minus__numerals_I1_J,axiom,
    ! [N: 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,bit0,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(N)))) ).

% or_minus_numerals(1)
tff(fact_4368_or__minus__numerals_I5_J,axiom,
    ! [N: 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,bit0,N)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(N)))) ).

% or_minus_numerals(5)
tff(fact_4369_or__nat__numerals_I4_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% or_nat_numerals(4)
tff(fact_4370_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_4371_or__nat__numerals_I3_J,axiom,
    ! [X: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,X))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,X)) ).

% or_nat_numerals(3)
tff(fact_4372_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),aa(num,num,bit0,Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_4373_or__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),aa(num,int,numeral_numeral(int),M)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,aa(num,num,bit0,N)))) ).

% or_minus_numerals(8)
tff(fact_4374_or__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,aa(num,num,bit0,N)))) ).

% or_minus_numerals(4)
tff(fact_4375_or__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bitM(N)))) ).

% or_minus_numerals(3)
tff(fact_4376_or__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))),aa(num,int,numeral_numeral(int),M)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bitM(N)))) ).

% or_minus_numerals(7)
tff(fact_4377_modulo__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] : modulo_modulo(code_integer,code_integer_of_int(Xa),code_integer_of_int(X)) = code_integer_of_int(modulo_modulo(int,Xa,X)) ).

% modulo_integer.abs_eq
tff(fact_4378_abs__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
       => ( aa(code_integer,code_integer,abs_abs(code_integer),K) = aa(code_integer,code_integer,uminus_uminus(code_integer),K) ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
       => ( aa(code_integer,code_integer,abs_abs(code_integer),K) = K ) ) ) ).

% abs_integer_code
tff(fact_4379_less__integer__code_I1_J,axiom,
    ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),zero_zero(code_integer))) ).

% less_integer_code(1)
tff(fact_4380_less__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),code_integer_of_int(Xa)),code_integer_of_int(X)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Xa),X)) ) ).

% less_integer.abs_eq
tff(fact_4381_or__not__num__neg_Osimps_I1_J,axiom,
    bit_or_not_num_neg(one2,one2) = one2 ).

% or_not_num_neg.simps(1)
tff(fact_4382_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),code_integer_of_int(Xa)),code_integer_of_int(X)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Xa),X)) ) ).

% less_eq_integer.abs_eq
tff(fact_4383_set__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5668285175392031749et_bit(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),N),aa(nat,nat,bit_se4730199178511100633sh_bit(nat,M),one_one(nat))) ).

% set_bit_nat_def
tff(fact_4384_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N: num] : bit_or_not_num_neg(aa(num,num,bit0,N),one2) = aa(num,num,bit0,one2) ).

% or_not_num_neg.simps(4)
tff(fact_4385_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N: num,M: num] : bit_or_not_num_neg(aa(num,num,bit0,N),aa(num,num,bit1,M)) = aa(num,num,bit0,bit_or_not_num_neg(N,M)) ).

% or_not_num_neg.simps(6)
tff(fact_4386_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] : bit_or_not_num_neg(one2,aa(num,num,bit1,M)) = aa(num,num,bit1,M) ).

% or_not_num_neg.simps(3)
tff(fact_4387_or__not__num__neg_Osimps_I7_J,axiom,
    ! [N: num] : bit_or_not_num_neg(aa(num,num,bit1,N),one2) = one2 ).

% or_not_num_neg.simps(7)
tff(fact_4388_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N: num,M: num] : bit_or_not_num_neg(aa(num,num,bit0,N),aa(num,num,bit0,M)) = bitM(bit_or_not_num_neg(N,M)) ).

% or_not_num_neg.simps(5)
tff(fact_4389_or__not__num__neg_Osimps_I9_J,axiom,
    ! [N: num,M: num] : bit_or_not_num_neg(aa(num,num,bit1,N),aa(num,num,bit1,M)) = bitM(bit_or_not_num_neg(N,M)) ).

% or_not_num_neg.simps(9)
tff(fact_4390_or__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = nat2(aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N))) ).

% or_nat_def
tff(fact_4391_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] : bit_or_not_num_neg(one2,aa(num,num,bit0,M)) = aa(num,num,bit1,M) ).

% or_not_num_neg.simps(2)
tff(fact_4392_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N: num,M: num] : bit_or_not_num_neg(aa(num,num,bit1,N),aa(num,num,bit0,M)) = bitM(bit_or_not_num_neg(N,M)) ).

% or_not_num_neg.simps(8)
tff(fact_4393_or__not__num__neg_Oelims,axiom,
    ! [X: num,Xa: num,Y: num] :
      ( ( bit_or_not_num_neg(X,Xa) = Y )
     => ( ( ( X = one2 )
         => ( ( Xa = one2 )
           => ( Y != one2 ) ) )
       => ( ( ( X = one2 )
           => ! [M5: num] :
                ( ( Xa = aa(num,num,bit0,M5) )
               => ( Y != aa(num,num,bit1,M5) ) ) )
         => ( ( ( X = one2 )
             => ! [M5: num] :
                  ( ( Xa = aa(num,num,bit1,M5) )
                 => ( Y != aa(num,num,bit1,M5) ) ) )
           => ( ( ? [N2: num] : X = aa(num,num,bit0,N2)
               => ( ( Xa = one2 )
                 => ( Y != aa(num,num,bit0,one2) ) ) )
             => ( ! [N2: num] :
                    ( ( X = aa(num,num,bit0,N2) )
                   => ! [M5: num] :
                        ( ( Xa = aa(num,num,bit0,M5) )
                       => ( Y != bitM(bit_or_not_num_neg(N2,M5)) ) ) )
               => ( ! [N2: num] :
                      ( ( X = aa(num,num,bit0,N2) )
                     => ! [M5: num] :
                          ( ( Xa = aa(num,num,bit1,M5) )
                         => ( Y != aa(num,num,bit0,bit_or_not_num_neg(N2,M5)) ) ) )
                 => ( ( ? [N2: num] : X = aa(num,num,bit1,N2)
                     => ( ( Xa = one2 )
                       => ( Y != one2 ) ) )
                   => ( ! [N2: num] :
                          ( ( X = aa(num,num,bit1,N2) )
                         => ! [M5: num] :
                              ( ( Xa = aa(num,num,bit0,M5) )
                             => ( Y != bitM(bit_or_not_num_neg(N2,M5)) ) ) )
                     => ~ ! [N2: num] :
                            ( ( X = aa(num,num,bit1,N2) )
                           => ! [M5: num] :
                                ( ( Xa = aa(num,num,bit1,M5) )
                               => ( Y != bitM(bit_or_not_num_neg(N2,M5)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_4394_numeral__or__not__num__eq,axiom,
    ! [M: num,N: num] : aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,N)) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% numeral_or_not_num_eq
tff(fact_4395_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(N,M))) ).

% int_numeral_not_or_num_neg
tff(fact_4396_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,N))) ).

% int_numeral_or_not_num_neg
tff(fact_4397_floor__real__def,axiom,
    ! [X: real] : archim6421214686448440834_floor(real,X) = the(int,aTP_Lamp_ij(real,fun(int,bool),X)) ).

% floor_real_def
tff(fact_4398_or__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))) ).

% or_Suc_0_eq
tff(fact_4399_Suc__0__or__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(bool,nat,zero_neq_one_of_bool(nat),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))) ).

% Suc_0_or_eq
tff(fact_4400_or__nat__rec,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(bool,nat,zero_neq_one_of_bool(nat),fdisj(aa(bool,bool,fNot,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),M)),aa(bool,bool,fNot,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ).

% or_nat_rec
tff(fact_4401_or__not__num__neg_Opelims,axiom,
    ! [X: num,Xa: num,Y: num] :
      ( ( bit_or_not_num_neg(X,Xa) = Y )
     => ( pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),product_Pair(num,num,X),Xa)))
       => ( ( ( X = one2 )
           => ( ( Xa = one2 )
             => ( ( Y = one2 )
               => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),product_Pair(num,num,one2),one2))) ) ) )
         => ( ( ( X = one2 )
             => ! [M5: num] :
                  ( ( Xa = aa(num,num,bit0,M5) )
                 => ( ( Y = aa(num,num,bit1,M5) )
                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit0,M5)))) ) ) )
           => ( ( ( X = one2 )
               => ! [M5: num] :
                    ( ( Xa = aa(num,num,bit1,M5) )
                   => ( ( Y = aa(num,num,bit1,M5) )
                     => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit1,M5)))) ) ) )
             => ( ! [N2: num] :
                    ( ( X = aa(num,num,bit0,N2) )
                   => ( ( Xa = one2 )
                     => ( ( Y = aa(num,num,bit0,one2) )
                       => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,N2)),one2))) ) ) )
               => ( ! [N2: num] :
                      ( ( X = aa(num,num,bit0,N2) )
                     => ! [M5: num] :
                          ( ( Xa = aa(num,num,bit0,M5) )
                         => ( ( Y = bitM(bit_or_not_num_neg(N2,M5)) )
                           => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,N2)),aa(num,num,bit0,M5)))) ) ) )
                 => ( ! [N2: num] :
                        ( ( X = aa(num,num,bit0,N2) )
                       => ! [M5: num] :
                            ( ( Xa = aa(num,num,bit1,M5) )
                           => ( ( Y = aa(num,num,bit0,bit_or_not_num_neg(N2,M5)) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,N2)),aa(num,num,bit1,M5)))) ) ) )
                   => ( ! [N2: num] :
                          ( ( X = aa(num,num,bit1,N2) )
                         => ( ( Xa = one2 )
                           => ( ( Y = one2 )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,N2)),one2))) ) ) )
                     => ( ! [N2: num] :
                            ( ( X = aa(num,num,bit1,N2) )
                           => ! [M5: num] :
                                ( ( Xa = aa(num,num,bit0,M5) )
                               => ( ( Y = bitM(bit_or_not_num_neg(N2,M5)) )
                                 => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,N2)),aa(num,num,bit0,M5)))) ) ) )
                       => ~ ! [N2: num] :
                              ( ( X = aa(num,num,bit1,N2) )
                             => ! [M5: num] :
                                  ( ( Xa = aa(num,num,bit1,M5) )
                                 => ( ( Y = bitM(bit_or_not_num_neg(N2,M5)) )
                                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,N2)),aa(num,num,bit1,M5)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
tff(fact_4402_or__int__unfold,axiom,
    ! [K: int,L: int] :
      ( ( ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
          | ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) )
       => ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,uminus_uminus(int),one_one(int)) ) )
      & ( ~ ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
            | ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) )
       => ( ( ( K = zero_zero(int) )
           => ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = L ) )
          & ( ( K != zero_zero(int) )
           => ( ( ( L = zero_zero(int) )
               => ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = K ) )
              & ( ( L != zero_zero(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(int,int,aa(int,fun(int,int),ord_max(int),modulo_modulo(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),modulo_modulo(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ) ) ) ) ).

% or_int_unfold
tff(fact_4403_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_4404_rec__nat__Suc__imp,axiom,
    ! [A: $tType,F3: fun(nat,A),F1: A,F2: fun(nat,fun(A,A)),N: nat] :
      ( ( F3 = rec_nat(A,F1,F2) )
     => ( aa(nat,A,F3,aa(nat,nat,suc,N)) = aa(A,A,aa(nat,fun(A,A),F2,N),aa(nat,A,F3,N)) ) ) ).

% rec_nat_Suc_imp
tff(fact_4405_of__bool__or__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: bool,Q: bool] : aa(bool,A,zero_neq_one_of_bool(A),fdisj(P,Q)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(bool,A,zero_neq_one_of_bool(A),P)),aa(bool,A,zero_neq_one_of_bool(A),Q)) ) ).

% of_bool_or_iff
tff(fact_4406_max__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),V) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),U) ) ) ) ) ).

% max_number_of(1)
tff(fact_4407_max__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),zero_zero(A)),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(3)
tff(fact_4408_max__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),X)),zero_zero(A)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(4)
tff(fact_4409_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_4410_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_4411_max__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(6)
tff(fact_4412_max__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_max(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),X) ) ).

% max_0_1(5)
tff(fact_4413_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(num,A,numeral_numeral(A),U) ) ) ) ) ).

% max_number_of(2)
tff(fact_4414_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),V) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) ) ) ) ).

% max_number_of(3)
tff(fact_4415_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) ) ) ) ).

% max_number_of(4)
tff(fact_4416_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X2: A,Xa2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Xa2))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X2),Xa2) = Xa2 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Xa2))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X2),Xa2) = X2 ) ) ) ) ).

% max_def_raw
tff(fact_4417_of__nat__max,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_max
tff(fact_4418_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,minus_minus(A,X),Z)),aa(A,A,minus_minus(A,Y),Z)) ) ).

% max_diff_distrib_left
tff(fact_4419_max__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ) ).

% max_def
tff(fact_4420_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = X ) ) ) ).

% max_absorb1
tff(fact_4421_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y) = Y ) ) ) ).

% max_absorb2
tff(fact_4422_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ).

% max_add_distrib_left
tff(fact_4423_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),aa(A,A,aa(A,fun(A,A),ord_max(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)) ) ).

% max_add_distrib_right
tff(fact_4424_subset__Collect__iff,axiom,
    ! [A: $tType,B3: set(A),A3: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ad(set(A),fun(fun(A,bool),fun(A,bool)),A3),P))))
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),B3))
           => pp(aa(A,bool,P,X4)) ) ) ) ).

% subset_Collect_iff
tff(fact_4425_subset__CollectI,axiom,
    ! [A: $tType,B3: set(A),A3: set(A),Q: fun(A,bool),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B3))
           => ( pp(aa(A,bool,Q,X3))
             => pp(aa(A,bool,P,X3)) ) )
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ad(set(A),fun(fun(A,bool),fun(A,bool)),B3),Q))),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ad(set(A),fun(fun(A,bool),fun(A,bool)),A3),P)))) ) ) ).

% subset_CollectI
tff(fact_4426_max__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),Z))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z)) ) ) ) ).

% max_less_iff_conj
tff(fact_4427_max_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb4
tff(fact_4428_max_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2)) ) ) ) ).

% max.bounded_iff
tff(fact_4430_max__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),Q3),zero_zero(extended_enat)) = Q3 ).

% max_enat_simps(2)
tff(fact_4431_max__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_max(extended_enat),zero_zero(extended_enat)),Q3) = Q3 ).

% max_enat_simps(3)
tff(fact_4432_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = A2 ) ) ) ).

% max.absorb1
tff(fact_4433_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) = B2 ) ) ) ).

% max.absorb2
tff(fact_4434_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N)) ).

% max_Suc_Suc
tff(fact_4435_max__0R,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),zero_zero(nat)) = N ).

% max_0R
tff(fact_4436_max__0L,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),zero_zero(nat)),N) = N ).

% max_0L
tff(fact_4437_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_4438_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_4439_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_4440_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_4441_max__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),pred_numeral(K))) ).

% max_Suc_numeral
tff(fact_4442_max__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),pred_numeral(K)),N)) ).

% max_numeral_Suc
tff(fact_4443_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N)),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Q3)) ).

% nat_add_max_left
tff(fact_4444_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q3)) ).

% nat_add_max_right
tff(fact_4445_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)) ).

% nat_mult_max_right
tff(fact_4446_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),N)),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q3)) ).

% nat_mult_max_left
tff(fact_4447_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,N),M)),M) = aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),N),M) ).

% nat_minus_add_max
tff(fact_4448_max__Suc2,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),M),aa(nat,nat,suc,N)) = case_nat(nat,aa(nat,nat,suc,N),aTP_Lamp_ik(nat,fun(nat,nat),N),M) ).

% max_Suc2
tff(fact_4449_max__Suc1,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,suc,N)),M) = case_nat(nat,aa(nat,nat,suc,N),aTP_Lamp_il(nat,fun(nat,nat),N),M) ).

% max_Suc1
tff(fact_4450_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2))) ) ) ).

% max.coboundedI2
tff(fact_4451_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2))) ) ) ).

% max.coboundedI1
tff(fact_4452_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_4453_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_4454_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),Y)) ) ) ) ).

% le_max_iff_disj
tff(fact_4455_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2))) ) ).

% max.cobounded2
tff(fact_4456_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2))) ) ).

% max.cobounded1
tff(fact_4457_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_4458_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2)) ) ) ) ).

% max.boundedI
tff(fact_4459_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2)) ) ) ) ).

% max.boundedE
tff(fact_4460_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) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% max.orderI
tff(fact_4461_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( A2 = aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2) ) ) ) ).

% max.orderE
tff(fact_4462_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,D2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_max(A),C2),D2)),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2))) ) ) ) ).

% max.mono
tff(fact_4463_max_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2))) ) ) ).

% max.strict_coboundedI2
tff(fact_4464_max_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2))) ) ) ).

% max.strict_coboundedI1
tff(fact_4465_max_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_4466_max_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_max(A),B2),C2)),A2))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A2)) ) ) ) ).

% max.strict_boundedE
tff(fact_4467_less__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),X))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),Y)) ) ) ) ).

% less_max_iff_disj
tff(fact_4468_or__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = N ) )
      & ( ( M != zero_zero(nat) )
       => ( ( ( N = zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = M ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = 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,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),modulo_modulo(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),divide_divide(nat,M,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),divide_divide(nat,N,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ) ) ) ) ).

% or_nat_unfold
tff(fact_4469_floor__rat__def,axiom,
    ! [X: rat] : archim6421214686448440834_floor(rat,X) = the(int,aTP_Lamp_im(rat,fun(int,bool),X)) ).

% floor_rat_def
tff(fact_4470_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = aa(bool,product_prod(code_integer,bool),product_Pair(code_integer,bool,divide_divide(code_integer,K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,bool,fNot,dvd_dvd(code_integer,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)),K))) ).

% bit_cut_integer_def
tff(fact_4471_divmod__integer__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,divide_divide(code_integer,K,L)),modulo_modulo(code_integer,K,L)) ).

% divmod_integer_def
tff(fact_4472_nat_Osplit__sels_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),F1: A,F2: fun(nat,A),Nat: nat] :
      ( pp(aa(A,bool,P,case_nat(A,F1,F2,Nat)))
    <=> ~ ( ( ( Nat = zero_zero(nat) )
            & ~ pp(aa(A,bool,P,F1)) )
          | ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
            & ~ pp(aa(A,bool,P,aa(nat,A,F2,pred(Nat)))) ) ) ) ).

% nat.split_sels(2)
tff(fact_4473_sgn__rat__def,axiom,
    ! [A2: rat] :
      ( ( ( A2 = zero_zero(rat) )
       => ( sgn_sgn(rat,A2) = zero_zero(rat) ) )
      & ( ( A2 != zero_zero(rat) )
       => ( ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A2))
           => ( sgn_sgn(rat,A2) = one_one(rat) ) )
          & ( ~ pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),A2))
           => ( sgn_sgn(rat,A2) = aa(rat,rat,uminus_uminus(rat),one_one(rat)) ) ) ) ) ) ).

% sgn_rat_def
tff(fact_4474_obtain__pos__sum,axiom,
    ! [R3: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R3))
     => ~ ! [S3: rat] :
            ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),S3))
           => ! [T4: rat] :
                ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),T4))
               => ( R3 != aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),S3),T4) ) ) ) ) ).

% obtain_pos_sum
tff(fact_4475_less__eq__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),X),Y))
    <=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),X),Y))
        | ( X = Y ) ) ) ).

% less_eq_rat_def
tff(fact_4476_abs__rat__def,axiom,
    ! [A2: rat] :
      ( ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),A2),zero_zero(rat)))
       => ( aa(rat,rat,abs_abs(rat),A2) = aa(rat,rat,uminus_uminus(rat),A2) ) )
      & ( ~ pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),A2),zero_zero(rat)))
       => ( aa(rat,rat,abs_abs(rat),A2) = A2 ) ) ) ).

% abs_rat_def
tff(fact_4477_pred__def,axiom,
    ! [Nat: nat] : pred(Nat) = case_nat(nat,zero_zero(nat),aTP_Lamp_fu(nat,nat),Nat) ).

% pred_def
tff(fact_4478_nat_Osplit__sels_I1_J,axiom,
    ! [A: $tType,P: fun(A,bool),F1: A,F2: fun(nat,A),Nat: nat] :
      ( pp(aa(A,bool,P,case_nat(A,F1,F2,Nat)))
    <=> ( ( ( Nat = zero_zero(nat) )
         => pp(aa(A,bool,P,F1)) )
        & ( ( Nat = aa(nat,nat,suc,pred(Nat)) )
         => pp(aa(A,bool,P,aa(nat,A,F2,pred(Nat)))) ) ) ) ).

% nat.split_sels(1)
tff(fact_4479_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      ( ( ( K = zero_zero(code_integer) )
       => ( code_bit_cut_integer(K) = aa(bool,product_prod(code_integer,bool),product_Pair(code_integer,bool,zero_zero(code_integer)),fFalse) ) )
      & ( ( K != zero_zero(code_integer) )
       => ( code_bit_cut_integer(K) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,bool),product_case_prod(code_integer,code_integer,product_prod(code_integer,bool),aTP_Lamp_in(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))) ) ) ) ).

% bit_cut_integer_code
tff(fact_4480_normalize__negative,axiom,
    ! [Q3: int,P2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Q3),zero_zero(int)))
     => ( normalize(aa(int,product_prod(int,int),product_Pair(int,int,P2),Q3)) = normalize(aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,uminus_uminus(int),P2)),aa(int,int,uminus_uminus(int),Q3))) ) ) ).

% normalize_negative
tff(fact_4481_divmod__abs__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_abs(K,L) = aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,divide_divide(code_integer,aa(code_integer,code_integer,abs_abs(code_integer),K),aa(code_integer,code_integer,abs_abs(code_integer),L))),modulo_modulo(code_integer,aa(code_integer,code_integer,abs_abs(code_integer),K),aa(code_integer,code_integer,abs_abs(code_integer),L))) ).

% divmod_abs_def
tff(fact_4482_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(X,Xa) = Y )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa),X))
         => ( Y = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xa),aa(nat,nat,minus_minus(nat,X),Xa)) ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa),X))
         => ( Y = nat_prod_decode_aux(aa(nat,nat,suc,X),aa(nat,nat,minus_minus(nat,Xa),aa(nat,nat,suc,X))) ) ) ) ) ).

% prod_decode_aux.elims
tff(fact_4483_normalize__denom__pos,axiom,
    ! [R3: product_prod(int,int),P2: int,Q3: int] :
      ( ( normalize(R3) = aa(int,product_prod(int,int),product_Pair(int,int,P2),Q3) )
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Q3)) ) ).

% normalize_denom_pos
tff(fact_4484_prod__decode__aux_Osimps,axiom,
    ! [M: nat,K: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),K))
       => ( nat_prod_decode_aux(K,M) = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,M),aa(nat,nat,minus_minus(nat,K),M)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),K))
       => ( nat_prod_decode_aux(K,M) = nat_prod_decode_aux(aa(nat,nat,suc,K),aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,K))) ) ) ) ).

% prod_decode_aux.simps
tff(fact_4485_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod(nat,nat)] :
      ( ( nat_prod_decode_aux(X,Xa) = Y )
     => ( pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa)))
       => ~ ( ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa),X))
               => ( Y = aa(nat,product_prod(nat,nat),product_Pair(nat,nat,Xa),aa(nat,nat,minus_minus(nat,X),Xa)) ) )
              & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Xa),X))
               => ( Y = nat_prod_decode_aux(aa(nat,nat,suc,X),aa(nat,nat,minus_minus(nat,Xa),aa(nat,nat,suc,X))) ) ) )
           => ~ pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),nat_pr5047031295181774490ux_rel),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa))) ) ) ) ).

% prod_decode_aux.pelims
tff(fact_4486_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( ( K = zero_zero(code_integer) )
       => ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,zero_zero(code_integer)),zero_zero(code_integer)) ) )
      & ( ( K != zero_zero(code_integer) )
       => ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),L))
           => ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),K))
               => ( code_divmod_integer(K,L) = code_divmod_abs(K,L) ) )
              & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),K))
               => ( code_divmod_integer(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_io(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)) ) ) ) )
          & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),L))
           => ( ( ( L = zero_zero(code_integer) )
               => ( code_divmod_integer(K,L) = aa(code_integer,product_prod(code_integer,code_integer),product_Pair(code_integer,code_integer,zero_zero(code_integer)),K) ) )
              & ( ( L != zero_zero(code_integer) )
               => ( code_divmod_integer(K,L) = product_apsnd(code_integer,code_integer,code_integer,uminus_uminus(code_integer),if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),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_ip(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_4487_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: set(nat),C2: fun(nat,A),D2: fun(nat,A)] :
          ( ( ( pp(aa(set(nat),bool,finite_finite(nat),A3))
              & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3)) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_iq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D2)),A3) = divide_divide(A,aa(nat,A,C2,zero_zero(nat)),aa(nat,A,D2,zero_zero(nat))) ) )
          & ( ~ ( pp(aa(set(nat),bool,finite_finite(nat),A3))
                & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3)) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_iq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D2)),A3) = zero_zero(A) ) ) ) ) ).

% sum_zero_power'
tff(fact_4488_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_4489_List_Ofinite__set,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(set(A),bool,finite_finite(A),aa(list(A),set(A),set2(A),Xs))) ).

% List.finite_set
tff(fact_4490_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : pp(aa(set(nat),bool,finite_finite(nat),set_or1337092689740270186AtMost(nat,L,U))) ).

% finite_atLeastAtMost
tff(fact_4491_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa(set(nat),bool,finite_finite(nat),set_or7035219750837199246ssThan(nat,L,U))) ).

% finite_atLeastLessThan
tff(fact_4492_finite__lessThan,axiom,
    ! [K: nat] : pp(aa(set(nat),bool,finite_finite(nat),aa(nat,set(nat),set_ord_lessThan(nat),K))) ).

% finite_lessThan
tff(fact_4493_finite__atMost,axiom,
    ! [K: nat] : pp(aa(set(nat),bool,finite_finite(nat),aa(nat,set(nat),set_ord_atMost(nat),K))) ).

% finite_atMost
tff(fact_4494_infinite__Icc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),set_or1337092689740270186AtMost(A,A2,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% infinite_Icc_iff
tff(fact_4495_infinite__Ico__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),set_or7035219750837199246ssThan(A,A2,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% infinite_Ico_iff
tff(fact_4496_prod_Oinfinite,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),G: fun(B,A)] :
          ( ~ pp(aa(set(B),bool,finite_finite(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3) = one_one(A) ) ) ) ).

% prod.infinite
tff(fact_4497_prod__eq__1__iff,axiom,
    ! [A: $tType,A3: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( ( aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F3),A3) = one_one(nat) )
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
           => ( aa(A,nat,F3,X4) = one_one(nat) ) ) ) ) ).

% prod_eq_1_iff
tff(fact_4498_prod_Odelta,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),A2: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S2))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_ir(B,fun(fun(B,A),fun(B,A)),A2),B2)),S2) = aa(B,A,B2,A2) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_ir(B,fun(fun(B,A),fun(B,A)),A2),B2)),S2) = one_one(A) ) ) ) ) ) ).

% prod.delta
tff(fact_4499_prod_Odelta_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),A2: B,B2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S2))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_is(B,fun(fun(B,A),fun(B,A)),A2),B2)),S2) = aa(B,A,B2,A2) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,A),fun(B,A),aTP_Lamp_is(B,fun(fun(B,A),fun(B,A)),A2),B2)),S2) = one_one(A) ) ) ) ) ) ).

% prod.delta'
tff(fact_4500_numeral__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [K: num,L: num] : divide_divide(A,aa(num,A,numeral_numeral(A),K),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,K,L)) ) ).

% numeral_div_numeral
tff(fact_4501_prod__pos__nat__iff,axiom,
    ! [A: $tType,A3: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,aa(fun(A,nat),fun(set(A),nat),groups7121269368397514597t_prod(A,nat),F3),A3)))
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(A,nat,F3,X4))) ) ) ) ).

% prod_pos_nat_iff
tff(fact_4502_fst__divmod__nat,axiom,
    ! [M: nat,N: nat] : aa(product_prod(nat,nat),nat,product_fst(nat,nat),divmod_nat(M,N)) = divide_divide(nat,M,N) ).

% fst_divmod_nat
tff(fact_4503_one__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : divide_divide(A,one_one(A),aa(num,A,numeral_numeral(A),N)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,one2,N)) ) ).

% one_div_numeral
tff(fact_4504_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: set(nat),C2: fun(nat,A)] :
          ( ( ( pp(aa(set(nat),bool,finite_finite(nat),A3))
              & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3)) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_it(fun(nat,A),fun(nat,A),C2)),A3) = aa(nat,A,C2,zero_zero(nat)) ) )
          & ( ~ ( pp(aa(set(nat),bool,finite_finite(nat),A3))
                & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3)) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_it(fun(nat,A),fun(nat,A),C2)),A3) = zero_zero(A) ) ) ) ) ).

% sum_zero_power
tff(fact_4505_infinite__Iio,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [A2: A] : ~ pp(aa(set(A),bool,finite_finite(A),aa(A,set(A),set_ord_lessThan(A),A2))) ) ).

% infinite_Iio
tff(fact_4506_finite__list,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ? [Xs2: list(A)] : aa(list(A),set(A),set2(A),Xs2) = A3 ) ).

% finite_list
tff(fact_4507_infinite__Iic,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [A2: A] : ~ pp(aa(set(A),bool,finite_finite(A),aa(A,set(A),set_ord_atMost(A),A2))) ) ).

% infinite_Iic
tff(fact_4508_finite__nat__set__iff__bounded__le,axiom,
    ! [N3: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),N3))
    <=> ? [M7: nat] :
        ! [X4: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),N3))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),M7)) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_4509_finite__less__ub,axiom,
    ! [F3: fun(nat,nat),U: nat] :
      ( ! [N2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),aa(nat,nat,F3,N2)))
     => pp(aa(set(nat),bool,finite_finite(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_iu(fun(nat,nat),fun(nat,fun(nat,bool)),F3),U)))) ) ).

% finite_less_ub
tff(fact_4510_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,bool),I2: nat] : pp(aa(set(nat),bool,finite_finite(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_iv(fun(nat,bool),fun(nat,fun(nat,bool)),P),I2)))) ).

% finite_M_bounded_by_nat
tff(fact_4511_bounded__nat__set__is__finite,axiom,
    ! [N3: set(nat),N: nat] :
      ( ! [X3: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),N3))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),N)) )
     => pp(aa(set(nat),bool,finite_finite(nat),N3)) ) ).

% bounded_nat_set_is_finite
tff(fact_4512_finite__nat__set__iff__bounded,axiom,
    ! [N3: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),N3))
    <=> ? [M7: nat] :
        ! [X4: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X4),N3))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),M7)) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_4513_finite__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => pp(aa(set(list(A)),bool,finite_finite(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_iw(set(A),fun(nat,fun(list(A),bool)),A3),N)))) ) ).

% finite_lists_length_eq
tff(fact_4514_sum__mono__inv,axiom,
    ! [A: $tType,I6: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [F3: fun(I6,A),I5: set(I6),G: fun(I6,A),I2: I6] :
          ( ( aa(set(I6),A,groups7311177749621191930dd_sum(I6,A,F3),I5) = aa(set(I6),A,groups7311177749621191930dd_sum(I6,A,G),I5) )
         => ( ! [I4: I6] :
                ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I4),I5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(I6,A,F3,I4)),aa(I6,A,G,I4))) )
           => ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),I2),I5))
             => ( pp(aa(set(I6),bool,finite_finite(I6),I5))
               => ( aa(I6,A,F3,I2) = aa(I6,A,G,I2) ) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_4515_infinite__Icc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ~ pp(aa(set(A),bool,finite_finite(A),set_or1337092689740270186AtMost(A,A2,B2))) ) ) ).

% infinite_Icc
tff(fact_4516_infinite__Ico,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ~ pp(aa(set(A),bool,finite_finite(A),set_or7035219750837199246ssThan(A,A2,B2))) ) ) ).

% infinite_Ico
tff(fact_4517_finite__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => pp(aa(set(list(A)),bool,finite_finite(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_ix(set(A),fun(nat,fun(list(A),bool)),A3),N)))) ) ).

% finite_lists_length_le
tff(fact_4518_sum_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),X: fun(B,A),Y: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_iy(set(B),fun(fun(B,A),fun(B,bool)),I5),X))))
         => ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_iy(set(B),fun(fun(B,A),fun(B,bool)),I5),Y))))
           => pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_iz(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I5),X),Y)))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_4519_prod_Ofinite__Collect__op,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),X: fun(B,A),Y: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ja(set(B),fun(fun(B,A),fun(B,bool)),I5),X))))
         => ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ja(set(B),fun(fun(B,A),fun(B,bool)),I5),Y))))
           => pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_jb(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),I5),X),Y)))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_4520_prod_Ointer__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),G: fun(B,A),P: fun(B,bool)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),collect(B,aa(fun(B,bool),fun(B,bool),aTP_Lamp_jc(set(B),fun(fun(B,bool),fun(B,bool)),A3),P))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,bool),fun(B,A),aTP_Lamp_jd(fun(B,A),fun(fun(B,bool),fun(B,A)),G),P)),A3) ) ) ) ).

% prod.inter_filter
tff(fact_4521_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : pp(aa(set(A),bool,finite_finite(A),collect(A,aa(A,fun(A,bool),aTP_Lamp_je(A,fun(A,fun(A,bool)),A2),B2)))) ) ).

% finite_int_segment
tff(fact_4522_fst__divmod,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,M,N)) = divide_divide(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N)) ) ).

% fst_divmod
tff(fact_4523_sum__le__included,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S: set(B),T2: set(C),G: fun(C,A),I2: fun(C,B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S))
         => ( pp(aa(set(C),bool,finite_finite(C),T2))
           => ( ! [X3: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X3),T2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(C,A,G,X3))) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S))
                   => ? [Xa2: C] :
                        ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),Xa2),T2))
                        & ( aa(C,B,I2,Xa2) = X3 )
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(C,A,G,Xa2))) ) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),S)),aa(set(C),A,groups7311177749621191930dd_sum(C,A,G),T2))) ) ) ) ) ) ).

% sum_le_included
tff(fact_4524_sum__nonneg__eq__0__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,X3))) )
           => ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A3) = zero_zero(A) )
            <=> ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
                 => ( aa(B,A,F3,X4) = zero_zero(A) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_4525_sum__strict__mono__ex1,axiom,
    ! [A: $tType,I6: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: set(I6),F3: fun(I6,A),G: fun(I6,A)] :
          ( pp(aa(set(I6),bool,finite_finite(I6),A3))
         => ( ! [X3: I6] :
                ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(I6,A,F3,X3)),aa(I6,A,G,X3))) )
           => ( ? [X2: I6] :
                  ( pp(aa(set(I6),bool,aa(I6,fun(set(I6),bool),member(I6),X2),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(I6,A,F3,X2)),aa(I6,A,G,X2))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(I6),A,groups7311177749621191930dd_sum(I6,A,F3),A3)),aa(set(I6),A,groups7311177749621191930dd_sum(I6,A,G),A3))) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_4526_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R: fun(A,fun(A,bool)),S2: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),R,zero_zero(A)),zero_zero(A)))
         => ( ! [X1: A,Y1: A,X23: A,Y23: A] :
                ( ( pp(aa(A,bool,aa(A,fun(A,bool),R,X1),X23))
                  & pp(aa(A,bool,aa(A,fun(A,bool),R,Y1),Y23)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),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))) )
           => ( pp(aa(set(B),bool,finite_finite(B),S2))
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S2))
                   => pp(aa(A,bool,aa(A,fun(A,bool),R,aa(B,A,H,X3)),aa(B,A,G,X3))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R,aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),S2)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),S2))) ) ) ) ) ) ).

% sum.related
tff(fact_4527_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R: fun(A,fun(A,bool)),S2: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),R,one_one(A)),one_one(A)))
         => ( ! [X1: A,Y1: A,X23: A,Y23: A] :
                ( ( pp(aa(A,bool,aa(A,fun(A,bool),R,X1),X23))
                  & pp(aa(A,bool,aa(A,fun(A,bool),R,Y1),Y23)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),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))) )
           => ( pp(aa(set(B),bool,finite_finite(B),S2))
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S2))
                   => pp(aa(A,bool,aa(A,fun(A,bool),R,aa(B,A,H,X3)),aa(B,A,G,X3))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),R,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2))) ) ) ) ) ) ).

% prod.related
tff(fact_4528_prod__dvd__prod__subset2,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [B3: set(B),A3: set(B),F3: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),B3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),B3))
           => ( ! [A5: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),A3))
                 => pp(dvd_dvd(A,aa(B,A,F3,A5),aa(B,A,G,A5))) )
             => pp(dvd_dvd(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A3),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),B3))) ) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_4529_prod__dvd__prod__subset,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [B3: set(B),A3: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),B3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),B3))
           => pp(dvd_dvd(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A3),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),B3))) ) ) ) ).

% prod_dvd_prod_subset
tff(fact_4530_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [S4: set(B),T5: set(C),S2: set(B),I2: fun(C,B),J: fun(B,C),T6: set(C),G: fun(B,A),H: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S4))
         => ( pp(aa(set(C),bool,finite_finite(C),T5))
           => ( ! [A5: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),aa(set(B),set(B),minus_minus(set(B),S2),S4)))
                 => ( aa(C,B,I2,aa(B,C,J,A5)) = A5 ) )
             => ( ! [A5: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),aa(set(B),set(B),minus_minus(set(B),S2),S4)))
                   => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),aa(B,C,J,A5)),aa(set(C),set(C),minus_minus(set(C),T6),T5))) )
               => ( ! [B5: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B5),aa(set(C),set(C),minus_minus(set(C),T6),T5)))
                     => ( aa(B,C,J,aa(C,B,I2,B5)) = B5 ) )
                 => ( ! [B5: C] :
                        ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B5),aa(set(C),set(C),minus_minus(set(C),T6),T5)))
                       => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(C,B,I2,B5)),aa(set(B),set(B),minus_minus(set(B),S2),S4))) )
                   => ( ! [A5: B] :
                          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),S4))
                         => ( aa(B,A,G,A5) = one_one(A) ) )
                     => ( ! [B5: C] :
                            ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B5),T5))
                           => ( aa(C,A,H,B5) = one_one(A) ) )
                       => ( ! [A5: B] :
                              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),S2))
                             => ( aa(C,A,H,aa(B,C,J,A5)) = aa(B,A,G,A5) ) )
                         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),H),T6) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_4531_sum__eq__Suc0__iff,axiom,
    ! [A: $tType,A3: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),A3) = aa(nat,nat,suc,zero_zero(nat)) )
      <=> ? [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
            & ( aa(A,nat,F3,X4) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),A3))
               => ( ( X4 != Xa4 )
                 => ( aa(A,nat,F3,Xa4) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_4532_sum__eq__1__iff,axiom,
    ! [A: $tType,A3: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),A3) = one_one(nat) )
      <=> ? [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
            & ( aa(A,nat,F3,X4) = one_one(nat) )
            & ! [Xa4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),A3))
               => ( ( X4 != Xa4 )
                 => ( aa(A,nat,F3,Xa4) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_4533_sum__nonneg__leq__bound,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S: set(B),F3: fun(B,A),B3: A,I2: B] :
          ( pp(aa(set(B),bool,finite_finite(B),S))
         => ( ! [I4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I4))) )
           => ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),S) = B3 )
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I2)),B3)) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_4534_sum__nonneg__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [S: set(B),F3: fun(B,A),I2: B] :
          ( pp(aa(set(B),bool,finite_finite(B),S))
         => ( ! [I4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),S))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I4))) )
           => ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),S) = zero_zero(A) )
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),S))
               => ( aa(B,A,F3,I2) = zero_zero(A) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_4535_prod_Osetdiff__irrelevant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),minus_minus(set(B),A3),collect(B,aTP_Lamp_jf(fun(B,A),fun(B,bool),G)))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_4536_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => pp(aa(set(nat),bool,finite_finite(nat),collect(nat,aTP_Lamp_jg(nat,fun(nat,bool),M)))) ) ).

% finite_divisors_nat
tff(fact_4537_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A] : pp(aa(set(A),bool,finite_finite(A),collect(A,aTP_Lamp_jh(A,fun(A,bool),A2)))) ) ).

% finite_abs_int_segment
tff(fact_4538_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N3: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N3),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)))
     => pp(aa(set(nat),bool,finite_finite(nat),N3)) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_4539_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N3: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
     => pp(aa(set(nat),bool,finite_finite(nat),N3)) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_4540_sum__pos2,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [I5: set(B),I2: B,F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),I5))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I5))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F3,I2)))
             => ( ! [I4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),I5))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I4))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),I5))) ) ) ) ) ) ).

% sum_pos2
tff(fact_4541_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),I2: A,F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite(A),I5))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I5))
           => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(A,B,F3,I2)))
             => ( ! [I4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I4),I5))
                   => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F3,I4))) )
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),I5))) ) ) ) ) ) ).

% less_1_prod2
tff(fact_4542_sum_Osame__carrier,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C5: set(B),A3: set(B),B3: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),C5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),C5))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),C5))
             => ( ! [A5: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),aa(set(B),set(B),minus_minus(set(B),C5),A3)))
                   => ( aa(B,A,G,A5) = zero_zero(A) ) )
               => ( ! [B5: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B5),aa(set(B),set(B),minus_minus(set(B),C5),B3)))
                     => ( aa(B,A,H,B5) = zero_zero(A) ) )
                 => ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),B3) )
                  <=> ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),C5) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),C5) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_4543_sum_Osame__carrierI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [C5: set(B),A3: set(B),B3: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),C5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),C5))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),C5))
             => ( ! [A5: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),aa(set(B),set(B),minus_minus(set(B),C5),A3)))
                   => ( aa(B,A,G,A5) = zero_zero(A) ) )
               => ( ! [B5: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B5),aa(set(B),set(B),minus_minus(set(B),C5),B3)))
                     => ( aa(B,A,H,B5) = zero_zero(A) ) )
                 => ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),C5) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),C5) )
                   => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),B3) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_4544_sum_Omono__neutral__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T6: set(B),S2: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T6),S2)))
                 => ( aa(B,A,G,X3) = zero_zero(A) ) )
             => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),S2) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),T6) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_4545_sum_Omono__neutral__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T6: set(B),S2: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T6),S2)))
                 => ( aa(B,A,G,X3) = zero_zero(A) ) )
             => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),T6) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),S2) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_4546_sum_Omono__neutral__cong__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T6: set(B),S2: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T6),S2)))
                 => ( aa(B,A,H,X3) = zero_zero(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S2))
                   => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
               => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),S2) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),T6) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_4547_sum_Omono__neutral__cong__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [T6: set(B),S2: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T6),S2)))
                 => ( aa(B,A,G,X3) = zero_zero(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S2))
                   => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
               => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),T6) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),S2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_4548_sum_Osubset__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [B3: set(B),A3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),A3))
         => ( pp(aa(set(B),bool,finite_finite(B),A3))
           => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),minus_minus(set(B),A3),B3))),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),B3)) ) ) ) ) ).

% sum.subset_diff
tff(fact_4549_sum__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [A3: set(B),B3: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),A3))
           => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),aa(set(B),set(B),minus_minus(set(B),A3),B3)) = aa(A,A,minus_minus(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A3)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),B3)) ) ) ) ) ).

% sum_diff
tff(fact_4550_prod_Osubset__diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [B3: set(B),A3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),A3))
         => ( pp(aa(set(B),bool,finite_finite(B),A3))
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),minus_minus(set(B),A3),B3))),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),B3)) ) ) ) ) ).

% prod.subset_diff
tff(fact_4551_prod_Osame__carrier,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C5: set(B),A3: set(B),B3: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),C5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),C5))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),C5))
             => ( ! [A5: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),aa(set(B),set(B),minus_minus(set(B),C5),A3)))
                   => ( aa(B,A,G,A5) = one_one(A) ) )
               => ( ! [B5: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B5),aa(set(B),set(B),minus_minus(set(B),C5),B3)))
                     => ( aa(B,A,H,B5) = one_one(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),B3) )
                  <=> ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),C5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),C5) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_4552_prod_Osame__carrierI,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C5: set(B),A3: set(B),B3: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),C5))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),C5))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),C5))
             => ( ! [A5: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),aa(set(B),set(B),minus_minus(set(B),C5),A3)))
                   => ( aa(B,A,G,A5) = one_one(A) ) )
               => ( ! [B5: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B5),aa(set(B),set(B),minus_minus(set(B),C5),B3)))
                     => ( aa(B,A,H,B5) = one_one(A) ) )
                 => ( ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),C5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),C5) )
                   => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),B3) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_4553_prod_Omono__neutral__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T6: set(B),S2: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T6),S2)))
                 => ( aa(B,A,G,X3) = one_one(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),T6) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_4554_prod_Omono__neutral__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T6: set(B),S2: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T6),S2)))
                 => ( aa(B,A,G,X3) = one_one(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),T6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_4555_prod_Omono__neutral__cong__left,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T6: set(B),S2: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T6),S2)))
                 => ( aa(B,A,H,X3) = one_one(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S2))
                   => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),T6) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_4556_prod_Omono__neutral__cong__right,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T6: set(B),S2: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T6))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T6))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T6),S2)))
                 => ( aa(B,A,G,X3) = one_one(A) ) )
             => ( ! [X3: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S2))
                   => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),T6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_4557_sum__diff__nat,axiom,
    ! [A: $tType,B3: set(A),A3: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite(A),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
       => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),aa(set(A),set(A),minus_minus(set(A),A3),B3)) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),A3)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),B3)) ) ) ) ).

% sum_diff_nat
tff(fact_4558_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N))
         => pp(aa(set(A),bool,finite_finite(A),collect(A,aTP_Lamp_ji(nat,fun(A,bool),N)))) ) ) ).

% finite_roots_unity
tff(fact_4559_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,A),S2: A,A3: set(nat),S4: A,F3: fun(nat,A)] :
          ( sums(A,G,S2)
         => ( pp(aa(set(nat),bool,finite_finite(nat),A3))
           => ( ( S4 = aa(A,A,aa(A,fun(A,A),plus_plus(A),S2),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_jj(fun(nat,A),fun(fun(nat,A),fun(nat,A)),G),F3)),A3)) )
             => sums(A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_jk(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),G),A3),F3),S4) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_4560_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [S4: set(B),T5: set(C),H: fun(B,C),S2: set(B),T6: set(C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S4))
         => ( pp(aa(set(C),bool,finite_finite(C),T5))
           => ( bij_betw(B,C,H,aa(set(B),set(B),minus_minus(set(B),S2),S4),aa(set(C),set(C),minus_minus(set(C),T6),T5))
             => ( ! [A5: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),S4))
                   => ( aa(C,A,G,aa(B,C,H,A5)) = one_one(A) ) )
               => ( ! [B5: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B5),T5))
                     => ( aa(C,A,G,B5) = one_one(A) ) )
                 => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(C,A),fun(B,A),aTP_Lamp_jl(fun(B,C),fun(fun(C,A),fun(B,A)),H),G)),S2) = aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),T6) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_4561_sum__mono2,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [B3: set(B),A3: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),B3))
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),B3))
           => ( ! [B5: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),B5),aa(set(B),set(B),minus_minus(set(B),B3),A3)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,B5))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A3)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),B3))) ) ) ) ) ).

% sum_mono2
tff(fact_4562_even__prod__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_parity(A)
     => ! [A3: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A3)))
          <=> ? [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
                & pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(B,A,F3,X4))) ) ) ) ) ).

% even_prod_iff
tff(fact_4563_sum__le__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A),I5: set(nat)] :
          ( summable(A,F3)
         => ( pp(aa(set(nat),bool,finite_finite(nat),I5))
           => ( ! [N2: nat] :
                  ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),aa(set(nat),set(nat),uminus_uminus(set(nat)),I5)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(nat,A,F3,N2))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F3),I5)),suminf(A,F3))) ) ) ) ) ).

% sum_le_suminf
tff(fact_4564_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B3: set(A),A3: set(A),B2: A,F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite(A),B3))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),aa(set(A),set(A),minus_minus(set(A),B3),A3)))
             => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),zero_zero(B)),aa(A,B,F3,B2)))
               => ( ! [X3: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B3))
                     => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,X3))) )
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),B3))) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_4565_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B3: set(A),A3: set(A),F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite(A),B3))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
           => ( ! [B5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B5),aa(set(A),set(A),minus_minus(set(A),B3),A3)))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),one_one(B)),aa(A,B,F3,B5))) )
             => ( ! [A5: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),A3))
                   => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,F3,A5))) )
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),A3)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),B3))) ) ) ) ) ) ).

% prod_mono2
tff(fact_4566_ln__prod,axiom,
    ! [A: $tType,I5: set(A),F3: fun(A,real)] :
      ( pp(aa(set(A),bool,finite_finite(A),I5))
     => ( ! [I4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I4),I5))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,F3,I4))) )
       => ( aa(real,real,ln_ln(real),aa(set(A),real,aa(fun(A,real),fun(set(A),real),groups7121269368397514597t_prod(A,real),F3),I5)) = aa(set(A),real,groups7311177749621191930dd_sum(A,real,aTP_Lamp_jm(fun(A,real),fun(A,real),F3)),I5) ) ) ) ).

% ln_prod
tff(fact_4567_even__set__encode__iff,axiom,
    ! [A3: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),A3))
     => ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(set(nat),nat,nat_set_encode,A3)))
      <=> ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3)) ) ) ).

% even_set_encode_iff
tff(fact_4568_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat] :
          ( pp(aa(set(A),bool,finite_finite(A),collect(A,aa(nat,fun(A,bool),aTP_Lamp_jn(fun(nat,A),fun(nat,fun(A,bool)),C2),N))))
        <=> ? [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),N))
              & ( aa(nat,A,C2,I3) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_4569_polyfun__roots__finite,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,N: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
           => pp(aa(set(A),bool,finite_finite(A),collect(A,aa(nat,fun(A,bool),aTP_Lamp_jn(fun(nat,A),fun(nat,fun(A,bool)),C2),N)))) ) ) ) ).

% polyfun_roots_finite
tff(fact_4570_finite__Collect__le__nat,axiom,
    ! [K: nat] : pp(aa(set(nat),bool,finite_finite(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_jo(nat,fun(nat,bool)),K)))) ).

% finite_Collect_le_nat
tff(fact_4571_finite__Collect__less__nat,axiom,
    ! [K: nat] : pp(aa(set(nat),bool,finite_finite(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ea(nat,fun(nat,bool)),K)))) ).

% finite_Collect_less_nat
tff(fact_4572_finite__Collect__subsets,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => pp(aa(set(set(A)),bool,finite_finite(set(A)),collect(set(A),aTP_Lamp_jp(set(A),fun(set(A),bool),A3)))) ) ).

% finite_Collect_subsets
tff(fact_4573_finite__Collect__bounded__ex,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),Q: fun(B,fun(A,bool))] :
      ( pp(aa(set(A),bool,finite_finite(A),collect(A,P)))
     => ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_jq(fun(A,bool),fun(fun(B,fun(A,bool)),fun(B,bool)),P),Q))))
      <=> ! [Y5: A] :
            ( pp(aa(A,bool,P,Y5))
           => pp(aa(set(B),bool,finite_finite(B),collect(B,aa(A,fun(B,bool),aTP_Lamp_jr(fun(B,fun(A,bool)),fun(A,fun(B,bool)),Q),Y5)))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_4574_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : pp(aa(set(int),bool,finite_finite(int),set_or1337092689740270186AtMost(int,L,U))) ).

% finite_atLeastAtMost_int
tff(fact_4575_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : pp(aa(set(int),bool,finite_finite(int),set_or7035219750837199246ssThan(int,L,U))) ).

% finite_atLeastLessThan_int
tff(fact_4576_finite__interval__int1,axiom,
    ! [A2: int,B2: int] : pp(aa(set(int),bool,finite_finite(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_js(int,fun(int,fun(int,bool)),A2),B2)))) ).

% finite_interval_int1
tff(fact_4577_finite__interval__int4,axiom,
    ! [A2: int,B2: int] : pp(aa(set(int),bool,finite_finite(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_jt(int,fun(int,fun(int,bool)),A2),B2)))) ).

% finite_interval_int4
tff(fact_4578_finite__interval__int2,axiom,
    ! [A2: int,B2: int] : pp(aa(set(int),bool,finite_finite(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_ju(int,fun(int,fun(int,bool)),A2),B2)))) ).

% finite_interval_int2
tff(fact_4579_finite__interval__int3,axiom,
    ! [A2: int,B2: int] : pp(aa(set(int),bool,finite_finite(int),collect(int,aa(int,fun(int,bool),aTP_Lamp_jv(int,fun(int,fun(int,bool)),A2),B2)))) ).

% finite_interval_int3
tff(fact_4580_finite__nth__roots,axiom,
    ! [N: nat,C2: complex] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => pp(aa(set(complex),bool,finite_finite(complex),collect(complex,aa(complex,fun(complex,bool),aTP_Lamp_ep(nat,fun(complex,fun(complex,bool)),N),C2)))) ) ).

% finite_nth_roots
tff(fact_4581_finite__maxlen,axiom,
    ! [A: $tType,M6: set(list(A))] :
      ( pp(aa(set(list(A)),bool,finite_finite(list(A)),M6))
     => ? [N2: nat] :
        ! [X2: list(A)] :
          ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),X2),M6))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),X2)),N2)) ) ) ).

% finite_maxlen
tff(fact_4582_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : pp(aa(set(int),bool,finite_finite(int),set_or7035219750837199246ssThan(int,zero_zero(int),U))) ).

% finite_atLeastZeroLessThan_int
tff(fact_4583_finite__has__minimal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A2))
                & ! [Xa2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),A3))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa2),X3))
                     => ( X3 = Xa2 ) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_4584_finite__has__maximal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3))
                & ! [Xa2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),A3))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa2))
                     => ( X3 = Xa2 ) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_4585_finite__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( pp(aa(set(A),bool,finite_finite(A),B3))
       => pp(aa(set(A),bool,finite_finite(A),A3)) ) ) ).

% finite_subset
tff(fact_4586_infinite__super,axiom,
    ! [A: $tType,S2: set(A),T6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),T6))
     => ( ~ pp(aa(set(A),bool,finite_finite(A),S2))
       => ~ pp(aa(set(A),bool,finite_finite(A),T6)) ) ) ).

% infinite_super
tff(fact_4587_rev__finite__subset,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
       => pp(aa(set(A),bool,finite_finite(A),A3)) ) ) ).

% rev_finite_subset
tff(fact_4588_finite__psubset__induct,axiom,
    ! [A: $tType,A3: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( ! [A7: set(A)] :
            ( pp(aa(set(A),bool,finite_finite(A),A7))
           => ( ! [B8: set(A)] :
                  ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),B8),A7))
                 => pp(aa(set(A),bool,P,B8)) )
             => pp(aa(set(A),bool,P,A7)) ) )
       => pp(aa(set(A),bool,P,A3)) ) ) ).

% finite_psubset_induct
tff(fact_4589_finite__image__set,axiom,
    ! [B: $tType,A: $tType,P: fun(A,bool),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite(A),collect(A,P)))
     => pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(A,B),fun(B,bool),aTP_Lamp_jw(fun(A,bool),fun(fun(A,B),fun(B,bool)),P),F3)))) ) ).

% finite_image_set
tff(fact_4590_finite__image__set2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(A,bool),Q: fun(B,bool),F3: fun(A,fun(B,C))] :
      ( pp(aa(set(A),bool,finite_finite(A),collect(A,P)))
     => ( pp(aa(set(B),bool,finite_finite(B),collect(B,Q)))
       => pp(aa(set(C),bool,finite_finite(C),collect(C,aa(fun(A,fun(B,C)),fun(C,bool),aa(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool)),aTP_Lamp_jx(fun(A,bool),fun(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool))),P),Q),F3)))) ) ) ).

% finite_image_set2
tff(fact_4591_finite__nat__iff__bounded__le,axiom,
    ! [S2: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),S2))
    <=> ? [K3: nat] : pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),S2),aa(nat,set(nat),set_ord_atMost(nat),K3))) ) ).

% finite_nat_iff_bounded_le
tff(fact_4592_finite__nat__iff__bounded,axiom,
    ! [S2: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),S2))
    <=> ? [K3: nat] : pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),S2),aa(nat,set(nat),set_ord_lessThan(nat),K3))) ) ).

% finite_nat_iff_bounded
tff(fact_4593_finite__nat__bounded,axiom,
    ! [S2: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),S2))
     => ? [K2: nat] : pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),S2),aa(nat,set(nat),set_ord_lessThan(nat),K2))) ) ).

% finite_nat_bounded
tff(fact_4594_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set(int)] :
      ( ~ pp(aa(set(int),bool,finite_finite(int),S2))
    <=> ! [M7: int] :
        ? [N5: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M7),aa(int,int,abs_abs(int),N5)))
          & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),N5),S2)) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_4595_infinite__int__iff__unbounded,axiom,
    ! [S2: set(int)] :
      ( ~ pp(aa(set(int),bool,finite_finite(int),S2))
    <=> ! [M7: int] :
        ? [N5: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),M7),aa(int,int,abs_abs(int),N5)))
          & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),N5),S2)) ) ) ).

% infinite_int_iff_unbounded
tff(fact_4596_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set(nat)] :
      ( ! [M5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),M5))
         => ? [N7: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N7))
              & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N7),S2)) ) )
     => ~ pp(aa(set(nat),bool,finite_finite(nat),S2)) ) ).

% unbounded_k_infinite
tff(fact_4597_infinite__nat__iff__unbounded,axiom,
    ! [S2: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite(nat),S2))
    <=> ! [M7: nat] :
        ? [N5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M7),N5))
          & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N5),S2)) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_4598_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set(nat)] :
      ( ~ pp(aa(set(nat),bool,finite_finite(nat),S2))
    <=> ! [M7: nat] :
        ? [N5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),N5))
          & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N5),S2)) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_4599_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,N: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => 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(N),real_V7770717601297561774m_norm(complex,C2)))),cis(divide_divide(real,arg(C2),aa(nat,real,semiring_1_of_nat(real),N))))),collect(complex,aTP_Lamp_eq(nat,fun(complex,bool),N)),collect(complex,aa(nat,fun(complex,bool),aTP_Lamp_jy(complex,fun(nat,fun(complex,bool)),C2),N))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_4600_in__finite__psubset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(product_prod(set(A),set(A))),bool,aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),bool),member(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),product_Pair(set(A),set(A),A3),B3)),finite_psubset(A)))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
        & pp(aa(set(A),bool,finite_finite(A),B3)) ) ) ).

% in_finite_psubset
tff(fact_4601_sum__count__set,axiom,
    ! [A: $tType,Xs: list(A),X6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X6))
     => ( pp(aa(set(A),bool,finite_finite(A),X6))
       => ( aa(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_4602_real__root__Suc__0,axiom,
    ! [X: real] : aa(real,real,root(aa(nat,nat,suc,zero_zero(nat))),X) = X ).

% real_root_Suc_0
tff(fact_4603_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(real,real,root(N),X) = aa(real,real,root(N),Y) )
      <=> ( X = Y ) ) ) ).

% real_root_eq_iff
tff(fact_4604_count__notin,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(A,nat,count_list(A,Xs),X) = zero_zero(nat) ) ) ).

% count_notin
tff(fact_4605_real__root__eq__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(real,real,root(N),X) = zero_zero(real) )
      <=> ( X = zero_zero(real) ) ) ) ).

% real_root_eq_0_iff
tff(fact_4606_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y)) ) ) ).

% real_root_less_iff
tff(fact_4607_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y)) ) ) ).

% real_root_le_iff
tff(fact_4608_real__root__one,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,root(N),one_one(real)) = one_one(real) ) ) ).

% real_root_one
tff(fact_4609_real__root__eq__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(real,real,root(N),X) = one_one(real) )
      <=> ( X = one_one(real) ) ) ) ).

% real_root_eq_1_iff
tff(fact_4610_real__root__gt__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y)) ) ) ).

% real_root_gt_0_iff
tff(fact_4611_real__root__lt__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),zero_zero(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real))) ) ) ).

% real_root_lt_0_iff
tff(fact_4612_real__root__ge__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y)) ) ) ).

% real_root_ge_0_iff
tff(fact_4613_real__root__le__0__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),zero_zero(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),zero_zero(real))) ) ) ).

% real_root_le_0_iff
tff(fact_4614_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),Y)) ) ) ).

% real_root_gt_1_iff
tff(fact_4615_real__root__lt__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),one_one(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real))) ) ) ).

% real_root_lt_1_iff
tff(fact_4616_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(real,real,root(N),Y)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),Y)) ) ) ).

% real_root_ge_1_iff
tff(fact_4617_real__root__le__1__iff,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),one_one(real)))
      <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real))) ) ) ).

% real_root_le_1_iff
tff(fact_4618_real__root__pow__pos2,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => ( aa(nat,real,power_power(real,aa(real,real,root(N),X)),N) = X ) ) ) ).

% real_root_pow_pos2
tff(fact_4619_real__root__pos__pos__le,axiom,
    ! [X: real,N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,root(N),X))) ) ).

% real_root_pos_pos_le
tff(fact_4620_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y))) ) ) ).

% real_root_less_mono
tff(fact_4621_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),Y))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),aa(real,real,root(N),Y))) ) ) ).

% real_root_le_mono
tff(fact_4622_real__root__power,axiom,
    ! [N: nat,X: real,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,root(N),aa(nat,real,power_power(real,X),K)) = aa(nat,real,power_power(real,aa(real,real,root(N),X)),K) ) ) ).

% real_root_power
tff(fact_4623_real__root__abs,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,root(N),aa(real,real,abs_abs(real),X)) = aa(real,real,abs_abs(real),aa(real,real,root(N),X)) ) ) ).

% real_root_abs
tff(fact_4624_sgn__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( sgn_sgn(real,aa(real,real,root(N),X)) = sgn_sgn(real,X) ) ) ).

% sgn_root
tff(fact_4625_real__root__gt__zero,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,root(N),X))) ) ) ).

% real_root_gt_zero
tff(fact_4626_real__root__strict__decreasing,axiom,
    ! [N: nat,N3: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N3))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N3),X)),aa(real,real,root(N),X))) ) ) ) ).

% real_root_strict_decreasing
tff(fact_4627_sqrt__def,axiom,
    sqrt = root(aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% sqrt_def
tff(fact_4628_root__abs__power,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,abs_abs(real),aa(real,real,root(N),aa(nat,real,power_power(real,Y),N))) = aa(real,real,abs_abs(real),Y) ) ) ).

% root_abs_power
tff(fact_4629_count__le__length,axiom,
    ! [A: $tType,Xs: list(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,count_list(A,Xs),X)),aa(list(A),nat,size_size(list(A)),Xs))) ).

% count_le_length
tff(fact_4630_real__root__pos__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,root(N),X))) ) ) ).

% real_root_pos_pos
tff(fact_4631_real__root__strict__increasing,axiom,
    ! [N: nat,N3: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),N3))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,root(N),X)),aa(real,real,root(N3),X))) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_4632_real__root__decreasing,axiom,
    ! [N: nat,N3: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N3))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),X))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N3),X)),aa(real,real,root(N),X))) ) ) ) ).

% real_root_decreasing
tff(fact_4633_real__root__pow__pos,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( aa(nat,real,power_power(real,aa(real,real,root(N),X)),N) = X ) ) ) ).

% real_root_pow_pos
tff(fact_4634_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y))
       => ( ( aa(nat,real,power_power(real,Y),N) = X )
         => ( aa(real,real,root(N),X) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_4635_real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
       => ( aa(real,real,root(N),aa(nat,real,power_power(real,X),N)) = X ) ) ) ).

% real_root_power_cancel
tff(fact_4636_odd__real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
     => ( aa(real,real,root(N),aa(nat,real,power_power(real,X),N)) = X ) ) ).

% odd_real_root_power_cancel
tff(fact_4637_odd__real__root__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
     => ( ( aa(nat,real,power_power(real,Y),N) = X )
       => ( aa(real,real,root(N),X) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_4638_odd__real__root__pow,axiom,
    ! [N: nat,X: real] :
      ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
     => ( aa(nat,real,power_power(real,aa(real,real,root(N),X)),N) = X ) ) ).

% odd_real_root_pow
tff(fact_4639_real__root__increasing,axiom,
    ! [N: nat,N3: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),N3))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,root(N),X)),aa(real,real,root(N3),X))) ) ) ) ) ).

% real_root_increasing
tff(fact_4640_root__sgn__power,axiom,
    ! [N: nat,Y: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,root(N),aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Y)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),Y)),N))) = Y ) ) ).

% root_sgn_power
tff(fact_4641_sgn__power__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,aa(real,real,root(N),X))),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),aa(real,real,root(N),X))),N)) = X ) ) ).

% sgn_power_root
tff(fact_4642_ln__root,axiom,
    ! [N: nat,B2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
       => ( aa(real,real,ln_ln(real),aa(real,real,root(N),B2)) = divide_divide(real,aa(real,real,ln_ln(real),B2),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ) ).

% ln_root
tff(fact_4643_log__root,axiom,
    ! [N: nat,A2: real,B2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
       => ( aa(real,real,log(B2),aa(real,real,root(N),A2)) = divide_divide(real,aa(real,real,log(B2),A2),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ) ).

% log_root
tff(fact_4644_log__base__root,axiom,
    ! [N: nat,B2: real,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
       => ( aa(real,real,log(aa(real,real,root(N),B2)),X) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(B2),X)) ) ) ) ).

% log_base_root
tff(fact_4645_split__root,axiom,
    ! [P: fun(real,bool),N: nat,X: real] :
      ( pp(aa(real,bool,P,aa(real,real,root(N),X)))
    <=> ( ( ( N = zero_zero(nat) )
         => pp(aa(real,bool,P,zero_zero(real))) )
        & ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
         => ! [Y5: real] :
              ( ( aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Y5)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),Y5)),N)) = X )
             => pp(aa(real,bool,P,Y5)) ) ) ) ) ).

% split_root
tff(fact_4646_finite__psubset__def,axiom,
    ! [A: $tType] : finite_psubset(A) = collect(product_prod(set(A),set(A)),product_case_prod(set(A),set(A),bool,aTP_Lamp_jz(set(A),fun(set(A),bool)))) ).

% finite_psubset_def
tff(fact_4647_root__powr__inverse,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => ( aa(real,real,root(N),X) = powr(real,X,divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ) ).

% root_powr_inverse
tff(fact_4648_set__encode__insert,axiom,
    ! [A3: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,finite_finite(nat),A3))
     => ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N),A3))
       => ( aa(set(nat),nat,nat_set_encode,aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),N),A3)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),aa(set(nat),nat,nat_set_encode,A3)) ) ) ) ).

% set_encode_insert
tff(fact_4649_even__sum__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_parity(A)
     => ! [A3: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A3)))
          <=> pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(set(B),nat,finite_card(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ka(set(B),fun(fun(B,A),fun(B,bool)),A3),F3))))) ) ) ) ).

% even_sum_iff
tff(fact_4650_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_4651_card__lessThan,axiom,
    ! [U: nat] : aa(set(nat),nat,finite_card(nat),aa(nat,set(nat),set_ord_lessThan(nat),U)) = U ).

% card_lessThan
tff(fact_4652_card__Collect__less__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ea(nat,fun(nat,bool)),N))) = N ).

% card_Collect_less_nat
tff(fact_4653_insert__subset,axiom,
    ! [A: $tType,X: A,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)),B3))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B3))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ) ).

% insert_subset
tff(fact_4654_card__atMost,axiom,
    ! [U: nat] : aa(set(nat),nat,finite_card(nat),aa(nat,set(nat),set_ord_atMost(nat),U)) = aa(nat,nat,suc,U) ).

% card_atMost
tff(fact_4655_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or7035219750837199246ssThan(nat,L,U)) = aa(nat,nat,minus_minus(nat,U),L) ).

% card_atLeastLessThan
tff(fact_4656_card__Collect__le__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_jo(nat,fun(nat,bool)),N))) = aa(nat,nat,suc,N) ).

% card_Collect_le_nat
tff(fact_4657_card__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or1337092689740270186AtMost(nat,L,U)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,U)),L) ).

% card_atLeastAtMost
tff(fact_4658_prod__constant,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Y: A,A3: set(B)] : aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aTP_Lamp_kb(A,fun(B,A),Y)),A3) = aa(nat,A,power_power(A,Y),aa(set(B),nat,finite_card(B),A3)) ) ).

% prod_constant
tff(fact_4659_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or7035219750837199246ssThan(int,L,U)) = nat2(aa(int,int,minus_minus(int,U),L)) ).

% card_atLeastLessThan_int
tff(fact_4660_sum_Oinsert,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
           => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3)) ) ) ) ) ).

% sum.insert
tff(fact_4661_card__insert__disjoint,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A3)) ) ) ) ).

% card_insert_disjoint
tff(fact_4662_numeral__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [K: num,L: num] : modulo_modulo(A,aa(num,A,numeral_numeral(A),K),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,K,L)) ) ).

% numeral_mod_numeral
tff(fact_4663_snd__divmod__nat,axiom,
    ! [M: nat,N: nat] : aa(product_prod(nat,nat),nat,product_snd(nat,nat),divmod_nat(M,N)) = modulo_modulo(nat,M,N) ).

% snd_divmod_nat
tff(fact_4664_card__Diff__insert,axiom,
    ! [A: $tType,A2: A,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),B3))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3))) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),B3))),one_one(nat)) ) ) ) ).

% card_Diff_insert
tff(fact_4665_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or1337092689740270186AtMost(int,L,U)) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,U),L)),one_one(int))) ).

% card_atLeastAtMost_int
tff(fact_4666_one__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : modulo_modulo(A,one_one(A),aa(num,A,numeral_numeral(A),N)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,one2,N)) ) ).

% one_mod_numeral
tff(fact_4667_card__insert__le,axiom,
    ! [A: $tType,A3: set(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)))) ).

% card_insert_le
tff(fact_4668_insert__mono,axiom,
    ! [A: $tType,C5: set(A),D4: set(A),A2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C5),D4))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),C5)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),D4))) ) ).

% insert_mono
tff(fact_4669_subset__insert,axiom,
    ! [A: $tType,X: A,A3: set(A),B3: set(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B3)))
      <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ) ).

% subset_insert
tff(fact_4670_subset__insertI,axiom,
    ! [A: $tType,B3: set(A),A2: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),B3))) ).

% subset_insertI
tff(fact_4671_subset__insertI2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),B2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),B3))) ) ).

% subset_insertI2
tff(fact_4672_n__subsets,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( aa(set(set(A)),nat,finite_card(set(A)),collect(set(A),aa(nat,fun(set(A),bool),aTP_Lamp_kc(set(A),fun(nat,fun(set(A),bool)),A3),K))) = aa(nat,nat,binomial(aa(set(A),nat,finite_card(A),A3)),K) ) ) ).

% n_subsets
tff(fact_4673_card__le__Suc__iff,axiom,
    ! [A: $tType,N: nat,A3: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(set(A),nat,finite_card(A),A3)))
    <=> ? [A6: A,B9: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A6),B9) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A6),B9))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(set(A),nat,finite_card(A),B9)))
          & pp(aa(set(A),bool,finite_finite(A),B9)) ) ) ).

% card_le_Suc_iff
tff(fact_4674_card__insert__if,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(set(A),nat,finite_card(A),A3) ) )
        & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),A3)) ) ) ) ) ).

% card_insert_if
tff(fact_4675_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) )
    <=> ? [B6: A,B9: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B6),B9) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B6),B9))
          & ( aa(set(A),nat,finite_card(A),B9) = K )
          & pp(aa(set(A),bool,finite_finite(A),B9)) ) ) ).

% card_Suc_eq_finite
tff(fact_4676_card__subset__eq,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
       => ( ( aa(set(A),nat,finite_card(A),A3) = aa(set(A),nat,finite_card(A),B3) )
         => ( A3 = B3 ) ) ) ) ).

% card_subset_eq
tff(fact_4677_infinite__arbitrarily__large,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( ~ pp(aa(set(A),bool,finite_finite(A),A3))
     => ? [B7: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),B7))
          & ( aa(set(A),nat,finite_card(A),B7) = N )
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B7),A3)) ) ) ).

% infinite_arbitrarily_large
tff(fact_4678_card__le__if__inj__on__rel,axiom,
    ! [B: $tType,A: $tType,B3: set(A),A3: set(B),R3: fun(B,fun(A,bool))] :
      ( pp(aa(set(A),bool,finite_finite(A),B3))
     => ( ! [A5: B] :
            ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A5),A3))
           => ? [B10: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B10),B3))
                & pp(aa(A,bool,aa(B,fun(A,bool),R3,A5),B10)) ) )
       => ( ! [A12: B,A23: B,B5: A] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A12),A3))
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A23),A3))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B5),B3))
                 => ( pp(aa(A,bool,aa(B,fun(A,bool),R3,A12),B5))
                   => ( pp(aa(A,bool,aa(B,fun(A,bool),R3,A23),B5))
                     => ( A12 = A23 ) ) ) ) ) )
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A3)),aa(set(A),nat,finite_card(A),B3))) ) ) ) ).

% card_le_if_inj_on_rel
tff(fact_4679_bij__betw__iff__card,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(B),bool,finite_finite(B),B3))
       => ( ? [F6: fun(A,B)] : bij_betw(A,B,F6,A3,B3)
        <=> ( aa(set(A),nat,finite_card(A),A3) = aa(set(B),nat,finite_card(B),B3) ) ) ) ) ).

% bij_betw_iff_card
tff(fact_4680_finite__same__card__bij,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(B),bool,finite_finite(B),B3))
       => ( ( aa(set(A),nat,finite_card(A),A3) = aa(set(B),nat,finite_card(B),B3) )
         => ? [H3: fun(A,B)] : bij_betw(A,B,H3,A3,B3) ) ) ) ).

% finite_same_card_bij
tff(fact_4681_subset__Diff__insert,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),X: A,C5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),minus_minus(set(A),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),C5))))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),minus_minus(set(A),B3),C5)))
        & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3)) ) ) ).

% subset_Diff_insert
tff(fact_4682_lessThan__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),K),aa(nat,set(nat),set_ord_lessThan(nat),K)) ).

% lessThan_Suc
tff(fact_4683_atMost__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,K)),aa(nat,set(nat),set_ord_atMost(nat),K)) ).

% atMost_Suc
tff(fact_4684_card__insert__le__m1,axiom,
    ! [A: $tType,N: nat,Y: set(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),Y)),aa(nat,nat,minus_minus(nat,N),one_one(nat))))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),Y))),N)) ) ) ).

% card_insert_le_m1
tff(fact_4685_card__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_iw(set(A),fun(nat,fun(list(A),bool)),A3),N))) = aa(nat,nat,power_power(nat,aa(set(A),nat,finite_card(A),A3)),N) ) ) ).

% card_lists_length_eq
tff(fact_4686_card__eq__sum,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),nat,finite_card(A),A3) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aTP_Lamp_kd(A,nat)),A3) ).

% card_eq_sum
tff(fact_4687_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_4688_card__2__iff_H,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S2) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
    <=> ? [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
          & ? [Xa4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),S2))
              & ( X4 != Xa4 )
              & ! [Xb4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xb4),S2))
                 => ( ( Xb4 = X4 )
                    | ( Xb4 = Xa4 ) ) ) ) ) ) ).

% card_2_iff'
tff(fact_4689_card__ge__0__finite,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A3)))
     => pp(aa(set(A),bool,finite_finite(A),A3)) ) ).

% card_ge_0_finite
tff(fact_4690_obtain__subset__with__card__n,axiom,
    ! [A: $tType,N: nat,S2: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(set(A),nat,finite_card(A),S2)))
     => ~ ! [T7: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T7),S2))
           => ( ( aa(set(A),nat,finite_card(A),T7) = N )
             => ~ pp(aa(set(A),bool,finite_finite(A),T7)) ) ) ) ).

% obtain_subset_with_card_n
tff(fact_4691_finite__if__finite__subsets__card__bdd,axiom,
    ! [A: $tType,F5: set(A),C5: nat] :
      ( ! [G2: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),G2),F5))
         => ( pp(aa(set(A),bool,finite_finite(A),G2))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),G2)),C5)) ) )
     => ( pp(aa(set(A),bool,finite_finite(A),F5))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),F5)),C5)) ) ) ).

% finite_if_finite_subsets_card_bdd
tff(fact_4692_card__seteq,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),B3)),aa(set(A),nat,finite_card(A),A3)))
         => ( A3 = B3 ) ) ) ) ).

% card_seteq
tff(fact_4693_card__mono,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3))) ) ) ).

% card_mono
tff(fact_4694_card__less__sym__Diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,finite_finite(A),B3))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),B3))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),B3),A3)))) ) ) ) ).

% card_less_sym_Diff
tff(fact_4695_card__le__sym__Diff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,finite_finite(A),B3))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),B3))),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),B3),A3)))) ) ) ) ).

% card_le_sym_Diff
tff(fact_4696_card__length,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),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_4697_ex__bij__betw__finite__nat,axiom,
    ! [A: $tType,M6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),M6))
     => ? [H3: fun(A,nat)] : bij_betw(A,nat,H3,M6,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M6))) ) ).

% ex_bij_betw_finite_nat
tff(fact_4698_psubset__card__mono,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3))) ) ) ).

% psubset_card_mono
tff(fact_4699_snd__divmod,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,M,N)) = modulo_modulo(A,aa(num,A,numeral_numeral(A),M),aa(num,A,numeral_numeral(A),N)) ) ).

% snd_divmod
tff(fact_4700_sum_Oinsert__if,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
             => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
             => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3)) ) ) ) ) ) ).

% sum.insert_if
tff(fact_4701_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,N)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ).

% atLeast0_atMost_Suc
tff(fact_4702_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),N),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ).

% atLeast0_lessThan_Suc
tff(fact_4703_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( set_or1337092689740270186AtMost(nat,M,N) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_4704_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),aa(nat,nat,suc,N)))
     => ( set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,N)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_4705_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) = set_or1337092689740270186AtMost(nat,M,N) ) ) ).

% atLeastAtMost_insertL
tff(fact_4706_lessThan__nat__numeral,axiom,
    ! [K: num] : aa(nat,set(nat),set_ord_lessThan(nat),aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),pred_numeral(K)),aa(nat,set(nat),set_ord_lessThan(nat),pred_numeral(K))) ).

% lessThan_nat_numeral
tff(fact_4707_card__less,axiom,
    ! [M6: set(nat),I2: nat] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),M6))
     => ( aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ke(set(nat),fun(nat,fun(nat,bool)),M6),I2))) != zero_zero(nat) ) ) ).

% card_less
tff(fact_4708_card__less__Suc,axiom,
    ! [M6: set(nat),I2: nat] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),M6))
     => ( aa(nat,nat,suc,aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_kf(set(nat),fun(nat,fun(nat,bool)),M6),I2)))) = aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ke(set(nat),fun(nat,fun(nat,bool)),M6),I2))) ) ) ).

% card_less_Suc
tff(fact_4709_card__less__Suc2,axiom,
    ! [M6: set(nat),I2: nat] :
      ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),M6))
     => ( aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_kf(set(nat),fun(nat,fun(nat,bool)),M6),I2))) = aa(set(nat),nat,finite_card(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ke(set(nat),fun(nat,fun(nat,bool)),M6),I2))) ) ) ).

% card_less_Suc2
tff(fact_4710_atMost__nat__numeral,axiom,
    ! [K: num] : aa(nat,set(nat),set_ord_atMost(nat),aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,set(nat),set_ord_atMost(nat),pred_numeral(K))) ).

% atMost_nat_numeral
tff(fact_4711_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] : aa(set(int),nat,finite_card(int),set_or7035219750837199246ssThan(int,zero_zero(int),U)) = nat2(U) ).

% card_atLeastZeroLessThan_int
tff(fact_4712_sum__Suc,axiom,
    ! [A: $tType,F3: fun(A,nat),A3: set(A)] : aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aTP_Lamp_kg(fun(A,nat),fun(A,nat),F3)),A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),A3)),aa(set(A),nat,finite_card(A),A3)) ).

% sum_Suc
tff(fact_4713_subset__card__intvl__is__intvl,axiom,
    ! [A3: set(nat),K: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),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_4714_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,groups7311177749621191930dd_sum(A,real,aTP_Lamp_kh(A,real)),A3) ).

% real_of_card
tff(fact_4715_sum__bounded__above,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & semiring_1(A) )
     => ! [A3: set(B),F3: fun(B,A),K5: A] :
          ( ! [I4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I4)),K5)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),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))),K5))) ) ) ).

% sum_bounded_above
tff(fact_4716_sum__bounded__below,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & semiring_1(A) )
     => ! [A3: set(B),K5: A,F3: fun(B,A)] :
          ( ! [I4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K5),aa(B,A,F3,I4))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),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))),K5)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A3))) ) ) ).

% sum_bounded_below
tff(fact_4717_card__le__Suc0__iff__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(nat,nat,suc,zero_zero(nat))))
      <=> ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
           => ! [Xa4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),A3))
               => ( X4 = Xa4 ) ) ) ) ) ).

% card_le_Suc0_iff_eq
tff(fact_4718_card__Diff__subset,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),B3)) = aa(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_4719_card__psubset,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(A),nat,finite_card(A),B3)))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3)) ) ) ) ).

% card_psubset
tff(fact_4720_diff__card__le__card__Diff,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),B3))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(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),minus_minus(set(A),A3),B3)))) ) ).

% diff_card_le_card_Diff
tff(fact_4721_card__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( aa(set(list(A)),nat,finite_card(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_ix(set(A),fun(nat,fun(list(A),bool)),A3),N))) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,power_power(nat,aa(set(A),nat,finite_card(A),A3))),aa(nat,set(nat),set_ord_atMost(nat),N)) ) ) ).

% card_lists_length_le
tff(fact_4722_ex__bij__betw__nat__finite,axiom,
    ! [A: $tType,M6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),M6))
     => ? [H3: fun(nat,A)] : bij_betw(nat,A,H3,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(A),nat,finite_card(A),M6)),M6) ) ).

% ex_bij_betw_nat_finite
tff(fact_4723_ex__bij__betw__nat__finite__1,axiom,
    ! [A: $tType,M6: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),M6))
     => ? [H3: fun(nat,A)] : bij_betw(nat,A,H3,set_or1337092689740270186AtMost(nat,one_one(nat),aa(set(A),nat,finite_card(A),M6)),M6) ) ).

% ex_bij_betw_nat_finite_1
tff(fact_4724_card__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [N: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),N))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),collect(A,aTP_Lamp_ji(nat,fun(A,bool),N)))),N)) ) ) ).

% card_roots_unity
tff(fact_4725_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N3: set(nat),N: nat] :
      ( pp(aa(set(nat),bool,aa(set(nat),fun(set(nat),bool),ord_less_eq(set(nat)),N3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),N3)),N)) ) ).

% subset_eq_atLeast0_lessThan_card
tff(fact_4726_card__sum__le__nat__sum,axiom,
    ! [S2: set(nat)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fu(nat,nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(set(nat),nat,finite_card(nat),S2)))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fu(nat,nat)),S2))) ).

% card_sum_le_nat_sum
tff(fact_4727_card__nth__roots,axiom,
    ! [C2: complex,N: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( aa(set(complex),nat,finite_card(complex),collect(complex,aa(nat,fun(complex,bool),aTP_Lamp_jy(complex,fun(nat,fun(complex,bool)),C2),N))) = N ) ) ) ).

% card_nth_roots
tff(fact_4728_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(set(complex),nat,finite_card(complex),collect(complex,aTP_Lamp_eq(nat,fun(complex,bool),N))) = N ) ) ).

% card_roots_unity_eq
tff(fact_4729_sum__norm__bound,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [S2: set(B),F3: fun(B,A),K5: real] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S2))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(B,A,F3,X3))),K5)) )
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),S2))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(set(B),nat,finite_card(B),S2))),K5))) ) ) ).

% sum_norm_bound
tff(fact_4730_prod__le__power,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: set(B),F3: fun(B,A),N: A,K: nat] :
          ( ! [I4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I4)))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I4)),N)) ) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),A3)),K))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A3)),aa(nat,A,power_power(A,N),K))) ) ) ) ) ).

% prod_le_power
tff(fact_4731_sum__bounded__above__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & semiring_1(A) )
     => ! [A3: set(B),F3: fun(B,A),K5: A] :
          ( ! [I4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,I4)),K5)) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(B),nat,finite_card(B),A3)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),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))),K5))) ) ) ) ).

% sum_bounded_above_strict
tff(fact_4732_polyfun__roots__card,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,N: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),collect(A,aa(nat,fun(A,bool),aTP_Lamp_jn(fun(nat,A),fun(nat,fun(A,bool)),C2),N)))),N)) ) ) ) ).

% polyfun_roots_card
tff(fact_4733_prod__gen__delta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),A2: B,B2: fun(B,A),C2: A] :
          ( pp(aa(set(B),bool,finite_finite(B),S2))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_ki(B,fun(fun(B,A),fun(A,fun(B,A))),A2),B2),C2)),S2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,B2,A2)),aa(nat,A,power_power(A,C2),aa(nat,nat,minus_minus(nat,aa(set(B),nat,finite_card(B),S2)),one_one(nat)))) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_ki(B,fun(fun(B,A),fun(A,fun(B,A))),A2),B2),C2)),S2) = aa(nat,A,power_power(A,C2),aa(set(B),nat,finite_card(B),S2)) ) ) ) ) ) ).

% prod_gen_delta
tff(fact_4734_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N),nat_set_decode(Z)))
     => ( nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)),Z)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),N),nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_4735_polyfun__rootbound,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,N: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
           => ( pp(aa(set(A),bool,finite_finite(A),collect(A,aa(nat,fun(A,bool),aTP_Lamp_jn(fun(nat,A),fun(nat,fun(A,bool)),C2),N))))
              & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),collect(A,aa(nat,fun(A,bool),aTP_Lamp_jn(fun(nat,A),fun(nat,fun(A,bool)),C2),N)))),N)) ) ) ) ) ).

% polyfun_rootbound
tff(fact_4736_in__set__enumerate__eq,axiom,
    ! [A: $tType,P2: product_prod(nat,A),N: nat,Xs: list(A)] :
      ( pp(aa(set(product_prod(nat,A)),bool,aa(product_prod(nat,A),fun(set(product_prod(nat,A)),bool),member(product_prod(nat,A)),P2),aa(list(product_prod(nat,A)),set(product_prod(nat,A)),set2(product_prod(nat,A)),enumerate(A,N,Xs))))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(product_prod(nat,A),nat,product_fst(nat,A),P2)))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(product_prod(nat,A),nat,product_fst(nat,A),P2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)))
        & ( aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,aa(product_prod(nat,A),nat,product_fst(nat,A),P2)),N)) = aa(product_prod(nat,A),A,product_snd(nat,A),P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_4737_card__lists__distinct__length__eq,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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),bool),aTP_Lamp_kj(set(A),fun(nat,fun(list(A),bool)),A3),K))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_fu(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(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_4738_card__lists__distinct__length__eq_H,axiom,
    ! [A: $tType,K: nat,A3: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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),bool),aTP_Lamp_kk(nat,fun(set(A),fun(list(A),bool)),K),A3))) = aa(set(nat),nat,aa(fun(nat,nat),fun(set(nat),nat),groups7121269368397514597t_prod(nat,nat),aTP_Lamp_fu(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(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_4739_length__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),nat,size_size(list(product_prod(nat,A))),enumerate(A,N,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_enumerate
tff(fact_4740_snd__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] : aa(product_prod(code_integer,code_integer),code_integer,product_snd(code_integer,code_integer),code_divmod_integer(K,L)) = modulo_modulo(code_integer,K,L) ).

% snd_divmod_integer
tff(fact_4741_snd__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] : aa(product_prod(code_integer,code_integer),code_integer,product_snd(code_integer,code_integer),code_divmod_abs(K,L)) = modulo_modulo(code_integer,aa(code_integer,code_integer,abs_abs(code_integer),K),aa(code_integer,code_integer,abs_abs(code_integer),L)) ).

% snd_divmod_abs
tff(fact_4742_finite__lists__distinct__length__eq,axiom,
    ! [A: $tType,A3: set(A),N: nat] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => pp(aa(set(list(A)),bool,finite_finite(list(A)),collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_kj(set(A),fun(nat,fun(list(A),bool)),A3),N)))) ) ).

% finite_lists_distinct_length_eq
tff(fact_4743_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( distinct(A,Xs)
         => distinct(A,Xs) ) ) ).

% sorted_list_of_set.distinct_if_distinct_map
tff(fact_4744_distinct__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : distinct(product_prod(nat,A),enumerate(A,N,Xs)) ).

% distinct_enumerate
tff(fact_4745_distinct__product,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( distinct(A,Xs)
     => ( distinct(B,Ys)
       => distinct(product_prod(A,B),product(A,B,Xs,Ys)) ) ) ).

% distinct_product
tff(fact_4746_finite__distinct__list,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ? [Xs2: list(A)] :
          ( ( aa(list(A),set(A),set2(A),Xs2) = A3 )
          & distinct(A,Xs2) ) ) ).

% finite_distinct_list
tff(fact_4747_subseqs__distinctD,axiom,
    ! [A: $tType,Ys: list(A),Xs: list(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Ys),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs))))
     => ( distinct(A,Xs)
       => distinct(A,Ys) ) ) ).

% subseqs_distinctD
tff(fact_4748_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_4749_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_4750_nth__eq__iff__index__eq,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,J: nat] :
      ( distinct(A,Xs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
         => ( ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Xs),J) )
          <=> ( I2 = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
tff(fact_4751_distinct__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
    <=> ! [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
         => ! [J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_4752_distinct__Ex1,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ? [X3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),aa(list(A),nat,size_size(list(A)),Xs)))
            & ( aa(nat,A,nth(A,Xs),X3) = X )
            & ! [Y3: nat] :
                ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Y3),aa(list(A),nat,size_size(list(A)),Xs)))
                  & ( aa(nat,A,nth(A,Xs),Y3) = X ) )
               => ( Y3 = X3 ) ) ) ) ) ).

% distinct_Ex1
tff(fact_4753_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),M),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)))
     => ( set_or1337092689740270186AtMost(int,M,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)) = aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)),set_or1337092689740270186AtMost(int,M,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_4754_bij__betw__nth,axiom,
    ! [A: $tType,Xs: list(A),A3: set(nat),B3: set(A)] :
      ( distinct(A,Xs)
     => ( ( A3 = aa(nat,set(nat),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_4755_nth__enumerate__eq,axiom,
    ! [A: $tType,M: nat,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,product_prod(nat,A),nth(product_prod(nat,A),enumerate(A,N,Xs)),M) = aa(A,product_prod(nat,A),product_Pair(nat,A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),aa(nat,A,nth(A,Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_4756_minus__one__mod__numeral,axiom,
    ! [N: num] : modulo_modulo(int,aa(int,int,uminus_uminus(int),one_one(int)),aa(num,int,numeral_numeral(int),N)) = adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,N))) ).

% minus_one_mod_numeral
tff(fact_4757_one__mod__minus__numeral,axiom,
    ! [N: num] : modulo_modulo(int,one_one(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,N)))) ).

% one_mod_minus_numeral
tff(fact_4758_minus__numeral__mod__numeral,axiom,
    ! [M: num,N: num] : modulo_modulo(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) = adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,M,N))) ).

% minus_numeral_mod_numeral
tff(fact_4759_numeral__mod__minus__numeral,axiom,
    ! [M: num,N: num] : modulo_modulo(int,aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,M,N)))) ).

% numeral_mod_minus_numeral
tff(fact_4760_Divides_Oadjust__mod__def,axiom,
    ! [R3: int,L: int] :
      ( ( ( R3 = zero_zero(int) )
       => ( adjust_mod(L,R3) = zero_zero(int) ) )
      & ( ( R3 != zero_zero(int) )
       => ( adjust_mod(L,R3) = aa(int,int,minus_minus(int,L),R3) ) ) ) ).

% Divides.adjust_mod_def
tff(fact_4761_and__int_Oelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa) = Y )
     => ( ( ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
            & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( Y = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),X)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),Xa))))) ) )
        & ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
              & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),X)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),Xa))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,X,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,Xa,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ) ).

% and_int.elims
tff(fact_4762_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      ( ( ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
          & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),L),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
       => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L))))) ) )
      & ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
            & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),L),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
       => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ).

% and_int.simps
tff(fact_4763_bezw_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod(int,int)] :
      ( ( bezw(X,Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)) ) )
        & ( ( Xa != zero_zero(nat) )
         => ( Y = aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Xa))))) ) ) ) ) ).

% bezw.elims
tff(fact_4764_empty__subsetI,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),bot_bot(set(A))),A3)) ).

% empty_subsetI
tff(fact_4765_subset__empty,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),bot_bot(set(A))))
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% subset_empty
tff(fact_4766_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% atLeastatMost_empty_iff
tff(fact_4767_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A2,B2) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_4768_atLeastatMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastatMost_empty
tff(fact_4769_singleton__insert__inj__eq_H,axiom,
    ! [A: $tType,A2: A,A3: set(A),B2: A] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) )
    <=> ( ( A2 = B2 )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_4770_singleton__insert__inj__eq,axiom,
    ! [A: $tType,B2: A,A2: A,A3: set(A)] :
      ( ( aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3) )
    <=> ( ( A2 = B2 )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))) ) ) ).

% singleton_insert_inj_eq
tff(fact_4771_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( set_or7035219750837199246ssThan(A,A2,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_4772_atLeastLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = bot_bot(set(A)) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% atLeastLessThan_empty_iff
tff(fact_4773_atLeastLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A2,B2) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% atLeastLessThan_empty_iff2
tff(fact_4774_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_4775_Diff__eq__empty__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( ( aa(set(A),set(A),minus_minus(set(A),A3),B3) = bot_bot(set(A)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ).

% Diff_eq_empty_iff
tff(fact_4776_atLeastAtMost__singleton__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( set_or1337092689740270186AtMost(A,A2,B2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),C2),bot_bot(set(A))) )
        <=> ( ( A2 = B2 )
            & ( B2 = C2 ) ) ) ) ).

% atLeastAtMost_singleton_iff
tff(fact_4777_atLeastAtMost__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : set_or1337092689740270186AtMost(A,A2,A2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ) ).

% atLeastAtMost_singleton
tff(fact_4778_single__Diff__lessThan,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A] : aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),K),bot_bot(set(A)))),aa(A,set(A),set_ord_lessThan(A),K)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),K),bot_bot(set(A))) ) ).

% single_Diff_lessThan
tff(fact_4779_subset__Compl__singleton,axiom,
    ! [A: $tType,A3: set(A),B2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))))
    <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),A3)) ) ).

% subset_Compl_singleton
tff(fact_4780_set__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( N != zero_zero(nat) )
     => ( aa(list(A),set(A),set2(A),replicate(A,N,X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_4781_not__psubset__empty,axiom,
    ! [A: $tType,A3: set(A)] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),bot_bot(set(A)))) ).

% not_psubset_empty
tff(fact_4782_bot_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( ( A2 != bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),A2)) ) ) ).

% bot.not_eq_extremum
tff(fact_4783_bot_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),bot_bot(A))) ) ).

% bot.extremum_strict
tff(fact_4784_bot_Oextremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),bot_bot(A)),A2)) ) ).

% bot.extremum
tff(fact_4785_bot_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),bot_bot(A)))
        <=> ( A2 = bot_bot(A) ) ) ) ).

% bot.extremum_unique
tff(fact_4786_bot_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),bot_bot(A)))
         => ( A2 = bot_bot(A) ) ) ) ).

% bot.extremum_uniqueI
tff(fact_4787_not__empty__eq__Iic__eq__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [H: A] : bot_bot(set(A)) != aa(A,set(A),set_ord_atMost(A),H) ) ).

% not_empty_eq_Iic_eq_empty
tff(fact_4788_lessThan__non__empty,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [X: A] : aa(A,set(A),set_ord_lessThan(A),X) != bot_bot(set(A)) ) ).

% lessThan_non_empty
tff(fact_4789_Iio__eq__empty__iff,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & order_bot(A) )
     => ! [N: A] :
          ( ( aa(A,set(A),set_ord_lessThan(A),N) = bot_bot(set(A)) )
        <=> ( N = bot_bot(A) ) ) ) ).

% Iio_eq_empty_iff
tff(fact_4790_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,minus_minus(A,X),Y) = bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ).

% diff_shunt_var
tff(fact_4791_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                & ! [Xa2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),A3))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa2),X3))
                     => ( X3 = Xa2 ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_4792_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                & ! [Xa2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),A3))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa2))
                     => ( X3 = Xa2 ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_4793_ex__min__if__finite,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [S2: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),S2))
         => ( ( S2 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                & ~ ? [Xa2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),S2))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Xa2),X3)) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_4794_infinite__growing,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X6: set(A)] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
               => ? [Xa2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),X6))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Xa2)) ) )
           => ~ pp(aa(set(A),bool,finite_finite(A),X6)) ) ) ) ).

% infinite_growing
tff(fact_4795_subset__singletonD,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))
     => ( ( A3 = bot_bot(set(A)) )
        | ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ) ) ).

% subset_singletonD
tff(fact_4796_subset__singleton__iff,axiom,
    ! [A: $tType,X6: set(A),A2: A] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))))
    <=> ( ( X6 = bot_bot(set(A)) )
        | ( X6 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ) ) ) ).

% subset_singleton_iff
tff(fact_4797_atLeastAtMost__singleton_H,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
         => ( set_or1337092689740270186AtMost(A,A2,B2) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A))) ) ) ) ).

% atLeastAtMost_singleton'
tff(fact_4798_subset__Compl__self__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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_4799_finite__ranking__induct,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [S2: set(B),P: fun(set(B),bool),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S2))
         => ( pp(aa(set(B),bool,P,bot_bot(set(B))))
           => ( ! [X3: B,S5: set(B)] :
                  ( pp(aa(set(B),bool,finite_finite(B),S5))
                 => ( ! [Y3: B] :
                        ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y3),S5))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,Y3)),aa(B,A,F3,X3))) )
                   => ( pp(aa(set(B),bool,P,S5))
                     => pp(aa(set(B),bool,P,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X3),S5))) ) ) )
             => pp(aa(set(B),bool,P,S2)) ) ) ) ) ).

% finite_ranking_induct
tff(fact_4800_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),P: fun(set(A),bool)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
           => ( ! [B5: A,A7: set(A)] :
                  ( pp(aa(set(A),bool,finite_finite(A),A7))
                 => ( ! [X2: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A7))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B5),X2)) )
                   => ( pp(aa(set(A),bool,P,A7))
                     => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B5),A7))) ) ) )
             => pp(aa(set(A),bool,P,A3)) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_4801_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),P: fun(set(A),bool)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
           => ( ! [B5: A,A7: set(A)] :
                  ( pp(aa(set(A),bool,finite_finite(A),A7))
                 => ( ! [X2: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A7))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),B5)) )
                   => ( pp(aa(set(A),bool,P,A7))
                     => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B5),A7))) ) ) )
             => pp(aa(set(A),bool,P,A3)) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_4802_sum__strict__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A3: set(B),F3: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ( A3 != bot_bot(set(B)) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X3)),aa(B,A,G,X3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A3)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3))) ) ) ) ) ).

% sum_strict_mono
tff(fact_4803_finite__subset__induct_H,axiom,
    ! [A: $tType,F5: set(A),A3: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite(A),F5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),F5),A3))
       => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
         => ( ! [A5: A,F7: set(A)] :
                ( pp(aa(set(A),bool,finite_finite(A),F7))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),A3))
                 => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),F7),A3))
                   => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),F7))
                     => ( pp(aa(set(A),bool,P,F7))
                       => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A5),F7))) ) ) ) ) )
           => pp(aa(set(A),bool,P,F5)) ) ) ) ) ).

% finite_subset_induct'
tff(fact_4804_finite__subset__induct,axiom,
    ! [A: $tType,F5: set(A),A3: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite(A),F5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),F5),A3))
       => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
         => ( ! [A5: A,F7: set(A)] :
                ( pp(aa(set(A),bool,finite_finite(A),F7))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),A3))
                 => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),F7))
                   => ( pp(aa(set(A),bool,P,F7))
                     => pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A5),F7))) ) ) ) )
           => pp(aa(set(A),bool,P,F5)) ) ) ) ) ).

% finite_subset_induct
tff(fact_4805_Diff__single__insert,axiom,
    ! [A: $tType,A3: set(A),X: A,B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),B3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B3))) ) ).

% Diff_single_insert
tff(fact_4806_subset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),X: A,B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B3)))
    <=> ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),B3)) )
        & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ) ) ).

% subset_insert_iff
tff(fact_4807_card__1__singletonE,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),A3) = one_one(nat) )
     => ~ ! [X3: A] : A3 != aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),bot_bot(set(A))) ) ).

% card_1_singletonE
tff(fact_4808_sum__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [I5: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),I5))
         => ( ( I5 != bot_bot(set(B)) )
           => ( ! [I4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),I5))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(B,A,F3,I4))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),I5))) ) ) ) ) ).

% sum_pos
tff(fact_4809_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite(A),I5))
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I4),I5))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(A,B,F3,I4))) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),one_one(B)),aa(set(A),B,aa(fun(A,B),fun(set(A),B),groups7121269368397514597t_prod(A,B),F3),I5))) ) ) ) ) ).

% less_1_prod
tff(fact_4810_card__gt__0__iff,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),A3)))
    <=> ( ( A3 != bot_bot(set(A)) )
        & pp(aa(set(A),bool,finite_finite(A),A3)) ) ) ).

% card_gt_0_iff
tff(fact_4811_finite__remove__induct,axiom,
    ! [A: $tType,B3: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite(A),B3))
     => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
       => ( ! [A7: set(A)] :
              ( pp(aa(set(A),bool,finite_finite(A),A7))
             => ( ( A7 != bot_bot(set(A)) )
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A7),B3))
                 => ( ! [X2: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A7))
                       => pp(aa(set(A),bool,P,aa(set(A),set(A),minus_minus(set(A),A7),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X2),bot_bot(set(A)))))) )
                   => pp(aa(set(A),bool,P,A7)) ) ) ) )
         => pp(aa(set(A),bool,P,B3)) ) ) ) ).

% finite_remove_induct
tff(fact_4812_remove__induct,axiom,
    ! [A: $tType,P: fun(set(A),bool),B3: set(A)] :
      ( pp(aa(set(A),bool,P,bot_bot(set(A))))
     => ( ( ~ pp(aa(set(A),bool,finite_finite(A),B3))
         => pp(aa(set(A),bool,P,B3)) )
       => ( ! [A7: set(A)] :
              ( pp(aa(set(A),bool,finite_finite(A),A7))
             => ( ( A7 != bot_bot(set(A)) )
               => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A7),B3))
                 => ( ! [X2: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A7))
                       => pp(aa(set(A),bool,P,aa(set(A),set(A),minus_minus(set(A),A7),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X2),bot_bot(set(A)))))) )
                   => pp(aa(set(A),bool,P,A7)) ) ) ) )
         => pp(aa(set(A),bool,P,B3)) ) ) ) ).

% remove_induct
tff(fact_4813_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)),insert(A),X4),bot_bot(set(A))) ) ).

% card_1_singleton_iff
tff(fact_4814_card__eq__SucD,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
     => ? [B5: A,B7: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B5),B7) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B5),B7))
          & ( aa(set(A),nat,finite_card(A),B7) = K )
          & ( ( K = zero_zero(nat) )
           => ( B7 = bot_bot(set(A)) ) ) ) ) ).

% card_eq_SucD
tff(fact_4815_card__Suc__eq,axiom,
    ! [A: $tType,A3: set(A),K: nat] :
      ( ( aa(set(A),nat,finite_card(A),A3) = aa(nat,nat,suc,K) )
    <=> ? [B6: A,B9: set(A)] :
          ( ( A3 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B6),B9) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B6),B9))
          & ( aa(set(A),nat,finite_card(A),B9) = K )
          & ( ( K = zero_zero(nat) )
           => ( B9 = bot_bot(set(A)) ) ) ) ) ).

% card_Suc_eq
tff(fact_4816_card__Diff1__le,axiom,
    ! [A: $tType,A3: set(A),X: A] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3))) ).

% card_Diff1_le
tff(fact_4817_finite__induct__select,axiom,
    ! [A: $tType,S2: set(A),P: fun(set(A),bool)] :
      ( pp(aa(set(A),bool,finite_finite(A),S2))
     => ( pp(aa(set(A),bool,P,bot_bot(set(A))))
       => ( ! [T7: set(A)] :
              ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),T7),S2))
             => ( pp(aa(set(A),bool,P,T7))
               => ? [X2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(set(A),set(A),minus_minus(set(A),S2),T7)))
                    & pp(aa(set(A),bool,P,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X2),T7))) ) ) )
         => pp(aa(set(A),bool,P,S2)) ) ) ) ).

% finite_induct_select
tff(fact_4818_psubset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),X: A,B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),B3)))
    <=> ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3)) )
        & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),B3))
         => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),B3)) )
            & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
             => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ) ) ) ) ).

% psubset_insert_iff
tff(fact_4819_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] : set_or7035219750837199246ssThan(A,A2,B2) = aa(set(A),set(A),minus_minus(set(A),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A)))) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_4820_set__replicate__Suc,axiom,
    ! [A: $tType,N: nat,X: A] : aa(list(A),set(A),set2(A),replicate(A,aa(nat,nat,suc,N),X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ).

% set_replicate_Suc
tff(fact_4821_set__replicate__conv__if,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(list(A),set(A),set2(A),replicate(A,N,X)) = bot_bot(set(A)) ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(list(A),set(A),set2(A),replicate(A,N,X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) ) ) ).

% set_replicate_conv_if
tff(fact_4822_prod__mono__strict,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_semidom(A)
     => ! [A3: set(B),F3: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ! [I4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),A3))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(B,A,F3,I4)))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,I4)),aa(B,A,G,I4))) ) )
           => ( ( A3 != bot_bot(set(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3))) ) ) ) ) ).

% prod_mono_strict
tff(fact_4823_card__2__iff,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S2) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) )
    <=> ? [X4: A,Y5: A] :
          ( ( S2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y5),bot_bot(set(A)))) )
          & ( X4 != Y5 ) ) ) ).

% card_2_iff
tff(fact_4824_sum_Oremove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),X: B,G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
           => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),minus_minus(set(B),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),bot_bot(set(B)))))) ) ) ) ) ).

% sum.remove
tff(fact_4825_sum_Oinsert__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A),X: B] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,G,X)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),minus_minus(set(B),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),bot_bot(set(B)))))) ) ) ) ).

% sum.insert_remove
tff(fact_4826_card__3__iff,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ( aa(set(A),nat,finite_card(A),S2) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
    <=> ? [X4: A,Y5: A,Z5: A] :
          ( ( S2 = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Z5),bot_bot(set(A))))) )
          & ( X4 != Y5 )
          & ( Y5 != Z5 )
          & ( X4 != Z5 ) ) ) ).

% card_3_iff
tff(fact_4827_odd__card__imp__not__empty,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),aa(set(A),nat,finite_card(A),A3)))
     => ( A3 != bot_bot(set(A)) ) ) ).

% odd_card_imp_not_empty
tff(fact_4828_card_Oremove,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),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),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ).

% card.remove
tff(fact_4829_card_Oinsert__remove,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ).

% card.insert_remove
tff(fact_4830_card__Suc__Diff1,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => ( aa(nat,nat,suc,aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) = aa(set(A),nat,finite_card(A),A3) ) ) ) ).

% card_Suc_Diff1
tff(fact_4831_card__Diff1__less,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3))) ) ) ).

% card_Diff1_less
tff(fact_4832_card__Diff2__less,axiom,
    ! [A: $tType,A3: set(A),X: A,Y: A] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A3))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3))) ) ) ) ).

% card_Diff2_less
tff(fact_4833_card__Diff1__less__iff,axiom,
    ! [A: $tType,A3: set(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))),aa(set(A),nat,finite_card(A),A3)))
    <=> ( pp(aa(set(A),bool,finite_finite(A),A3))
        & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3)) ) ) ).

% card_Diff1_less_iff
tff(fact_4834_card__Diff__singleton,axiom,
    ! [A: $tType,X: A,A3: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),A3)),one_one(nat)) ) ) ).

% card_Diff_singleton
tff(fact_4835_card__Diff__singleton__if,axiom,
    ! [A: $tType,X: A,A3: set(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,finite_card(A),A3)),one_one(nat)) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => ( aa(set(A),nat,finite_card(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) = aa(set(A),nat,finite_card(A),A3) ) ) ) ).

% card_Diff_singleton_if
tff(fact_4836_sum_Odelta__remove,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),A2: B,B2: fun(B,A),C2: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S2))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_kl(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A2),B2),C2)),S2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,B2,A2)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,C2),aa(set(B),set(B),minus_minus(set(B),S2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A2),bot_bot(set(B)))))) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),S2))
             => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_kl(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),A2),B2),C2)),S2) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,C2),aa(set(B),set(B),minus_minus(set(B),S2),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A2),bot_bot(set(B))))) ) ) ) ) ) ).

% sum.delta_remove
tff(fact_4837_simp__from__to,axiom,
    ! [J: int,I2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2))
       => ( set_or1337092689740270186AtMost(int,I2,J) = bot_bot(set(int)) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2))
       => ( set_or1337092689740270186AtMost(int,I2,J) = aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),I2),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)) ) ) ) ).

% simp_from_to
tff(fact_4838_member__le__sum,axiom,
    ! [B: $tType,C: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I2: C,A3: set(C),F3: fun(C,B)] :
          ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),I2),A3))
         => ( ! [X3: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X3),aa(set(C),set(C),minus_minus(set(C),A3),aa(set(C),set(C),aa(C,fun(set(C),set(C)),insert(C),I2),bot_bot(set(C))))))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(C,B,F3,X3))) )
           => ( pp(aa(set(C),bool,finite_finite(C),A3))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(C,B,F3,I2)),aa(set(C),B,groups7311177749621191930dd_sum(C,B,F3),A3))) ) ) ) ) ).

% member_le_sum
tff(fact_4839_sum__bounded__above__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_field(A)
     => ! [A3: set(B),F3: fun(B,A),K5: A] :
          ( ! [I4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I4)),divide_divide(A,K5,aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,finite_card(B),A3))))) )
         => ( pp(aa(set(B),bool,finite_finite(B),A3))
           => ( ( A3 != bot_bot(set(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A3)),K5)) ) ) ) ) ).

% sum_bounded_above_divide
tff(fact_4840_prod__diff1,axiom,
    ! [A: $tType,B: $tType] :
      ( semidom_divide(A)
     => ! [A3: set(B),F3: fun(B,A),A2: B] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ( aa(B,A,F3,A2) != zero_zero(A) )
           => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),A3))
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),aa(set(B),set(B),minus_minus(set(B),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A2),bot_bot(set(B))))) = divide_divide(A,aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A3),aa(B,A,F3,A2)) ) )
              & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),A3))
               => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),aa(set(B),set(B),minus_minus(set(B),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),A2),bot_bot(set(B))))) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A3) ) ) ) ) ) ) ).

% prod_diff1
tff(fact_4841_sinh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sinh(A,X) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),exp(A,X)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),one_one(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(A,A,uminus_uminus(A),one_one(A))),bot_bot(set(A)))))) ) ) ).

% sinh_zero_iff
tff(fact_4842_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Y))
     => ( bezw(X,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,X,Y)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Y))))) ) ) ).

% bezw_non_0
tff(fact_4843_bezw_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero(nat) )
       => ( bezw(X,Y) = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)) ) )
      & ( ( Y != zero_zero(nat) )
       => ( bezw(X,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,X,Y)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,modulo_modulo(nat,X,Y)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Y))))) ) ) ) ).

% bezw.simps
tff(fact_4844_bezw_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod(int,int)] :
      ( ( bezw(X,Xa) = Y )
     => ( pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),bezw_rel),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa)))
       => ~ ( ( ( ( Xa = zero_zero(nat) )
               => ( Y = aa(int,product_prod(int,int),product_Pair(int,int,one_one(int)),zero_zero(int)) ) )
              & ( ( Xa != zero_zero(nat) )
               => ( Y = aa(int,product_prod(int,int),product_Pair(int,int,aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xa,modulo_modulo(nat,X,Xa)))),aa(nat,int,semiring_1_of_nat(int),divide_divide(nat,X,Xa))))) ) ) )
           => ~ pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),bezw_rel),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa))) ) ) ) ).

% bezw.pelims
tff(fact_4845_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),product_Pair(int,int,K),L)))
     => ( ( ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
            & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),L),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L))))) ) )
        & ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
              & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),L),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
         => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),K)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,K,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) ) ) ).

% and_int.psimps
tff(fact_4846_and__int_Opelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),X),Xa) = Y )
     => ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),product_Pair(int,int,X),Xa)))
       => ~ ( ( ( ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
                  & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
               => ( Y = aa(int,int,uminus_uminus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),X)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),Xa))))) ) )
              & ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
                    & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Xa),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
               => ( Y = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(bool,int,zero_neq_one_of_bool(int),fconj(aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),X)),aa(bool,bool,fNot,dvd_dvd(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)),Xa))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),divide_divide(int,X,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,Xa,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))))) ) ) )
           => ~ pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),product_Pair(int,int,X),Xa))) ) ) ) ).

% and_int.pelims
tff(fact_4847_bit__0__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( bit_se5641148757651400278ts_bit(A,zero_zero(A)) = bot_bot(fun(nat,bool)) ) ) ).

% bit_0_eq
tff(fact_4848_lessThan__0,axiom,
    aa(nat,set(nat),set_ord_lessThan(nat),zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_4849_atLeastLessThan__singleton,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,M)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),M),bot_bot(set(nat))) ).

% atLeastLessThan_singleton
tff(fact_4850_atMost__0,axiom,
    aa(nat,set(nat),set_ord_atMost(nat),zero_zero(nat)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat))) ).

% atMost_0
tff(fact_4851_bot__enat__def,axiom,
    bot_bot(extended_enat) = zero_zero(extended_enat) ).

% bot_enat_def
tff(fact_4852_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_4853_atLeastLessThan0,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,zero_zero(nat)) = bot_bot(set(nat)) ).

% atLeastLessThan0
tff(fact_4854_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( aa(nat,set(nat),set_ord_lessThan(nat),N) = bot_bot(set(nat)) )
    <=> ( N = zero_zero(nat) ) ) ).

% lessThan_empty_iff
tff(fact_4855_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),N),set_or7035219750837199246ssThan(nat,M,N)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
       => ( set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N)) = bot_bot(set(nat)) ) ) ) ).

% atLeastLessThanSuc
tff(fact_4856_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(set(nat),set(nat),minus_minus(set(nat),aa(nat,set(nat),set_ord_lessThan(nat),N)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_4857_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(set(nat),set(nat),minus_minus(set(nat),aa(nat,set(nat),set_ord_atMost(nat),N)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_4858_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),pred_numeral(K)))
       => ( set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),pred_numeral(K)),set_or7035219750837199246ssThan(nat,M,pred_numeral(K))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),pred_numeral(K)))
       => ( set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = bot_bot(set(nat)) ) ) ) ).

% atLeastLessThan_nat_numeral
tff(fact_4859_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,bool))] :
      ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),product_Pair(int,int,A0),A1)))
     => ( ! [K2: int,L4: int] :
            ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),product_Pair(int,int,K2),L4)))
           => ( ( ~ ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),K2),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))))
                    & pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),L4),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),zero_zero(int)),aa(set(int),set(int),aa(int,fun(set(int),set(int)),insert(int),aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))) )
               => pp(aa(int,bool,aa(int,fun(int,bool),P,divide_divide(int,K2,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2)))),divide_divide(int,L4,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))))) )
             => pp(aa(int,bool,aa(int,fun(int,bool),P,K2),L4)) ) )
       => pp(aa(int,bool,aa(int,fun(int,bool),P,A0),A1)) ) ) ).

% and_int.pinduct
tff(fact_4860_normalize__def,axiom,
    ! [P2: product_prod(int,int)] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P2)))
       => ( normalize(P2) = aa(int,product_prod(int,int),product_Pair(int,int,divide_divide(int,aa(product_prod(int,int),int,product_fst(int,int),P2),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),aa(product_prod(int,int),int,product_snd(int,int),P2)))),divide_divide(int,aa(product_prod(int,int),int,product_snd(int,int),P2),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),aa(product_prod(int,int),int,product_snd(int,int),P2)))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P2)))
       => ( ( ( aa(product_prod(int,int),int,product_snd(int,int),P2) = zero_zero(int) )
           => ( normalize(P2) = aa(int,product_prod(int,int),product_Pair(int,int,zero_zero(int)),one_one(int)) ) )
          & ( ( aa(product_prod(int,int),int,product_snd(int,int),P2) != zero_zero(int) )
           => ( normalize(P2) = aa(int,product_prod(int,int),product_Pair(int,int,divide_divide(int,aa(product_prod(int,int),int,product_fst(int,int),P2),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),P2)),aa(product_prod(int,int),int,product_snd(int,int),P2))))),divide_divide(int,aa(product_prod(int,int),int,product_snd(int,int),P2),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),P2)),aa(product_prod(int,int),int,product_snd(int,int),P2))))) ) ) ) ) ) ).

% normalize_def
tff(fact_4861_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F5: set(A),I5: set(A),F3: fun(A,B),I2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),F5))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_km(set(A),fun(fun(A,B),fun(A,bool)),I5),F3))),F5))
           => ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I5))
               => ( groups1027152243600224163dd_sum(A,B,F3,aa(set(A),set(A),minus_minus(set(A),I5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I2),bot_bot(set(A))))) = aa(B,B,minus_minus(B,groups1027152243600224163dd_sum(A,B,F3,I5)),aa(A,B,F3,I2)) ) )
              & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),I5))
               => ( groups1027152243600224163dd_sum(A,B,F3,aa(set(A),set(A),minus_minus(set(A),I5),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),I2),bot_bot(set(A))))) = groups1027152243600224163dd_sum(A,B,F3,I5) ) ) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_4862_distinct__union,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( distinct(A,union(A,Xs,Ys))
    <=> distinct(A,Ys) ) ).

% distinct_union
tff(fact_4863_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_4864_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_4865_gcd__add1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,N: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),M),N)),N) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N) ) ).

% gcd_add1
tff(fact_4866_gcd__add2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,N: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),aa(A,A,aa(A,fun(A,A),plus_plus(A),M),N)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N) ) ).

% gcd_add2
tff(fact_4867_gcd__exp,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A2: A,N: nat,B2: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(nat,A,power_power(A,A2),N)),aa(nat,A,power_power(A,B2),N)) = aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),B2)),N) ) ).

% gcd_exp
tff(fact_4868_gcd__neg__numeral__2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A,N: 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),N))) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),A2),aa(num,A,numeral_numeral(A),N)) ) ).

% gcd_neg_numeral_2
tff(fact_4869_gcd__neg__numeral__1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [N: 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),N))),A2) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),aa(num,A,numeral_numeral(A),N)),A2) ) ).

% gcd_neg_numeral_1
tff(fact_4870_is__unit__gcd__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( pp(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_4871_gcd__pos__int,axiom,
    ! [M: int,N: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),M),N)))
    <=> ( ( M != zero_zero(int) )
        | ( N != zero_zero(int) ) ) ) ).

% gcd_pos_int
tff(fact_4872_sum_Oinsert_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),P2: fun(B,A),I2: B] :
          ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_iy(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I5))
             => ( groups1027152243600224163dd_sum(B,A,P2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),I2),I5)) = groups1027152243600224163dd_sum(B,A,P2,I5) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I5))
             => ( groups1027152243600224163dd_sum(B,A,P2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),I2),I5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,P2,I2)),groups1027152243600224163dd_sum(B,A,P2,I5)) ) ) ) ) ) ).

% sum.insert'
tff(fact_4873_gcd__red__int,axiom,
    ! [X: int,Y: int] : aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Y),modulo_modulo(int,X,Y)) ).

% gcd_red_int
tff(fact_4874_gcd__add__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,K: A,N: A] : aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),M)),N)) = aa(A,A,aa(A,fun(A,A),gcd_gcd(A),M),N) ) ).

% gcd_add_mult
tff(fact_4875_gcd__ge__0__int,axiom,
    ! [X: int,Y: int] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y))) ).

% gcd_ge_0_int
tff(fact_4876_gcd__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(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_4877_gcd__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(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_4878_gcd__div__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(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_4879_gcd__div__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(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_4880_sum_Odistrib__triv_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),I5))
         => ( groups1027152243600224163dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fb(fun(B,A),fun(fun(B,A),fun(B,A)),G),H),I5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups1027152243600224163dd_sum(B,A,G,I5)),groups1027152243600224163dd_sum(B,A,H,I5)) ) ) ) ).

% sum.distrib_triv'
tff(fact_4881_gcd__le1__int,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A2),B2)),A2)) ) ).

% gcd_le1_int
tff(fact_4882_gcd__le2__int,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A2),B2)),B2)) ) ).

% gcd_le2_int
tff(fact_4883_gcd__cases__int,axiom,
    ! [X: int,Y: int,P: fun(int,bool)] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
       => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
         => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y))) ) )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),X))
         => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y),zero_zero(int)))
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),aa(int,int,uminus_uminus(int),Y)))) ) )
       => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),zero_zero(int)))
           => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),Y))
             => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),Y))) ) )
         => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),zero_zero(int)))
             => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Y),zero_zero(int)))
               => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),aa(int,int,uminus_uminus(int),X)),aa(int,int,uminus_uminus(int),Y)))) ) )
           => pp(aa(int,bool,P,aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y))) ) ) ) ) ).

% gcd_cases_int
tff(fact_4884_gcd__unique__int,axiom,
    ! [D2: int,A2: int,B2: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),D2))
        & pp(dvd_dvd(int,D2,A2))
        & pp(dvd_dvd(int,D2,B2))
        & ! [E3: int] :
            ( ( pp(dvd_dvd(int,E3,A2))
              & pp(dvd_dvd(int,E3,B2)) )
           => pp(dvd_dvd(int,E3,D2)) ) )
    <=> ( D2 = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),A2),B2) ) ) ).

% gcd_unique_int
tff(fact_4885_gcd__non__0__int,axiom,
    ! [Y: int,X: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),Y))
     => ( aa(int,int,aa(int,fun(int,int),gcd_gcd(int),X),Y) = aa(int,int,aa(int,fun(int,int),gcd_gcd(int),Y),modulo_modulo(int,X,Y)) ) ) ).

% gcd_non_0_int
tff(fact_4886_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),if(int,aa(int,bool,aa(int,fun(int,bool),fequal(int),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_4887_sum_Omono__neutral__cong__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),T6: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T6))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T6),S2)))
               => ( aa(B,A,G,X3) = zero_zero(A) ) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S2))
                 => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
             => ( groups1027152243600224163dd_sum(B,A,G,T6) = groups1027152243600224163dd_sum(B,A,H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_4888_sum_Omono__neutral__cong__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),T6: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T6))
         => ( ! [I4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),aa(set(B),set(B),minus_minus(set(B),T6),S2)))
               => ( aa(B,A,H,I4) = zero_zero(A) ) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S2))
                 => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
             => ( groups1027152243600224163dd_sum(B,A,G,S2) = groups1027152243600224163dd_sum(B,A,H,T6) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_4889_sum_Omono__neutral__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),T6: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T6))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T6),S2)))
               => ( aa(B,A,G,X3) = zero_zero(A) ) )
           => ( groups1027152243600224163dd_sum(B,A,G,T6) = groups1027152243600224163dd_sum(B,A,G,S2) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_4890_sum_Omono__neutral__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),T6: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T6))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T6),S2)))
               => ( aa(B,A,G,X3) = zero_zero(A) ) )
           => ( groups1027152243600224163dd_sum(B,A,G,S2) = groups1027152243600224163dd_sum(B,A,G,T6) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_4891_sum_Odistrib_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [I5: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_iy(set(B),fun(fun(B,A),fun(B,bool)),I5),G))))
         => ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_iy(set(B),fun(fun(B,A),fun(B,bool)),I5),H))))
           => ( groups1027152243600224163dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fb(fun(B,A),fun(fun(B,A),fun(B,A)),G),H),I5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups1027152243600224163dd_sum(B,A,G,I5)),groups1027152243600224163dd_sum(B,A,H,I5)) ) ) ) ) ).

% sum.distrib'
tff(fact_4892_set__update__distinct,axiom,
    ! [A: $tType,Xs: list(A),N: nat,X: A] :
      ( distinct(A,Xs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(list(A),set(A),set2(A),list_update(A,Xs,N,X)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(nat,A,nth(A,Xs),N)),bot_bot(set(A))))) ) ) ) ).

% set_update_distinct
tff(fact_4893_card__Pow,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( aa(set(set(A)),nat,finite_card(set(A)),pow2(A,A3)) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(set(A),nat,finite_card(A),A3)) ) ) ).

% card_Pow
tff(fact_4894_size__prod__simp,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,nat),G: fun(B,nat),P2: product_prod(A,B)] : basic_BNF_size_prod(A,B,F3,G,P2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F3,aa(product_prod(A,B),A,product_fst(A,B),P2))),aa(B,nat,G,aa(product_prod(A,B),B,product_snd(A,B),P2)))),aa(nat,nat,suc,zero_zero(nat))) ).

% size_prod_simp
tff(fact_4895_gcd__1__nat,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),one_one(nat)) = one_one(nat) ).

% gcd_1_nat
tff(fact_4896_list__update__overwrite,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,X: A,Y: A] : list_update(A,list_update(A,Xs,I2,X),I2,Y) = list_update(A,Xs,I2,Y) ).

% list_update_overwrite
tff(fact_4897_gcd__Suc__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,suc,zero_zero(nat)) ).

% gcd_Suc_0
tff(fact_4898_gcd__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N)))
    <=> ( ( M != zero_zero(nat) )
        | ( N != zero_zero(nat) ) ) ) ).

% gcd_pos_nat
tff(fact_4899_PowI,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),A3),pow2(A,B3))) ) ).

% PowI
tff(fact_4900_Pow__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),A3),pow2(A,B3)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ).

% Pow_iff
tff(fact_4901_length__list__update,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,X: A] : aa(list(A),nat,size_size(list(A)),list_update(A,Xs,I2,X)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_list_update
tff(fact_4902_list__update__id,axiom,
    ! [A: $tType,Xs: list(A),I2: nat] : list_update(A,Xs,I2,aa(nat,A,nth(A,Xs),I2)) = Xs ).

% list_update_id
tff(fact_4903_nth__list__update__neq,axiom,
    ! [A: $tType,I2: nat,J: nat,Xs: list(A),X: A] :
      ( ( I2 != J )
     => ( aa(nat,A,nth(A,list_update(A,Xs,I2,X)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_4904_list__update__beyond,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I2))
     => ( list_update(A,Xs,I2,X) = Xs ) ) ).

% list_update_beyond
tff(fact_4905_nth__list__update__eq,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I2,X)),I2) = X ) ) ).

% nth_list_update_eq
tff(fact_4906_set__swap,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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,I2,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I2))) = aa(list(A),set(A),set2(A),Xs) ) ) ) ).

% set_swap
tff(fact_4907_distinct__swap,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( distinct(A,list_update(A,list_update(A,Xs,I2,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I2)))
        <=> distinct(A,Xs) ) ) ) ).

% distinct_swap
tff(fact_4908_gcd__red__nat,axiom,
    ! [X: nat,Y: nat] : aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y),modulo_modulo(nat,X,Y)) ).

% gcd_red_nat
tff(fact_4909_list__update__swap,axiom,
    ! [A: $tType,I2: nat,I7: nat,Xs: list(A),X: A,X5: A] :
      ( ( I2 != I7 )
     => ( list_update(A,list_update(A,Xs,I2,X),I7,X5) = list_update(A,list_update(A,Xs,I7,X5),I2,X) ) ) ).

% list_update_swap
tff(fact_4910_Pow__def,axiom,
    ! [A: $tType,A3: set(A)] : pow2(A,A3) = collect(set(A),aTP_Lamp_jp(set(A),fun(set(A),bool),A3)) ).

% Pow_def
tff(fact_4911_PowD,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),A3),pow2(A,B3)))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ).

% PowD
tff(fact_4912_Pow__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),pow2(A,A3)),pow2(A,B3))) ) ).

% Pow_mono
tff(fact_4913_gcd__le1__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),A2)) ) ).

% gcd_le1_nat
tff(fact_4914_gcd__le2__nat,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero(nat) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)),B2)) ) ).

% gcd_le2_nat
tff(fact_4915_gcd__diff1__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,minus_minus(nat,M),N)),N) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N) ) ) ).

% gcd_diff1_nat
tff(fact_4916_gcd__diff2__nat,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),aa(nat,nat,minus_minus(nat,N),M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N) ) ) ).

% gcd_diff2_nat
tff(fact_4917_gcd__non__0__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero(nat) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y),modulo_modulo(nat,X,Y)) ) ) ).

% gcd_non_0_nat
tff(fact_4918_gcd__nat_Osimps,axiom,
    ! [Y: nat,X: nat] :
      ( ( ( Y = zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = X ) )
      & ( ( Y != zero_zero(nat) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Y) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Y),modulo_modulo(nat,X,Y)) ) ) ) ).

% gcd_nat.simps
tff(fact_4919_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Xa) = Y )
     => ( ( ( Xa = zero_zero(nat) )
         => ( Y = X ) )
        & ( ( Xa != zero_zero(nat) )
         => ( Y = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),modulo_modulo(nat,X,Xa)) ) ) ) ) ).

% gcd_nat.elims
tff(fact_4920_set__update__subsetI,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A),X: A,I2: nat] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A3))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I2,X))),A3)) ) ) ).

% set_update_subsetI
tff(fact_4921_bezout__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [X3: nat,Y4: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)),aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2)) ) ).

% bezout_nat
tff(fact_4922_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A2: nat] :
    ? [X3: nat,Y4: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3)))
        & ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y4)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) )
      | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3)))
        & ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y4)) = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),A2),B2) ) ) ) ).

% bezout_gcd_nat'
tff(fact_4923_set__update__subset__insert,axiom,
    ! [A: $tType,Xs: list(A),I2: nat,X: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I2,X))),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(list(A),set(A),set2(A),Xs)))) ).

% set_update_subset_insert
tff(fact_4924_set__update__memI,axiom,
    ! [A: $tType,N: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),list_update(A,Xs,N,X)))) ) ).

% set_update_memI
tff(fact_4925_nth__list__update,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),J: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( ( ( I2 = J )
         => ( aa(nat,A,nth(A,list_update(A,Xs,I2,X)),J) = X ) )
        & ( ( I2 != J )
         => ( aa(nat,A,nth(A,list_update(A,Xs,I2,X)),J) = aa(nat,A,nth(A,Xs),J) ) ) ) ) ).

% nth_list_update
tff(fact_4926_list__update__same__conv,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( ( list_update(A,Xs,I2,X) = Xs )
      <=> ( aa(nat,A,nth(A,Xs),I2) = X ) ) ) ).

% list_update_same_conv
tff(fact_4927_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),if(code_integer,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),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_4928_less__by__empty,axiom,
    ! [A: $tType,A3: set(product_prod(A,A)),B3: set(product_prod(A,A))] :
      ( ( A3 = bot_bot(set(product_prod(A,A))) )
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),A3),B3)) ) ).

% less_by_empty
tff(fact_4929_distinct__list__update,axiom,
    ! [A: $tType,Xs: list(A),A2: A,I2: nat] :
      ( distinct(A,Xs)
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(nat,A,nth(A,Xs),I2)),bot_bot(set(A))))))
       => distinct(A,list_update(A,Xs,I2,A2)) ) ) ).

% distinct_list_update
tff(fact_4930_nat__descend__induct,axiom,
    ! [N: nat,P: fun(nat,bool),M: nat] :
      ( ! [K2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),K2))
         => pp(aa(nat,bool,P,K2)) )
     => ( ! [K2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K2),N))
           => ( ! [I: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K2),I))
                 => pp(aa(nat,bool,P,I)) )
             => pp(aa(nat,bool,P,K2)) ) )
       => pp(aa(nat,bool,P,M)) ) ) ).

% nat_descend_induct
tff(fact_4931_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),X),Xa) = Y )
     => ( pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),gcd_nat_rel),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa)))
       => ~ ( ( ( ( Xa = zero_zero(nat) )
               => ( Y = X ) )
              & ( ( Xa != zero_zero(nat) )
               => ( Y = aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),Xa),modulo_modulo(nat,X,Xa)) ) ) )
           => ~ pp(aa(product_prod(nat,nat),bool,accp(product_prod(nat,nat),gcd_nat_rel),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),Xa))) ) ) ) ).

% gcd_nat.pelims
tff(fact_4932_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_4933_finite__enumerate,axiom,
    ! [S2: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),S2))
     => ? [R2: fun(nat,nat)] :
          ( strict_mono_on(nat,nat,R2,aa(nat,set(nat),set_ord_lessThan(nat),aa(set(nat),nat,finite_card(nat),S2)))
          & ! [N7: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N7),aa(set(nat),nat,finite_card(nat),S2)))
             => pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,R2,N7)),S2)) ) ) ) ).

% finite_enumerate
tff(fact_4934_drop__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).

% drop_bit_of_0
tff(fact_4935_drop__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,M),aa(A,A,bit_se4197421643247451524op_bit(A,N),A2)) = aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),A2) ) ).

% drop_bit_drop_bit
tff(fact_4936_drop__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_se4197421643247451524op_bit(A,N),A2)),aa(A,A,bit_se4197421643247451524op_bit(A,N),B2)) ) ).

% drop_bit_and
tff(fact_4937_drop__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se4197421643247451524op_bit(A,N),A2)),aa(A,A,bit_se4197421643247451524op_bit(A,N),B2)) ) ).

% drop_bit_or
tff(fact_4938_drop__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A,B2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_se4197421643247451524op_bit(A,N),A2)),aa(A,A,bit_se4197421643247451524op_bit(A,N),B2)) ) ).

% drop_bit_xor
tff(fact_4939_drop__bit__of__bool,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,B2: bool] : aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(bool,A,zero_neq_one_of_bool(A),B2)) = aa(bool,A,zero_neq_one_of_bool(A),fconj(aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat)),B2)) ) ).

% drop_bit_of_bool
tff(fact_4940_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se4197421643247451524op_bit(int,N),K)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),K)) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_4941_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),aa(int,int,bit_se4197421643247451524op_bit(int,N),K)),zero_zero(int)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),K),zero_zero(int))) ) ).

% drop_bit_negative_int_iff
tff(fact_4942_drop__bit__minus__one,axiom,
    ! [N: nat] : aa(int,int,bit_se4197421643247451524op_bit(int,N),aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% drop_bit_minus_one
tff(fact_4943_drop__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) = aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_Suc_bit0
tff(fact_4944_drop__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_Suc_bit1
tff(fact_4945_drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,N),one_one(A)) = aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ) ).

% drop_bit_of_1
tff(fact_4946_drop__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit0,K))) = aa(A,A,bit_se4197421643247451524op_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_numeral_bit0
tff(fact_4947_drop__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,bit_se4197421643247451524op_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_numeral_bit1
tff(fact_4948_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,K)))) = aa(int,int,bit_se4197421643247451524op_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_4949_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),aa(num,num,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_4950_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_se4197421643247451524op_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,bit_se4197421643247451524op_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_4951_drop__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,M: nat] : aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se4197421643247451524op_bit(nat,N),M)) ) ).

% drop_bit_of_nat
tff(fact_4952_of__nat__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se4197421643247451524op_bit(nat,M),N)) = aa(A,A,bit_se4197421643247451524op_bit(A,M),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_drop_bit
tff(fact_4953_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) = A2 )
        <=> ( aa(A,A,bit_se4197421643247451524op_bit(A,N),A2) = zero_zero(A) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_4954_drop__bit__push__bit__int,axiom,
    ! [M: nat,N: nat,K: int] : aa(int,int,bit_se4197421643247451524op_bit(int,M),aa(int,int,bit_se4730199178511100633sh_bit(int,N),K)) = aa(int,int,bit_se4197421643247451524op_bit(int,aa(nat,nat,minus_minus(nat,M),N)),aa(int,int,bit_se4730199178511100633sh_bit(int,aa(nat,nat,minus_minus(nat,N),M)),K)) ).

% drop_bit_push_bit_int
tff(fact_4955_take__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_se4197421643247451524op_bit(A,N),A2)) = aa(A,A,bit_se4197421643247451524op_bit(A,N),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),A2)) ) ).

% take_bit_drop_bit
tff(fact_4956_drop__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,minus_minus(nat,N),M)),aa(A,A,bit_se4197421643247451524op_bit(A,M),A2)) ) ).

% drop_bit_take_bit
tff(fact_4957_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] : divide_divide(A,A2,aa(A,A,bit_se4730199178511100633sh_bit(A,N),one_one(A))) = aa(A,A,bit_se4197421643247451524op_bit(A,N),A2) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_4958_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_se4197421643247451524op_bit(A,N),A2)),one_one(A)) = one_one(A) ) ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_4959_bits__ident,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,bit_se4197421643247451524op_bit(A,N),A2))),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) = A2 ) ).

% bits_ident
tff(fact_4960_stable__imp__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( ( divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) = A2 )
         => ( aa(A,A,bit_se4197421643247451524op_bit(A,N),A2) = A2 ) ) ) ).

% stable_imp_drop_bit_eq
tff(fact_4961_drop__bit__half,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,N),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) = divide_divide(A,aa(A,A,bit_se4197421643247451524op_bit(A,N),A2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))) ) ).

% drop_bit_half
tff(fact_4962_drop__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_se4197421643247451524op_bit(int,N),K) = divide_divide(int,K,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),N)) ).

% drop_bit_int_def
tff(fact_4963_drop__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,suc,N)),A2) = aa(A,A,bit_se4197421643247451524op_bit(A,N),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ).

% drop_bit_Suc
tff(fact_4964_drop__bit__eq__div,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : aa(A,A,bit_se4197421643247451524op_bit(A,N),A2) = divide_divide(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) ) ).

% drop_bit_eq_div
tff(fact_4965_even__drop__bit__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,bit_se4197421643247451524op_bit(A,N),A2)))
        <=> ~ pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N)) ) ) ).

% even_drop_bit_iff_not_bit
tff(fact_4966_bit__iff__odd__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(nat,bool,bit_se5641148757651400278ts_bit(A,A2),N))
        <=> ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(A,A,bit_se4197421643247451524op_bit(A,N),A2))) ) ) ).

% bit_iff_odd_drop_bit
tff(fact_4967_slice__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,M: nat,A2: A] : aa(A,A,bit_se4730199178511100633sh_bit(A,N),aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_se4197421643247451524op_bit(A,N),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),M),N))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)))) ) ).

% slice_eq_mask
tff(fact_4968_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(A,A,bit_se4197421643247451524op_bit(A,N),A2) = A2 ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(A,A,bit_se4197421643247451524op_bit(A,N),A2) = aa(A,A,bit_se4197421643247451524op_bit(A,aa(nat,nat,minus_minus(nat,N),one_one(nat))),divide_divide(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))) ) ) ) ) ).

% drop_bit_rec
tff(fact_4969_strict__mono__on__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F3: fun(A,B),A3: set(A)] :
          ( strict_mono_on(A,B,F3,A3)
        <=> ! [R5: A,S6: A] :
              ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),R5),A3))
                & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),S6),A3))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),R5),S6)) )
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,R5)),aa(A,B,F3,S6))) ) ) ) ).

% strict_mono_on_def
tff(fact_4970_strict__mono__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [A3: set(A),F3: fun(A,B)] :
          ( ! [R2: A,S3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),R2),A3))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),S3),A3))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),R2),S3))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,R2)),aa(A,B,F3,S3))) ) ) )
         => strict_mono_on(A,B,F3,A3) ) ) ).

% strict_mono_onI
tff(fact_4971_strict__mono__onD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & ord(B) )
     => ! [F3: fun(A,B),A3: set(A),R3: A,S: A] :
          ( strict_mono_on(A,B,F3,A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),R3),A3))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),S),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),R3),S))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,R3)),aa(A,B,F3,S))) ) ) ) ) ) ).

% strict_mono_onD
tff(fact_4972_drop__bit__of__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,bit_se4197421643247451524op_bit(nat,N),aa(nat,nat,suc,zero_zero(nat))) = aa(bool,nat,zero_neq_one_of_bool(nat),aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),N),zero_zero(nat))) ).

% drop_bit_of_Suc_0
tff(fact_4973_drop__bit__nat__eq,axiom,
    ! [N: nat,K: int] : aa(nat,nat,bit_se4197421643247451524op_bit(nat,N),nat2(K)) = nat2(aa(int,int,bit_se4197421643247451524op_bit(int,N),K)) ).

% drop_bit_nat_eq
tff(fact_4974_drop__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,bit_se4197421643247451524op_bit(nat,N),M) = divide_divide(nat,M,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)) ).

% drop_bit_nat_def
tff(fact_4975_strict__mono__on__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & preorder(B) )
     => ! [F3: fun(A,B),A3: set(A),X: A,Y: A] :
          ( strict_mono_on(A,B,F3,A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),A3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,Y))) ) ) ) ) ) ).

% strict_mono_on_leD
tff(fact_4976_nth__rotate1,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),rotate1(A),Xs)),N) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,suc,N),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate1
tff(fact_4977_set__remove1__eq,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( aa(list(A),set(A),set2(A),remove1(A,X,Xs)) = aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ) ) ).

% set_remove1_eq
tff(fact_4978_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or5935395276787703475ssThan(int,L,U)) = nat2(aa(int,int,minus_minus(int,U),aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)))) ).

% card_greaterThanLessThan_int
tff(fact_4979_greaterThanLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),set_or5935395276787703475ssThan(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),I2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),I2),U)) ) ) ) ).

% greaterThanLessThan_iff
tff(fact_4980_in__set__remove1,axiom,
    ! [A: $tType,A2: A,B2: A,Xs: list(A)] :
      ( ( A2 != B2 )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(list(A),set(A),set2(A),remove1(A,B2,Xs))))
      <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% in_set_remove1
tff(fact_4981_set__rotate1,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),rotate1(A),Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% set_rotate1
tff(fact_4982_length__rotate1,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),rotate1(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_rotate1
tff(fact_4983_distinct1__rotate,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),rotate1(A),Xs))
    <=> distinct(A,Xs) ) ).

% distinct1_rotate
tff(fact_4984_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : pp(aa(set(int),bool,finite_finite(int),set_or5935395276787703475ssThan(int,L,U))) ).

% finite_greaterThanLessThan_int
tff(fact_4985_greaterThanLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),K))
         => ( set_or5935395276787703475ssThan(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanLessThan_empty
tff(fact_4986_greaterThanLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or5935395276787703475ssThan(A,A2,B2) = bot_bot(set(A)) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% greaterThanLessThan_empty_iff
tff(fact_4987_greaterThanLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A2,B2) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% greaterThanLessThan_empty_iff2
tff(fact_4988_infinite__Ioo__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),set_or5935395276787703475ssThan(A,A2,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% infinite_Ioo_iff
tff(fact_4989_rotate1__length01,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
     => ( aa(list(A),list(A),rotate1(A),Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_4990_distinct__remove1,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => distinct(A,remove1(A,X,Xs)) ) ).

% distinct_remove1
tff(fact_4991_remove1__commute,axiom,
    ! [A: $tType,X: A,Y: A,Zs: list(A)] : remove1(A,X,remove1(A,Y,Zs)) = remove1(A,Y,remove1(A,X,Zs)) ).

% remove1_commute
tff(fact_4992_notin__set__remove1,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),remove1(A,Y,Xs)))) ) ).

% notin_set_remove1
tff(fact_4993_remove1__idem,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( remove1(A,X,Xs) = Xs ) ) ).

% remove1_idem
tff(fact_4994_infinite__Ioo,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ~ pp(aa(set(A),bool,finite_finite(A),set_or5935395276787703475ssThan(A,A2,B2))) ) ) ).

% infinite_Ioo
tff(fact_4995_set__remove1__subset,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),remove1(A,X,Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% set_remove1_subset
tff(fact_4996_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or5935395276787703475ssThan(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_4997_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_4998_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_4999_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_5000_atLeastAtMost__diff__ends,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(set(A),set(A),minus_minus(set(A),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),bot_bot(set(A))))) = set_or5935395276787703475ssThan(A,A2,B2) ) ).

% atLeastAtMost_diff_ends
tff(fact_5001_length__remove1,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(list(A),nat,size_size(list(A)),remove1(A,X,Xs)) = aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(list(A),nat,size_size(list(A)),remove1(A,X,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).

% length_remove1
tff(fact_5002_image2__def,axiom,
    ! [A: $tType,B: $tType,C: $tType,A3: set(C),F3: fun(C,A),G: fun(C,B)] : bNF_Greatest_image2(C,A,B,A3,F3,G) = collect(product_prod(A,B),aa(fun(C,B),fun(product_prod(A,B),bool),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool)),aTP_Lamp_kn(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool))),A3),F3),G)) ).

% image2_def
tff(fact_5003_xor__minus__numerals_I1_J,axiom,
    ! [N: 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),N))),K) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),neg_numeral_sub(int,N,one2)),K)) ).

% xor_minus_numerals(1)
tff(fact_5004_xor__minus__numerals_I2_J,axiom,
    ! [K: int,N: 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),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),neg_numeral_sub(int,N,one2))) ).

% xor_minus_numerals(2)
tff(fact_5005_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : pp(aa(set(nat),bool,finite_finite(nat),set_or5935395276787703475ssThan(nat,L,U))) ).

% finite_greaterThanLessThan
tff(fact_5006_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_5007_diff__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,M,N) ) ).

% diff_numeral_simps(1)
tff(fact_5008_card__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or5935395276787703475ssThan(nat,L,U)) = aa(nat,nat,minus_minus(nat,U),aa(nat,nat,suc,L)) ).

% card_greaterThanLessThan
tff(fact_5009_sub__num__simps_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,aa(num,num,bit0,K),aa(num,num,bit0,L)) = neg_numeral_dbl(A,neg_numeral_sub(A,K,L)) ) ).

% sub_num_simps(6)
tff(fact_5010_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_5011_semiring__norm_I167_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W)),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,W,V)),Y) ) ).

% semiring_norm(167)
tff(fact_5012_semiring__norm_I166_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),neg_numeral_sub(A,V,W)),Y) ) ).

% semiring_norm(166)
tff(fact_5013_add__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,N,M) ) ).

% add_neg_numeral_simps(2)
tff(fact_5014_add__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,M,N) ) ).

% add_neg_numeral_simps(1)
tff(fact_5015_diff__numeral__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,N,M) ) ).

% diff_numeral_simps(4)
tff(fact_5016_sub__num__simps_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,aa(num,num,bit0,K),aa(num,num,bit1,L)) = neg_numeral_dbl_dec(A,neg_numeral_sub(A,K,L)) ) ).

% sub_num_simps(7)
tff(fact_5017_sub__num__simps_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,aa(num,num,bit1,K),aa(num,num,bit0,L)) = neg_numeral_dbl_inc(A,neg_numeral_sub(A,K,L)) ) ).

% sub_num_simps(8)
tff(fact_5018_diff__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),M)),one_one(A)) = neg_numeral_sub(A,M,one2) ) ).

% diff_numeral_special(2)
tff(fact_5019_diff__numeral__special_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,minus_minus(A,one_one(A)),aa(num,A,numeral_numeral(A),N)) = neg_numeral_sub(A,one2,N) ) ).

% diff_numeral_special(1)
tff(fact_5020_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),aa(num,num,bit0,K)) ) ).

% sub_num_simps(5)
tff(fact_5021_not__minus__numeral__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = neg_numeral_sub(A,N,one2) ) ).

% not_minus_numeral_eq
tff(fact_5022_sub__num__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_sub(A,aa(num,num,bit0,K),one2) = aa(num,A,numeral_numeral(A),bitM(K)) ) ).

% sub_num_simps(4)
tff(fact_5023_add__neg__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: 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),N)) = neg_numeral_sub(A,N,one2) ) ).

% add_neg_numeral_special(4)
tff(fact_5024_add__neg__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,M,one2) ) ).

% add_neg_numeral_special(3)
tff(fact_5025_add__neg__numeral__special_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) = neg_numeral_sub(A,one2,M) ) ).

% add_neg_numeral_special(2)
tff(fact_5026_add__neg__numeral__special_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: 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),M))) = neg_numeral_sub(A,one2,M) ) ).

% add_neg_numeral_special(1)
tff(fact_5027_diff__numeral__special_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) = neg_numeral_sub(A,one2,M) ) ).

% diff_numeral_special(8)
tff(fact_5028_diff__numeral__special_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(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),N))) = neg_numeral_sub(A,N,one2) ) ).

% diff_numeral_special(7)
tff(fact_5029_minus__sub__one__diff__one,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),neg_numeral_sub(A,M,one2))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) ) ).

% minus_sub_one_diff_one
tff(fact_5030_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),aa(num,num,bit0,L))) ) ).

% sub_num_simps(3)
tff(fact_5031_sub__num__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [L: num] : neg_numeral_sub(A,one2,aa(num,num,bit0,L)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bitM(L))) ) ).

% sub_num_simps(2)
tff(fact_5032_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_5033_neg__numeral__class_Osub__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num,L: num] : neg_numeral_sub(A,K,L) = aa(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_5034_tanh__real__bounds,axiom,
    ! [X: real] : pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,tanh(real),X)),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)))) ).

% tanh_real_bounds
tff(fact_5035_sub__non__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),neg_numeral_sub(A,N,M)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),M),N)) ) ) ).

% sub_non_negative
tff(fact_5036_sub__non__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),neg_numeral_sub(A,N,M)),zero_zero(A)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less_eq(num),N),M)) ) ) ).

% sub_non_positive
tff(fact_5037_sub__positive,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),neg_numeral_sub(A,N,M)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),M),N)) ) ) ).

% sub_positive
tff(fact_5038_sub__negative,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,M: num] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),neg_numeral_sub(A,N,M)),zero_zero(A)))
        <=> pp(aa(num,bool,aa(num,fun(num,bool),ord_less(num),N),M)) ) ) ).

% sub_negative
tff(fact_5039_sub__inc__One__eq,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : neg_numeral_sub(A,inc(N),one2) = aa(num,A,numeral_numeral(A),N) ) ).

% sub_inc_One_eq
tff(fact_5040_minus__numeral__eq__not__sub__one,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) = aa(A,A,bit_ri4277139882892585799ns_not(A),neg_numeral_sub(A,N,one2)) ) ).

% minus_numeral_eq_not_sub_one
tff(fact_5041_sub__BitM__One__eq,axiom,
    ! [N: num] : neg_numeral_sub(int,bitM(N),one2) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),neg_numeral_sub(int,N,one2)) ).

% sub_BitM_One_eq
tff(fact_5042_div__add__self1__no__field,axiom,
    ! [B: $tType,A: $tType] :
      ( ( euclid4440199948858584721cancel(A)
        & field(B) )
     => ! [X: B,B2: A,A2: A] :
          ( nO_MATCH(B,A,X,B2)
         => ( ( 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_no_field
tff(fact_5043_div__add__self2__no__field,axiom,
    ! [B: $tType,A: $tType] :
      ( ( euclid4440199948858584721cancel(A)
        & field(B) )
     => ! [X: B,B2: A,A2: A] :
          ( nO_MATCH(B,A,X,B2)
         => ( ( 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_no_field
tff(fact_5044_exE__some,axiom,
    ! [A: $tType,P: fun(A,bool),C2: A] :
      ( ? [X_12: A] : pp(aa(A,bool,P,X_12))
     => ( ( C2 = fChoice(A,P) )
       => pp(aa(A,bool,P,C2)) ) ) ).

% exE_some
tff(fact_5045_scale__right__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,Y: A,A2: real] :
          ( nO_MATCH(A,real,divide_divide(A,X,Y),A2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ) ) ).

% scale_right_distrib_NO_MATCH
tff(fact_5046_scale__right__diff__distrib__NO__MATCH,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,Y: A,A2: real] :
          ( nO_MATCH(A,real,divide_divide(A,X,Y),A2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,minus_minus(A,X),Y)) = aa(A,A,minus_minus(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ) ) ).

% scale_right_diff_distrib_NO_MATCH
tff(fact_5047_distrib__right__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring(A)
     => ! [X: B,Y: B,C2: A,A2: A,B2: A] :
          ( nO_MATCH(B,A,divide_divide(B,X,Y),C2)
         => ( 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_NO_MATCH
tff(fact_5048_distrib__left__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring(A)
     => ! [X: B,Y: B,A2: A,B2: A,C2: A] :
          ( nO_MATCH(B,A,divide_divide(B,X,Y),A2)
         => ( 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_NO_MATCH
tff(fact_5049_right__diff__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( ring(A)
     => ! [X: B,Y: B,A2: A,B2: A,C2: A] :
          ( nO_MATCH(B,A,divide_divide(B,X,Y),A2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,minus_minus(A,B2),C2)) = aa(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_NO_MATCH
tff(fact_5050_left__diff__distrib__NO__MATCH,axiom,
    ! [B: $tType,A: $tType] :
      ( ring(A)
     => ! [X: B,Y: B,C2: A,A2: A,B2: A] :
          ( nO_MATCH(B,A,divide_divide(B,X,Y),C2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),B2)),C2) = aa(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_NO_MATCH
tff(fact_5051_power__minus_H,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,N: nat] :
          ( nO_MATCH(A,A,one_one(A),X)
         => ( aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),X)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,power_power(A,X),N)) ) ) ) ).

% power_minus'
tff(fact_5052_scale__left__distrib__NO__MATCH,axiom,
    ! [C: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,Y: A,C2: C,A2: real,B2: real] :
          ( nO_MATCH(A,C,divide_divide(A,X,Y),C2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,B2),X)) ) ) ) ).

% scale_left_distrib_NO_MATCH
tff(fact_5053_scale__left__diff__distrib__NO__MATCH,axiom,
    ! [C: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [X: A,Y: A,C2: C,A2: real,B2: real] :
          ( nO_MATCH(A,C,divide_divide(A,X,Y),C2)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,A2),B2)),X) = aa(A,A,minus_minus(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),X)),aa(A,A,real_V8093663219630862766scaleR(A,B2),X)) ) ) ) ).

% scale_left_diff_distrib_NO_MATCH
tff(fact_5054_dependent__nat__choice,axiom,
    ! [A: $tType,P: fun(nat,fun(A,bool)),Q: fun(nat,fun(A,fun(A,bool)))] :
      ( ? [X_12: A] : pp(aa(A,bool,aa(nat,fun(A,bool),P,zero_zero(nat)),X_12))
     => ( ! [X3: A,N2: nat] :
            ( pp(aa(A,bool,aa(nat,fun(A,bool),P,N2),X3))
           => ? [Y3: A] :
                ( pp(aa(A,bool,aa(nat,fun(A,bool),P,aa(nat,nat,suc,N2)),Y3))
                & pp(aa(A,bool,aa(A,fun(A,bool),aa(nat,fun(A,fun(A,bool)),Q,N2),X3),Y3)) ) )
       => ? [F4: fun(nat,A)] :
          ! [N7: nat] :
            ( pp(aa(A,bool,aa(nat,fun(A,bool),P,N7),aa(nat,A,F4,N7)))
            & pp(aa(A,bool,aa(A,fun(A,bool),aa(nat,fun(A,fun(A,bool)),Q,N7),aa(nat,A,F4,N7)),aa(nat,A,F4,aa(nat,nat,suc,N7)))) ) ) ) ).

% dependent_nat_choice
tff(fact_5055_bounded__linear__axioms__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( real_V4916620083959148203axioms(A,B,F3)
        <=> ? [K6: real] :
            ! [X4: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X4))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X4)),K6))) ) ) ).

% bounded_linear_axioms_def
tff(fact_5056_bounded__linear__axioms_Ointro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( ? [K8: real] :
            ! [X3: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X3))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X3)),K8)))
         => real_V4916620083959148203axioms(A,B,F3) ) ) ).

% bounded_linear_axioms.intro
tff(fact_5057_horner__sum__eq__sum__funpow,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F3,A2,Xs) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_ko(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F3),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum_funpow
tff(fact_5058_Suc__funpow,axiom,
    ! [N: nat] : aa(fun(nat,nat),fun(nat,nat),aa(nat,fun(fun(nat,nat),fun(nat,nat)),compow(fun(nat,nat)),N),suc) = aa(nat,fun(nat,nat),plus_plus(nat),N) ).

% Suc_funpow
tff(fact_5059_funpow__0,axiom,
    ! [A: $tType,F3: fun(A,A),X: A] : aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),zero_zero(nat)),F3),X) = X ).

% funpow_0
tff(fact_5060_bij__betw__funpow,axiom,
    ! [A: $tType,F3: fun(A,A),S2: set(A),N: nat] :
      ( bij_betw(A,A,F3,S2,S2)
     => bij_betw(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),S2,S2) ) ).

% bij_betw_funpow
tff(fact_5061_funpow__mod__eq,axiom,
    ! [A: $tType,N: nat,F3: fun(A,A),X: A,M: nat] :
      ( ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),X) = X )
     => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),modulo_modulo(nat,M,N)),F3),X) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F3),X) ) ) ).

% funpow_mod_eq
tff(fact_5062_funpow__swap1,axiom,
    ! [A: $tType,F3: fun(A,A),N: nat,X: A] : aa(A,A,F3,aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),X)) = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),aa(A,A,F3,X)) ).

% funpow_swap1
tff(fact_5063_funpow__mult,axiom,
    ! [A: $tType,N: nat,M: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F3)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),F3) ).

% funpow_mult
tff(fact_5064_funpow__times__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [F3: fun(A,nat),X: A] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(A,nat,F3,X)),aa(A,fun(A,A),times_times(A),X)) = aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,X),aa(A,nat,F3,X))) ) ).

% funpow_times_power
tff(fact_5065_of__nat__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),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),aa(A,fun(A,A),plus_plus(A),one_one(A))),zero_zero(A)) ) ).

% of_nat_def
tff(fact_5066_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_5067_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_5068_relpowp__fun__conv,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Y: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Y))
    <=> ? [F6: fun(nat,A)] :
          ( ( aa(nat,A,F6,zero_zero(nat)) = X )
          & ( aa(nat,A,F6,N) = Y )
          & ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
             => pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,F6,I3)),aa(nat,A,F6,aa(nat,nat,suc,I3)))) ) ) ) ).

% relpowp_fun_conv
tff(fact_5069_relpowp__1,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool))] : aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),one_one(nat)),P) = P ).

% relpowp_1
tff(fact_5070_Nat_Ofunpow__code__def,axiom,
    ! [A: $tType] : funpow(A) = compow(fun(A,A)) ).

% Nat.funpow_code_def
tff(fact_5071_relpowp__Suc__E,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Z: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),P),X),Z))
     => ~ ! [Y4: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Y4))
           => ~ pp(aa(A,bool,aa(A,fun(A,bool),P,Y4),Z)) ) ) ).

% relpowp_Suc_E
tff(fact_5072_relpowp__Suc__I,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Y: A,Z: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Y))
     => ( pp(aa(A,bool,aa(A,fun(A,bool),P,Y),Z))
       => pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),P),X),Z)) ) ) ).

% relpowp_Suc_I
tff(fact_5073_relpowp__Suc__D2,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Z: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),P),X),Z))
     => ? [Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y4))
          & pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),Y4),Z)) ) ) ).

% relpowp_Suc_D2
tff(fact_5074_relpowp__Suc__E2,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Z: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),P),X),Z))
     => ~ ! [Y4: A] :
            ( pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y4))
           => ~ pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),Y4),Z)) ) ) ).

% relpowp_Suc_E2
tff(fact_5075_relpowp__Suc__I2,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),X: A,Y: A,N: nat,Z: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y))
     => ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),Y),Z))
       => pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),aa(nat,nat,suc,N)),P),X),Z)) ) ) ).

% relpowp_Suc_I2
tff(fact_5076_relpowp__E,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Z: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Z))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y4: A,M5: nat] :
              ( ( N = aa(nat,nat,suc,M5) )
             => ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),M5),P),X),Y4))
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),P,Y4),Z)) ) ) ) ) ).

% relpowp_E
tff(fact_5077_relpowp__E2,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(A,bool)),X: A,Z: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),P),X),Z))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y4: A,M5: nat] :
              ( ( N = aa(nat,nat,suc,M5) )
             => ( pp(aa(A,bool,aa(A,fun(A,bool),P,X),Y4))
               => ~ pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),M5),P),Y4),Z)) ) ) ) ) ).

% relpowp_E2
tff(fact_5078_relpowp__bot,axiom,
    ! [A: $tType,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aa(nat,fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),compow(fun(A,fun(A,bool))),N),bot_bot(fun(A,fun(A,bool)))) = bot_bot(fun(A,fun(A,bool))) ) ) ).

% relpowp_bot
tff(fact_5079_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),zero_zero(code_integer)))
     => ( code_nat_of_integer(K) = zero_zero(nat) ) ) ).

% nat_of_integer_non_positive
tff(fact_5080_max__nat_Osemilattice__neutr__order__axioms,axiom,
    semila1105856199041335345_order(nat,ord_max(nat),zero_zero(nat),aTP_Lamp_jo(nat,fun(nat,bool)),aTP_Lamp_ea(nat,fun(nat,bool))) ).

% max_nat.semilattice_neutr_order_axioms
tff(fact_5081_card__UNION,axiom,
    ! [A: $tType,A3: set(set(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite(set(A)),A3))
     => ( ! [X3: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),A3))
           => pp(aa(set(A),bool,finite_finite(A),X3)) )
       => ( aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)) = nat2(aa(set(set(set(A))),int,groups7311177749621191930dd_sum(set(set(A)),int,aTP_Lamp_kp(set(set(A)),int)),collect(set(set(A)),aTP_Lamp_kq(set(set(A)),fun(set(set(A)),bool),A3)))) ) ) ) ).

% card_UNION
tff(fact_5082_Sup__lessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [Y: A] : aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_lessThan(A),Y)) = Y ) ).

% Sup_lessThan
tff(fact_5083_Sup__atMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Y: A] : aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_atMost(A),Y)) = Y ) ).

% Sup_atMost
tff(fact_5084_Sup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,X,Y)) = Y ) ) ) ).

% Sup_atLeastAtMost
tff(fact_5085_Inf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,X,Y)) = X ) ) ) ).

% Inf_atLeastAtMost
tff(fact_5086_Sup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,X,Y)) = Y ) ) ) ).

% Sup_atLeastLessThan
tff(fact_5087_Inf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,X,Y)) = X ) ) ) ).

% Inf_atLeastLessThan
tff(fact_5088_Inf__atMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] : aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_atMost(A),X)) = bot_bot(A) ) ).

% Inf_atMost
tff(fact_5089_Sup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,X,Y)) = Y ) ) ) ).

% Sup_greaterThanLessThan
tff(fact_5090_Inf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,X,Y)) = X ) ) ) ).

% Inf_greaterThanLessThan
tff(fact_5091_card__Union__le__sum__card,axiom,
    ! [A: $tType,U2: set(set(A))] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),U2))),aa(set(set(A)),nat,groups7311177749621191930dd_sum(set(A),nat,finite_card(A)),U2))) ).

% card_Union_le_sum_card
tff(fact_5092_cInf__abs__ge,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),A2: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X3)),A2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Inf_Inf(A),S2))),A2)) ) ) ) ).

% cInf_abs_ge
tff(fact_5093_card__Union__le__sum__card__weak,axiom,
    ! [A: $tType,U2: set(set(A))] :
      ( ! [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),U2))
         => pp(aa(set(A),bool,finite_finite(A),X3)) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),U2))),aa(set(set(A)),nat,groups7311177749621191930dd_sum(set(A),nat,finite_card(A)),U2))) ) ).

% card_Union_le_sum_card_weak
tff(fact_5094_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] : code_nat_of_integer(aa(num,code_integer,numeral_numeral(code_integer),K)) = aa(num,nat,numeral_numeral(nat),K) ).

% nat_of_integer_code_post(3)
tff(fact_5095_nat__of__integer__code__post_I2_J,axiom,
    code_nat_of_integer(one_one(code_integer)) = one_one(nat) ).

% nat_of_integer_code_post(2)
tff(fact_5096_cInf__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),L: A,E2: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X3),L))),E2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(set(A),A,complete_Inf_Inf(A),S2)),L))),E2)) ) ) ) ).

% cInf_asclose
tff(fact_5097_cSup__asclose,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),L: A,E2: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,X3),L))),E2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(set(A),A,complete_Sup_Sup(A),S2)),L))),E2)) ) ) ) ).

% cSup_asclose
tff(fact_5098_finite__subset__Union,axiom,
    ! [A: $tType,A3: set(A),B11: set(set(A))] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B11)))
       => ~ ! [F8: set(set(A))] :
              ( pp(aa(set(set(A)),bool,finite_finite(set(A)),F8))
             => ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),F8),B11))
               => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F8))) ) ) ) ) ).

% finite_subset_Union
tff(fact_5099_cInf__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or5935395276787703475ssThan(A,Y,X)) = Y ) ) ) ).

% cInf_greaterThanLessThan
tff(fact_5100_cSup__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or5935395276787703475ssThan(A,Y,X)) = X ) ) ) ).

% cSup_greaterThanLessThan
tff(fact_5101_cInf__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or7035219750837199246ssThan(A,Y,X)) = Y ) ) ) ).

% cInf_atLeastLessThan
tff(fact_5102_cSup__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or1337092689740270186AtMost(A,Y,X)) = X ) ) ) ).

% cSup_atLeastAtMost
tff(fact_5103_cInf__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or1337092689740270186AtMost(A,Y,X)) = Y ) ) ) ).

% cInf_atLeastAtMost
tff(fact_5104_cSup__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or7035219750837199246ssThan(A,Y,X)) = X ) ) ) ).

% cSup_atLeastLessThan
tff(fact_5105_ex__gt__or__lt,axiom,
    ! [A: $tType] :
      ( condit5016429287641298734tinuum(A)
     => ! [A2: A] :
        ? [B5: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B5))
          | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B5),A2)) ) ) ).

% ex_gt_or_lt
tff(fact_5106_complete__interval,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [A2: A,B2: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( pp(aa(A,bool,P,A2))
           => ( ~ pp(aa(A,bool,P,B2))
             => ? [C4: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C4))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C4),B2))
                  & ! [X2: A] :
                      ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X2))
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X2),C4)) )
                     => pp(aa(A,bool,P,X2)) )
                  & ! [D6: A] :
                      ( ! [X3: A] :
                          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3))
                            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),D6)) )
                         => pp(aa(A,bool,P,X3)) )
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D6),C4)) ) ) ) ) ) ) ).

% complete_interval
tff(fact_5107_cSup__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_bot(A) )
     => ! [X6: set(A),A2: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A2)) )
         => ( ! [Y4: A] :
                ( ! [X2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),X6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y4)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),Y4)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X6) = A2 ) ) ) ) ).

% cSup_eq
tff(fact_5108_cSup__eq__maximum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z: A,X6: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),X6))
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),X6) = Z ) ) ) ) ).

% cSup_eq_maximum
tff(fact_5109_cInf__eq,axiom,
    ! [A: $tType] :
      ( ( condit1219197933456340205attice(A)
        & no_top(A) )
     => ! [X6: set(A),A2: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3)) )
         => ( ! [Y4: A] :
                ( ! [X2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),X6))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X2)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),A2)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X6) = A2 ) ) ) ) ).

% cInf_eq
tff(fact_5110_cInf__eq__minimum,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Z: A,X6: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z),X6))
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X3)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),X6) = Z ) ) ) ) ).

% cInf_eq_minimum
tff(fact_5111_cSup__least,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),Z: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),X6)),Z)) ) ) ) ).

% cSup_least
tff(fact_5112_cSup__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A2: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A2)) )
           => ( ! [Y4: A] :
                  ( ! [X2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),X6))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Y4)) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),Y4)) )
             => ( aa(set(A),A,complete_Sup_Sup(A),X6) = A2 ) ) ) ) ) ).

% cSup_eq_non_empty
tff(fact_5113_le__cSup__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),X6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),X6))) ) ) ) ).

% le_cSup_finite
tff(fact_5114_less__cSupD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Z: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),aa(set(A),A,complete_Sup_Sup(A),X6)))
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),X3)) ) ) ) ) ).

% less_cSupD
tff(fact_5115_less__cSupE,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [Y: A,X6: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(set(A),A,complete_Sup_Sup(A),X6)))
         => ( ( X6 != bot_bot(set(A)) )
           => ~ ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
                 => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X3)) ) ) ) ) ).

% less_cSupE
tff(fact_5116_finite__imp__Sup__less,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),X: A,A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),X6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),A2)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X6)),A2)) ) ) ) ) ).

% finite_imp_Sup_less
tff(fact_5117_cInf__greatest,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),Z: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),X6))) ) ) ) ).

% cInf_greatest
tff(fact_5118_cInf__eq__non__empty,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),A2: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3)) )
           => ( ! [Y4: A] :
                  ( ! [X2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),X6))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X2)) )
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),A2)) )
             => ( aa(set(A),A,complete_Inf_Inf(A),X6) = A2 ) ) ) ) ) ).

% cInf_eq_non_empty
tff(fact_5119_cInf__le__finite,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [X6: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),X6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),X6)),X)) ) ) ) ).

% cInf_le_finite
tff(fact_5120_cInf__lessD,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),Z: A] :
          ( ( X6 != bot_bot(set(A)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),X6)),Z))
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Z)) ) ) ) ) ).

% cInf_lessD
tff(fact_5121_finite__imp__less__Inf,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),X: A,A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),X6))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),X3)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(set(A),A,complete_Inf_Inf(A),X6))) ) ) ) ) ).

% finite_imp_less_Inf
tff(fact_5122_finite__Sup__less__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),X6)),A2))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),A2)) ) ) ) ) ) ).

% finite_Sup_less_iff
tff(fact_5123_finite__less__Inf__iff,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [X6: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),X6))
         => ( ( X6 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(set(A),A,complete_Inf_Inf(A),X6)))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),X6))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),X4)) ) ) ) ) ) ).

% finite_less_Inf_iff
tff(fact_5124_cSup__abs__le,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linordered_idom(A) )
     => ! [S2: set(A),A2: A] :
          ( ( S2 != bot_bot(set(A)) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),X3)),A2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(A),A,complete_Sup_Sup(A),S2))),A2)) ) ) ) ).

% cSup_abs_le
tff(fact_5125_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] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),X4))
             => ? [Xa4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Xa4),X4)) ) ) ) ) ).

% Inf_eq_bot_iff
tff(fact_5126_Inf__le__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( ( A3 != bot_bot(set(A)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ).

% Inf_le_Sup
tff(fact_5127_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),zero_zero(code_integer)))
       => ( code_nat_of_integer(K) = zero_zero(nat) ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),zero_zero(code_integer)))
       => ( code_nat_of_integer(K) = aa(product_prod(code_integer,code_integer),nat,product_case_prod(code_integer,code_integer,nat,aTP_Lamp_kr(code_integer,fun(code_integer,nat))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))) ) ) ) ).

% nat_of_integer_code
tff(fact_5128_Sup__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),X: A] :
          ( ! [Y4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) )
         => ( ! [Y4: A] :
                ( ! [Z3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z3),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z3),Y4)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),A3) = X ) ) ) ) ).

% Sup_eqI
tff(fact_5129_Sup__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( ! [A5: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),A3))
             => ? [X2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),B3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),X2)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3))) ) ) ).

% Sup_mono
tff(fact_5130_Sup__least,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),Z: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),Z)) ) ) ).

% Sup_least
tff(fact_5131_Sup__upper,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,A3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ).

% Sup_upper
tff(fact_5132_Sup__le__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),B2))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),B2)) ) ) ) ).

% Sup_le_iff
tff(fact_5133_Sup__upper2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A3: set(A),V: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),U),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V),U))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ) ).

% Sup_upper2
tff(fact_5134_less__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A2: A,S2: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),S2)))
        <=> ? [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),X4)) ) ) ) ).

% less_Sup_iff
tff(fact_5135_Inf__eqI,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),X: A] :
          ( ! [I4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),I4)) )
         => ( ! [Y4: A] :
                ( ! [I: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),I)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),A3) = X ) ) ) ) ).

% Inf_eqI
tff(fact_5136_Inf__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: set(A),A3: set(A)] :
          ( ! [B5: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B5),B3))
             => ? [X2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),B5)) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B3))) ) ) ).

% Inf_mono
tff(fact_5137_Inf__lower,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,A3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),X)) ) ) ).

% Inf_lower
tff(fact_5138_Inf__lower2,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,A3: set(A),V: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),U),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),V)) ) ) ) ).

% Inf_lower2
tff(fact_5139_le__Inf__iff,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B2: A,A3: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(set(A),A,complete_Inf_Inf(A),A3)))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X4)) ) ) ) ).

% le_Inf_iff
tff(fact_5140_Inf__greatest,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),Z: A] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),A3))) ) ) ).

% Inf_greatest
tff(fact_5141_Inf__less__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [S2: set(A),A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S2)),A2))
        <=> ? [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),A2)) ) ) ) ).

% Inf_less_iff
tff(fact_5142_Union__least,axiom,
    ! [A: $tType,A3: set(set(A)),C5: set(A)] :
      ( ! [X7: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X7),A3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X7),C5)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)),C5)) ) ).

% Union_least
tff(fact_5143_Union__upper,axiom,
    ! [A: $tType,B3: set(A),A3: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),B3),A3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3))) ) ).

% Union_upper
tff(fact_5144_Union__subsetI,axiom,
    ! [A: $tType,A3: set(set(A)),B3: set(set(A))] :
      ( ! [X3: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X3),A3))
         => ? [Y3: set(A)] :
              ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Y3),B3))
              & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X3),Y3)) ) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3))) ) ).

% Union_subsetI
tff(fact_5145_Inter__lower,axiom,
    ! [A: $tType,B3: set(A),A3: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),B3),A3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3)),B3)) ) ).

% Inter_lower
tff(fact_5146_Inter__greatest,axiom,
    ! [A: $tType,A3: set(set(A)),C5: set(A)] :
      ( ! [X7: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X7),A3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C5),X7)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C5),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3))) ) ).

% Inter_greatest
tff(fact_5147_Union__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,set(A)),P: fun(B,bool)] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),collect(set(A),aa(fun(B,bool),fun(set(A),bool),aTP_Lamp_ks(fun(B,set(A)),fun(fun(B,bool),fun(set(A),bool)),F3),P))) = collect(A,aa(fun(B,bool),fun(A,bool),aTP_Lamp_kt(fun(B,set(A)),fun(fun(B,bool),fun(A,bool)),F3),P)) ).

% Union_SetCompr_eq
tff(fact_5148_le__Sup__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [X: A,A3: set(A)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A3)))
        <=> ! [Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X))
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X4)) ) ) ) ) ).

% le_Sup_iff
tff(fact_5149_Inf__le__iff,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),X))
        <=> ! [Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y5))
             => ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y5)) ) ) ) ) ).

% Inf_le_iff
tff(fact_5150_less__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),U: A] :
          ( ! [V3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V3)) )
         => ( ( A3 != bot_bot(set(A)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ) ).

% less_eq_Sup
tff(fact_5151_Sup__subset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3))) ) ) ).

% Sup_subset_mono
tff(fact_5152_Inf__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),U: A] :
          ( ! [V3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),V3),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V3),U)) )
         => ( ( A3 != bot_bot(set(A)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),U)) ) ) ) ).

% Inf_less_eq
tff(fact_5153_Inf__superset__mono,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: set(A),A3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B3))) ) ) ).

% Inf_superset_mono
tff(fact_5154_Union__mono,axiom,
    ! [A: $tType,A3: set(set(A)),B3: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),A3),B3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3))) ) ).

% Union_mono
tff(fact_5155_Inter__anti__mono,axiom,
    ! [A: $tType,B3: set(set(A)),A3: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),B3),A3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B3))) ) ).

% Inter_anti_mono
tff(fact_5156_Inter__subset,axiom,
    ! [A: $tType,A3: set(set(A)),B3: set(A)] :
      ( ! [X7: set(A)] :
          ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X7),A3))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X7),B3)) )
     => ( ( A3 != bot_bot(set(set(A))) )
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3)),B3)) ) ) ).

% Inter_subset
tff(fact_5157_subset__Pow__Union,axiom,
    ! [A: $tType,A3: set(set(A))] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),A3),pow2(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)))) ).

% subset_Pow_Union
tff(fact_5158_set__removeAll,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),removeAll(A,X),Xs)) = aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ).

% set_removeAll
tff(fact_5159_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_5160_int__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
       => ( code_int_of_integer(K) = aa(int,int,uminus_uminus(int),code_int_of_integer(aa(code_integer,code_integer,uminus_uminus(code_integer),K))) ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),zero_zero(code_integer)))
       => ( ( ( K = zero_zero(code_integer) )
           => ( code_int_of_integer(K) = zero_zero(int) ) )
          & ( ( K != zero_zero(code_integer) )
           => ( code_int_of_integer(K) = aa(product_prod(code_integer,code_integer),int,product_case_prod(code_integer,code_integer,int,aTP_Lamp_ku(code_integer,fun(code_integer,int))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))) ) ) ) ) ) ).

% int_of_integer_code
tff(fact_5161_removeAll__id,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(A),removeAll(A,X),Xs) = Xs ) ) ).

% removeAll_id
tff(fact_5162_modulo__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] : code_int_of_integer(modulo_modulo(code_integer,X,Xa)) = modulo_modulo(int,code_int_of_integer(X),code_int_of_integer(Xa)) ).

% modulo_integer.rep_eq
tff(fact_5163_distinct__removeAll,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => distinct(A,aa(list(A),list(A),removeAll(A,X),Xs)) ) ).

% distinct_removeAll
tff(fact_5164_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),K),L))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),code_int_of_integer(K)),code_int_of_integer(L))) ) ).

% integer_less_iff
tff(fact_5165_less__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),X),Xa))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),code_int_of_integer(X)),code_int_of_integer(Xa))) ) ).

% less_integer.rep_eq
tff(fact_5166_less__eq__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),X),Xa))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),code_int_of_integer(X)),code_int_of_integer(Xa))) ) ).

% less_eq_integer.rep_eq
tff(fact_5167_integer__less__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),L))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),code_int_of_integer(K)),code_int_of_integer(L))) ) ).

% integer_less_eq_iff
tff(fact_5168_length__removeAll__less__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),removeAll(A,X),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_removeAll_less_eq
tff(fact_5169_distinct__remove1__removeAll,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( remove1(A,X,Xs) = aa(list(A),list(A),removeAll(A,X),Xs) ) ) ).

% distinct_remove1_removeAll
tff(fact_5170_length__removeAll__less,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),removeAll(A,X),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% length_removeAll_less
tff(fact_5171_times__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),times_times(int),abs_Integ(Xa)),abs_Integ(X)) = 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_kw(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X)) ).

% times_int.abs_eq
tff(fact_5172_arg__min__if__finite_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S2: set(A),F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite(A),S2))
         => ( ( S2 != bot_bot(set(A)) )
           => ~ ? [X2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),S2))
                  & pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X2)),aa(A,B,F3,lattic7623131987881927897min_on(A,B,F3,S2)))) ) ) ) ) ).

% arg_min_if_finite(2)
tff(fact_5173_arg__min__least,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S2: set(A),Y: A,F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite(A),S2))
         => ( ( S2 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),S2))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,lattic7623131987881927897min_on(A,B,F3,S2))),aa(A,B,F3,Y))) ) ) ) ) ).

% arg_min_least
tff(fact_5174_arg__min__if__finite_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( order(B)
     => ! [S2: set(A),F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite(A),S2))
         => ( ( S2 != bot_bot(set(A)) )
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),lattic7623131987881927897min_on(A,B,F3,S2)),S2)) ) ) ) ).

% arg_min_if_finite(1)
tff(fact_5175_one__int__def,axiom,
    one_one(int) = abs_Integ(aa(nat,product_prod(nat,nat),product_Pair(nat,nat,one_one(nat)),zero_zero(nat))) ).

% one_int_def
tff(fact_5176_less__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),X: product_prod(nat,nat)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),abs_Integ(Xa)),abs_Integ(X)))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_ky(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa),X)) ) ).

% less_int.abs_eq
tff(fact_5177_less__eq__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),X: product_prod(nat,nat)] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),abs_Integ(Xa)),abs_Integ(X)))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_la(nat,fun(nat,fun(product_prod(nat,nat),bool)))),Xa),X)) ) ).

% less_eq_int.abs_eq
tff(fact_5178_plus__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,aa(int,fun(int,int),plus_plus(int),abs_Integ(Xa)),abs_Integ(X)) = 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_lc(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X)) ).

% plus_int.abs_eq
tff(fact_5179_minus__int_Oabs__eq,axiom,
    ! [Xa: product_prod(nat,nat),X: product_prod(nat,nat)] : aa(int,int,minus_minus(int,abs_Integ(Xa)),abs_Integ(X)) = 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_le(nat,fun(nat,fun(product_prod(nat,nat),product_prod(nat,nat))))),Xa),X)) ).

% minus_int.abs_eq
tff(fact_5180_insert__subsetI,axiom,
    ! [A: $tType,X: A,A3: set(A),X6: set(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),A3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),X6)),A3)) ) ) ).

% insert_subsetI
tff(fact_5181_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( num_of_nat(aa(nat,nat,suc,N)) = inc(num_of_nat(N)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( num_of_nat(aa(nat,nat,suc,N)) = one2 ) ) ) ).

% num_of_nat.simps(2)
tff(fact_5182_pred__nat__def,axiom,
    pred_nat = collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_lf(nat,fun(nat,bool)))) ).

% pred_nat_def
tff(fact_5183_num__of__nat__numeral__eq,axiom,
    ! [Q3: num] : num_of_nat(aa(num,nat,numeral_numeral(nat),Q3)) = Q3 ).

% num_of_nat_numeral_eq
tff(fact_5184_num__of__nat_Osimps_I1_J,axiom,
    num_of_nat(zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_5185_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(num,nat,numeral_numeral(nat),num_of_nat(N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_5186_num__of__nat__One,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),one_one(nat)))
     => ( num_of_nat(N) = one2 ) ) ).

% num_of_nat_One
tff(fact_5187_prop__restrict,axiom,
    ! [A: $tType,X: A,Z6: set(A),X6: set(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),Z6))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Z6),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ad(set(A),fun(fun(A,bool),fun(A,bool)),X6),P))))
       => pp(aa(A,bool,P,X)) ) ) ).

% prop_restrict
tff(fact_5188_Collect__restrict,axiom,
    ! [A: $tType,X6: set(A),P: fun(A,bool)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ad(set(A),fun(fun(A,bool),fun(A,bool)),X6),P))),X6)) ).

% Collect_restrict
tff(fact_5189_numeral__num__of__nat__unfold,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( ( N = zero_zero(nat) )
           => ( aa(num,A,numeral_numeral(A),num_of_nat(N)) = one_one(A) ) )
          & ( ( N != zero_zero(nat) )
           => ( aa(num,A,numeral_numeral(A),num_of_nat(N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ) ) ) ).

% numeral_num_of_nat_unfold
tff(fact_5190_num__of__nat__double,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N)) = aa(num,num,bit0,num_of_nat(N)) ) ) ).

% num_of_nat_double
tff(fact_5191_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),num_of_nat(M)),num_of_nat(N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_5192_subset__emptyI,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [X3: A] : ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),bot_bot(set(A)))) ) ).

% subset_emptyI
tff(fact_5193_less__eq__int_Orep__eq,axiom,
    ! [X: int,Xa: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_la(nat,fun(nat,fun(product_prod(nat,nat),bool)))),rep_Integ(X)),rep_Integ(Xa))) ) ).

% less_eq_int.rep_eq
tff(fact_5194_less__int_Orep__eq,axiom,
    ! [X: int,Xa: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),X),Xa))
    <=> pp(aa(product_prod(nat,nat),bool,aa(product_prod(nat,nat),fun(product_prod(nat,nat),bool),product_case_prod(nat,nat,fun(product_prod(nat,nat),bool),aTP_Lamp_ky(nat,fun(nat,fun(product_prod(nat,nat),bool)))),rep_Integ(X)),rep_Integ(Xa))) ) ).

% less_int.rep_eq
tff(fact_5195_eq__numeral__iff__iszero_I7_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = one_one(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),one2))) ) ) ).

% eq_numeral_iff_iszero(7)
tff(fact_5196_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_5197_eq__iff__iszero__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: A,Y: A] :
          ( ( X = Y )
        <=> ring_1_iszero(A,aa(A,A,minus_minus(A,X),Y)) ) ) ).

% eq_iff_iszero_diff
tff(fact_5198_not__iszero__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ~ ring_1_iszero(A,one_one(A)) ) ).

% not_iszero_1
tff(fact_5199_iszero__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ring_1_iszero(A,zero_zero(A)) ) ).

% iszero_0
tff(fact_5200_iszero__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] :
          ( ring_1_iszero(A,Z)
        <=> ( Z = zero_zero(A) ) ) ) ).

% iszero_def
tff(fact_5201_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_5202_eq__numeral__iff__iszero_I9_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num] :
          ( ( aa(num,A,numeral_numeral(A),X) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),X)) ) ) ).

% eq_numeral_iff_iszero(9)
tff(fact_5203_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_5204_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_5205_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_5206_eq__numeral__iff__iszero_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num,Y: num] :
          ( ( aa(num,A,numeral_numeral(A),X) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,X,Y)) ) ) ).

% eq_numeral_iff_iszero(1)
tff(fact_5207_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_5208_eq__numeral__iff__iszero_I11_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = zero_zero(A) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),X)) ) ) ).

% eq_numeral_iff_iszero(11)
tff(fact_5209_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_5210_eq__numeral__iff__iszero_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num,Y: num] :
          ( ( aa(num,A,numeral_numeral(A),X) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y))) ) ) ).

% eq_numeral_iff_iszero(2)
tff(fact_5211_eq__numeral__iff__iszero_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num,Y: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = aa(num,A,numeral_numeral(A),Y) )
        <=> ring_1_iszero(A,aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),X),Y))) ) ) ).

% eq_numeral_iff_iszero(3)
tff(fact_5212_eq__numeral__iff__iszero_I4_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num,Y: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),X)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Y)) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,Y,X)) ) ) ).

% eq_numeral_iff_iszero(4)
tff(fact_5213_eq__numeral__iff__iszero_I5_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [X: num] :
          ( ( aa(num,A,numeral_numeral(A),X) = one_one(A) )
        <=> ring_1_iszero(A,neg_numeral_sub(A,X,one2)) ) ) ).

% eq_numeral_iff_iszero(5)
tff(fact_5214_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_5215_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_5216_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ vEBT_VEBT_valid(X,Xa)
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa = one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList3: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList3,Summary3) )
             => ( ( Deg2 = Xa )
                & ! [X3: vEBT_VEBT] :
                    ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                   => vEBT_VEBT_valid(X3,divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
                & vEBT_VEBT_valid(Summary3,aa(nat,nat,minus_minus(nat,Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
                & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,minus_minus(nat,Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_lg(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList3),Summary3)),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_5217_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_valid(X,Xa)
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( Xa != one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList3: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList3,Summary3) )
             => ~ ( ( Deg2 = Xa )
                  & ! [X2: vEBT_VEBT] :
                      ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                     => vEBT_VEBT_valid(X2,divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
                  & vEBT_VEBT_valid(Summary3,aa(nat,nat,minus_minus(nat,Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
                  & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,minus_minus(nat,Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                  & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_lg(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList3),Summary3)),Mima)) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_5218_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: bool] :
      ( ( vEBT_VEBT_valid(X,Xa)
      <=> pp(Y) )
     => ( ( ? [Uu2: bool,Uv2: bool] : X = vEBT_Leaf(Uu2,Uv2)
         => ( pp(Y)
          <=> ( Xa != one_one(nat) ) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList3: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
              ( ( X = vEBT_Node(Mima,Deg2,TreeList3,Summary3) )
             => ( pp(Y)
              <=> ~ ( ( Deg2 = Xa )
                    & ! [X4: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                       => vEBT_VEBT_valid(X4,divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
                    & vEBT_VEBT_valid(Summary3,aa(nat,nat,minus_minus(nat,Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
                    & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,minus_minus(nat,Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                    & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_lg(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList3),Summary3)),Mima)) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_5219_option_Osimps_I4_J,axiom,
    ! [A: $tType,B: $tType,F1: B,F2: fun(A,B)] : case_option(B,A,F1,F2,none(A)) = F1 ).

% option.simps(4)
tff(fact_5220_option_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option = none(A) )
    <=> pp(case_option(bool,A,fTrue,aTP_Lamp_lh(A,bool),Option)) ) ).

% option.disc_eq_case(1)
tff(fact_5221_option_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
    <=> pp(case_option(bool,A,fFalse,aTP_Lamp_li(A,bool),Option)) ) ).

% option.disc_eq_case(2)
tff(fact_5222_case__optionE,axiom,
    ! [A: $tType,P: bool,Q: fun(A,bool),X: option(A)] :
      ( pp(case_option(bool,A,P,Q,X))
     => ( ( ( X = none(A) )
         => ~ pp(P) )
       => ~ ! [Y4: A] :
              ( ( X = aa(A,option(A),some(A),Y4) )
             => ~ pp(aa(A,bool,Q,Y4)) ) ) ) ).

% case_optionE
tff(fact_5223_Sup__eq__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] : aa(set(A),A,complete_Sup_Sup(A),A3) = aa(set(A),A,complete_Inf_Inf(A),collect(A,aTP_Lamp_lj(set(A),fun(A,bool),A3))) ) ).

% Sup_eq_Inf
tff(fact_5224_Inf__eq__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] : aa(set(A),A,complete_Inf_Inf(A),A3) = aa(set(A),A,complete_Sup_Sup(A),collect(A,aTP_Lamp_lk(set(A),fun(A,bool),A3))) ) ).

% Inf_eq_Sup
tff(fact_5225_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option(product_prod(nat,nat)),Deg: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT,Deg4: nat] :
      ( vEBT_VEBT_valid(vEBT_Node(Mima2,Deg,TreeList,Summary),Deg4)
    <=> ( ( Deg = Deg4 )
        & ! [X4: vEBT_VEBT] :
            ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList)))
           => vEBT_VEBT_valid(X4,divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
        & vEBT_VEBT_valid(Summary,aa(nat,nat,minus_minus(nat,Deg),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
        & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,minus_minus(nat,Deg),divide_divide(nat,Deg,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
        & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_lg(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg),TreeList),Summary)),Mima2)) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_5226_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat,Y: bool] :
      ( ( vEBT_VEBT_valid(X,Xa)
      <=> pp(Y) )
     => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa)))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( ( pp(Y)
                <=> ( Xa = one_one(nat) ) )
               => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,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)),Xa))) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList3: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList3,Summary3) )
               => ( ( pp(Y)
                  <=> ( ( Deg2 = Xa )
                      & ! [X4: vEBT_VEBT] :
                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                         => vEBT_VEBT_valid(X4,divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
                      & vEBT_VEBT_valid(Summary3,aa(nat,nat,minus_minus(nat,Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
                      & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,minus_minus(nat,Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_lg(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList3),Summary3)),Mima)) ) )
                 => ~ pp(aa(product_prod(vEBT_VEBT,nat),bool,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,Deg2,TreeList3,Summary3)),Xa))) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_5227_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( vEBT_VEBT_valid(X,Xa)
     => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa)))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,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)),Xa)))
               => ( Xa != one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList3: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList3,Summary3) )
               => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,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,Deg2,TreeList3,Summary3)),Xa)))
                 => ~ ( ( Deg2 = Xa )
                      & ! [X2: vEBT_VEBT] :
                          ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X2),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                         => vEBT_VEBT_valid(X2,divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
                      & vEBT_VEBT_valid(Summary3,aa(nat,nat,minus_minus(nat,Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
                      & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,minus_minus(nat,Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                      & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_lg(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList3),Summary3)),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_5228_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa: nat] :
      ( ~ vEBT_VEBT_valid(X,Xa)
     => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel),aa(nat,product_prod(vEBT_VEBT,nat),product_Pair(vEBT_VEBT,nat,X),Xa)))
       => ( ! [Uu2: bool,Uv2: bool] :
              ( ( X = vEBT_Leaf(Uu2,Uv2) )
             => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,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)),Xa)))
               => ( Xa = one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList3: list(vEBT_VEBT),Summary3: vEBT_VEBT] :
                ( ( X = vEBT_Node(Mima,Deg2,TreeList3,Summary3) )
               => ( pp(aa(product_prod(vEBT_VEBT,nat),bool,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,Deg2,TreeList3,Summary3)),Xa)))
                 => ( ( Deg2 = Xa )
                    & ! [X3: vEBT_VEBT] :
                        ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3)))
                       => vEBT_VEBT_valid(X3,divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))) )
                    & vEBT_VEBT_valid(Summary3,aa(nat,nat,minus_minus(nat,Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))
                    & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,minus_minus(nat,Deg2),divide_divide(nat,Deg2,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) )
                    & pp(case_option(bool,product_prod(nat,nat),fconj(aa(bool,bool,fNot,aa(fun(nat,bool),bool,fEx(nat),aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Summary3))),fAll(vEBT_VEBT,combs(vEBT_VEBT,bool,bool,combb(bool,fun(bool,bool),vEBT_VEBT,fimplies,combc(vEBT_VEBT,set(vEBT_VEBT),bool,member(vEBT_VEBT),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList3))),combb(bool,bool,vEBT_VEBT,fNot,combb(fun(nat,bool),bool,vEBT_VEBT,fEx(nat),vEBT_V8194947554948674370ptions))))),product_case_prod(nat,nat,bool,aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_lg(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Deg2),TreeList3),Summary3)),Mima)) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_5229_Sup__int__def,axiom,
    ! [X6: set(int)] : aa(set(int),int,complete_Sup_Sup(int),X6) = the(int,aTP_Lamp_ll(set(int),fun(int,bool),X6)) ).

% Sup_int_def
tff(fact_5230_Union__maximal__sets,axiom,
    ! [A: $tType,F9: set(set(A))] :
      ( pp(aa(set(set(A)),bool,finite_finite(set(A)),F9))
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),collect(set(A),aTP_Lamp_lm(set(set(A)),fun(set(A),bool),F9))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),F9) ) ) ).

% Union_maximal_sets
tff(fact_5231_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = case_option(int,num,zero_zero(int),aTP_Lamp_ln(num,fun(num,int),M),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_5232_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool)] : order_Greatest(A,P) = the(A,aTP_Lamp_lo(fun(A,bool),fun(A,bool),P)) ) ).

% Greatest_def
tff(fact_5233_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(N))) ).

% and_minus_numerals(7)
tff(fact_5234_take__bit__num__simps_I1_J,axiom,
    ! [M: num] : bit_take_bit_num(zero_zero(nat),M) = none(num) ).

% take_bit_num_simps(1)
tff(fact_5235_take__bit__num__simps_I2_J,axiom,
    ! [N: nat] : bit_take_bit_num(aa(nat,nat,suc,N),one2) = aa(num,option(num),some(num),one2) ).

% take_bit_num_simps(2)
tff(fact_5236_take__bit__num__simps_I5_J,axiom,
    ! [R3: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R3),one2) = aa(num,option(num),some(num),one2) ).

% take_bit_num_simps(5)
tff(fact_5237_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N),aa(num,num,bit0,M)) = case_option(option(num),num,none(num),aTP_Lamp_lp(num,option(num)),bit_take_bit_num(N,M)) ).

% take_bit_num_simps(3)
tff(fact_5238_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N),aa(num,num,bit1,M)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(N,M))) ).

% take_bit_num_simps(4)
tff(fact_5239_take__bit__num__simps_I6_J,axiom,
    ! [R3: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R3),aa(num,num,bit0,M)) = case_option(option(num),num,none(num),aTP_Lamp_lp(num,option(num)),bit_take_bit_num(pred_numeral(R3),M)) ).

% take_bit_num_simps(6)
tff(fact_5240_take__bit__num__simps_I7_J,axiom,
    ! [R3: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R3),aa(num,num,bit1,M)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(pred_numeral(R3),M))) ).

% take_bit_num_simps(7)
tff(fact_5241_take__bit__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,N: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),M)),aa(num,A,numeral_numeral(A),N)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N)) ) ).

% take_bit_numeral_numeral
tff(fact_5242_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,aa(num,num,bit0,N))) ).

% and_minus_numerals(8)
tff(fact_5243_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,aa(num,num,bit0,N))) ).

% and_minus_numerals(4)
tff(fact_5244_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,N)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(N))) ).

% and_minus_numerals(3)
tff(fact_5245_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one2,one2) = none(num) ).

% and_not_num.simps(1)
tff(fact_5246_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,aa(num,num,bit0,M)) = case_nat(option(num),none(num),aTP_Lamp_lq(num,fun(nat,option(num)),M),N) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_5247_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N: nat] : bit_take_bit_num(N,one2) = case_nat(option(num),none(num),aTP_Lamp_lr(nat,option(num)),N) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_5248_and__not__num_Osimps_I2_J,axiom,
    ! [N: num] : bit_and_not_num(one2,aa(num,num,bit0,N)) = aa(num,option(num),some(num),one2) ).

% and_not_num.simps(2)
tff(fact_5249_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] : bit_and_not_num(aa(num,num,bit0,M),one2) = aa(num,option(num),some(num),aa(num,num,bit0,M)) ).

% and_not_num.simps(4)
tff(fact_5250_GreatestI__ex__nat,axiom,
    ! [P: fun(nat,bool),B2: nat] :
      ( ? [X_12: nat] : pp(aa(nat,bool,P,X_12))
     => ( ! [Y4: nat] :
            ( pp(aa(nat,bool,P,Y4))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y4),B2)) )
       => pp(aa(nat,bool,P,order_Greatest(nat,P))) ) ) ).

% GreatestI_ex_nat
tff(fact_5251_Greatest__le__nat,axiom,
    ! [P: fun(nat,bool),K: nat,B2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [Y4: nat] :
            ( pp(aa(nat,bool,P,Y4))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y4),B2)) )
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),order_Greatest(nat,P))) ) ) ).

% Greatest_le_nat
tff(fact_5252_GreatestI__nat,axiom,
    ! [P: fun(nat,bool),K: nat,B2: nat] :
      ( pp(aa(nat,bool,P,K))
     => ( ! [Y4: nat] :
            ( pp(aa(nat,bool,P,Y4))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Y4),B2)) )
       => pp(aa(nat,bool,P,order_Greatest(nat,P))) ) ) ).

% GreatestI_nat
tff(fact_5253_and__not__num_Osimps_I3_J,axiom,
    ! [N: num] : bit_and_not_num(one2,aa(num,num,bit1,N)) = none(num) ).

% and_not_num.simps(3)
tff(fact_5254_take__bit__num__eq__Some__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: num,Q3: num] :
          ( ( bit_take_bit_num(M,N) = aa(num,option(num),some(num),Q3) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),Q3) ) ) ) ).

% take_bit_num_eq_Some_imp
tff(fact_5255_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,aa(num,num,bit1,M)) = case_nat(option(num),none(num),aTP_Lamp_ls(num,fun(nat,option(num)),M),N) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_5256_Greatest__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),X: A] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y4: A] :
                ( pp(aa(A,bool,P,Y4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) )
           => ( order_Greatest(A,P) = X ) ) ) ) ).

% Greatest_equality
tff(fact_5257_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,bool),X: A,Q: fun(A,bool)] :
          ( pp(aa(A,bool,P,X))
         => ( ! [Y4: A] :
                ( pp(aa(A,bool,P,Y4))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) )
           => ( ! [X3: A] :
                  ( pp(aa(A,bool,P,X3))
                 => ( ! [Y3: A] :
                        ( pp(aa(A,bool,P,Y3))
                       => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y3),X3)) )
                   => pp(aa(A,bool,Q,X3)) ) )
             => pp(aa(A,bool,Q,order_Greatest(A,P))) ) ) ) ) ).

% GreatestI2_order
tff(fact_5258_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] : bit_and_not_num(aa(num,num,bit1,M),one2) = aa(num,option(num),some(num),aa(num,num,bit0,M)) ).

% and_not_num.simps(7)
tff(fact_5259_take__bit__num__eq__None__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: num] :
          ( ( bit_take_bit_num(M,N) = none(num) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).

% take_bit_num_eq_None_imp
tff(fact_5260_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N: num,Q3: num] :
      ( ( bit_and_not_num(M,N) = aa(num,option(num),some(num),Q3) )
    <=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = aa(num,int,numeral_numeral(int),Q3) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_5261_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa(num,num,bit1,M),aa(num,num,bit0,N)) = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_lt(num,option(num)),bit_and_not_num(M,N)) ).

% and_not_num.simps(8)
tff(fact_5262_and__not__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num(M,N) = none(num) )
    <=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = zero_zero(int) ) ) ).

% and_not_num_eq_None_iff
tff(fact_5263_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(N,M)) ).

% int_numeral_not_and_num
tff(fact_5264_int__numeral__and__not__num,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,N)) ).

% int_numeral_and_not_num
tff(fact_5265_take__bit__num__def,axiom,
    ! [N: nat,M: num] :
      ( ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(num,nat,numeral_numeral(nat),M)) = zero_zero(nat) )
       => ( bit_take_bit_num(N,M) = none(num) ) )
      & ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(num,nat,numeral_numeral(nat),M)) != zero_zero(nat) )
       => ( bit_take_bit_num(N,M) = aa(num,option(num),some(num),num_of_nat(aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(num,nat,numeral_numeral(nat),M)))) ) ) ) ).

% take_bit_num_def
tff(fact_5266_Ball__Collect,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,bool)] :
      ( ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
         => pp(aa(A,bool,P,X4)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),collect(A,P))) ) ).

% Ball_Collect
tff(fact_5267_prod_Oinsert_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),P2: fun(B,A),I2: B] :
          ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ja(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))))
         => ( ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I5))
             => ( groups1962203154675924110t_prod(B,A,P2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),I2),I5)) = groups1962203154675924110t_prod(B,A,P2,I5) ) )
            & ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),I5))
             => ( groups1962203154675924110t_prod(B,A,P2,aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),I2),I5)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,P2,I2)),groups1962203154675924110t_prod(B,A,P2,I5)) ) ) ) ) ) ).

% prod.insert'
tff(fact_5268_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) = remove1(A,X,aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
tff(fact_5269_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( aa(list(A),set(A),set2(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) = A3 ) ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_5270_prod_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P2: fun(B,A)] : groups1962203154675924110t_prod(B,A,P2,bot_bot(set(B))) = one_one(A) ) ).

% prod.empty'
tff(fact_5271_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : aa(list(A),nat,size_size(list(A)),aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) = aa(set(A),nat,finite_card(A),A3) ) ).

% sorted_list_of_set.length_sorted_key_list_of_set
tff(fact_5272_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( ( aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = aa(set(A),list(A),linord4507533701916653071of_set(A),B3) )
         => ( pp(aa(set(A),bool,finite_finite(A),A3))
           => ( pp(aa(set(A),bool,finite_finite(A),B3))
             => ( A3 = B3 ) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
tff(fact_5273_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : distinct(A,aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) ) ).

% sorted_list_of_set.distinct_sorted_key_list_of_set
tff(fact_5274_prod_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),I5: set(B)] : groups1962203154675924110t_prod(B,A,G,collect(B,aa(set(B),fun(B,bool),aTP_Lamp_lu(fun(B,A),fun(set(B),fun(B,bool)),G),I5))) = groups1962203154675924110t_prod(B,A,G,I5) ) ).

% prod.non_neutral'
tff(fact_5275_ge__eq__refl,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool)),X: A] :
      ( pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),R))
     => pp(aa(A,bool,aa(A,fun(A,bool),R,X),X)) ) ).

% ge_eq_refl
tff(fact_5276_refl__ge__eq,axiom,
    ! [A: $tType,R: fun(A,fun(A,bool))] :
      ( ! [X3: A] : pp(aa(A,bool,aa(A,fun(A,bool),R,X3),X3))
     => pp(aa(fun(A,fun(A,bool)),bool,aa(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),bool),ord_less_eq(fun(A,fun(A,bool))),fequal(A)),R)) ) ).

% refl_ge_eq
tff(fact_5277_prod_Omono__neutral__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),T6: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T6))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T6),S2)))
               => ( aa(B,A,G,X3) = one_one(A) ) )
           => ( groups1962203154675924110t_prod(B,A,G,S2) = groups1962203154675924110t_prod(B,A,G,T6) ) ) ) ) ).

% prod.mono_neutral_left'
tff(fact_5278_prod_Omono__neutral__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),T6: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T6))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T6),S2)))
               => ( aa(B,A,G,X3) = one_one(A) ) )
           => ( groups1962203154675924110t_prod(B,A,G,T6) = groups1962203154675924110t_prod(B,A,G,S2) ) ) ) ) ).

% prod.mono_neutral_right'
tff(fact_5279_prod_Omono__neutral__cong__left_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),T6: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T6))
         => ( ! [I4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),aa(set(B),set(B),minus_minus(set(B),T6),S2)))
               => ( aa(B,A,H,I4) = one_one(A) ) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S2))
                 => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
             => ( groups1962203154675924110t_prod(B,A,G,S2) = groups1962203154675924110t_prod(B,A,H,T6) ) ) ) ) ) ).

% prod.mono_neutral_cong_left'
tff(fact_5280_prod_Omono__neutral__cong__right_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),T6: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),S2),T6))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),minus_minus(set(B),T6),S2)))
               => ( aa(B,A,G,X3) = one_one(A) ) )
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),S2))
                 => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
             => ( groups1962203154675924110t_prod(B,A,G,T6) = groups1962203154675924110t_prod(B,A,H,S2) ) ) ) ) ) ).

% prod.mono_neutral_cong_right'
tff(fact_5281_prod_Odistrib_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),G: fun(B,A),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ja(set(B),fun(fun(B,A),fun(B,bool)),I5),G))))
         => ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ja(set(B),fun(fun(B,A),fun(B,bool)),I5),H))))
           => ( groups1962203154675924110t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_lv(fun(B,A),fun(fun(B,A),fun(B,A)),G),H),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),groups1962203154675924110t_prod(B,A,G,I5)),groups1962203154675924110t_prod(B,A,H,I5)) ) ) ) ) ).

% prod.distrib'
tff(fact_5282_prod_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [I5: set(B),P2: fun(B,A)] :
          ( ( pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ja(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))))
           => ( groups1962203154675924110t_prod(B,A,P2,I5) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),P2),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ja(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))) ) )
          & ( ~ pp(aa(set(B),bool,finite_finite(B),collect(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_ja(set(B),fun(fun(B,A),fun(B,bool)),I5),P2))))
           => ( groups1962203154675924110t_prod(B,A,P2,I5) = one_one(A) ) ) ) ) ).

% prod.G_def
tff(fact_5283_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,minus_minus(nat,J),aa(nat,nat,suc,I2))))
     => ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,I2,J))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_5284_and__not__num_Oelims,axiom,
    ! [X: num,Xa: num,Y: option(num)] :
      ( ( bit_and_not_num(X,Xa) = Y )
     => ( ( ( X = one2 )
         => ( ( Xa = one2 )
           => ( Y != none(num) ) ) )
       => ( ( ( X = one2 )
           => ( ? [N2: num] : Xa = aa(num,num,bit0,N2)
             => ( Y != aa(num,option(num),some(num),one2) ) ) )
         => ( ( ( X = one2 )
             => ( ? [N2: num] : Xa = aa(num,num,bit1,N2)
               => ( Y != none(num) ) ) )
           => ( ! [M5: num] :
                  ( ( X = aa(num,num,bit0,M5) )
                 => ( ( Xa = one2 )
                   => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M5)) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ! [N2: num] :
                        ( ( Xa = aa(num,num,bit0,N2) )
                       => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N2)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N2: num] :
                          ( ( Xa = aa(num,num,bit1,N2) )
                         => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N2)) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa(num,num,bit1,M5) )
                       => ( ( Xa = one2 )
                         => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M5)) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ! [N2: num] :
                              ( ( Xa = aa(num,num,bit0,N2) )
                             => ( Y != case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_lt(num,option(num)),bit_and_not_num(M5,N2)) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N2: num] :
                                ( ( Xa = aa(num,num,bit1,N2) )
                               => ( Y != aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
tff(fact_5285_Bit__Operations_Otake__bit__num__code,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,M) = aa(product_prod(nat,num),option(num),product_case_prod(nat,num,option(num),aTP_Lamp_lz(nat,fun(num,option(num)))),aa(num,product_prod(nat,num),product_Pair(nat,num,N),M)) ).

% Bit_Operations.take_bit_num_code
tff(fact_5286_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ma(A,A)),X),aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert_remove
tff(fact_5287_None__eq__map__option__iff,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),X: option(B)] :
      ( ( none(A) = aa(option(B),option(A),map_option(B,A,F3),X) )
    <=> ( X = none(B) ) ) ).

% None_eq_map_option_iff
tff(fact_5288_map__option__is__None,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Opt: option(B)] :
      ( ( aa(option(B),option(A),map_option(B,A,F3),Opt) = none(A) )
    <=> ( Opt = none(B) ) ) ).

% map_option_is_None
tff(fact_5289_option_Omap__disc__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A2: option(A)] :
      ( ( aa(option(A),option(B),map_option(A,B,F3),A2) = none(B) )
    <=> ( A2 = none(A) ) ) ).

% option.map_disc_iff
tff(fact_5290_remove1__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [X: B,F3: fun(B,A),Xs: list(B)] : remove1(B,X,aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),Xs)) = Xs ) ).

% remove1_insort_key
tff(fact_5291_length__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),X: B,Xs: list(B)] : aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),Xs)) = aa(nat,nat,suc,aa(list(B),nat,size_size(list(B)),Xs)) ) ).

% length_insort
tff(fact_5292_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ma(A,A)),X),aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_insert
tff(fact_5293_option_Osimps_I8_J,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B)] : aa(option(A),option(B),map_option(A,B,F3),none(A)) = none(B) ).

% option.simps(8)
tff(fact_5294_insort__key__left__comm,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),X: B,Y: B,Xs: list(B)] :
          ( ( aa(B,A,F3,X) != aa(B,A,F3,Y) )
         => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),Y),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),Y),Xs)) ) ) ) ).

% insort_key_left_comm
tff(fact_5295_num_Ocase__distrib,axiom,
    ! [A: $tType,B: $tType,H: fun(A,B),F1: A,F2: fun(num,A),F32: fun(num,A),Num: num] : aa(A,B,H,case_num(A,F1,F2,F32,Num)) = case_num(B,aa(A,B,H,F1),aa(fun(num,A),fun(num,B),aTP_Lamp_mb(fun(A,B),fun(fun(num,A),fun(num,B)),H),F2),aa(fun(num,A),fun(num,B),aTP_Lamp_mb(fun(A,B),fun(fun(num,A),fun(num,B)),H),F32),Num) ).

% num.case_distrib
tff(fact_5296_insort__left__comm,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ma(A,A)),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ma(A,A)),Y),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ma(A,A)),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ma(A,A)),X),Xs)) ) ).

% insort_left_comm
tff(fact_5297_and__not__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa(num,num,bit0,M),aa(num,num,bit0,N)) = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M,N)) ).

% and_not_num.simps(5)
tff(fact_5298_set__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),X: B,Xs: list(B)] : aa(list(B),set(B),set2(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),Xs)) = aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),aa(list(B),set(B),set2(B),Xs)) ) ).

% set_insort_key
tff(fact_5299_distinct__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),X: B,Xs: list(B)] :
          ( distinct(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),Xs))
        <=> ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),aa(list(B),set(B),set2(B),Xs)))
            & distinct(B,Xs) ) ) ) ).

% distinct_insort
tff(fact_5300_verit__eq__simplify_I17_J,axiom,
    ! [A: $tType,F1: A,F2: fun(num,A),F32: fun(num,A),X22: num] : case_num(A,F1,F2,F32,aa(num,num,bit0,X22)) = aa(num,A,F2,X22) ).

% verit_eq_simplify(17)
tff(fact_5301_verit__eq__simplify_I16_J,axiom,
    ! [A: $tType,F1: A,F2: fun(num,A),F32: fun(num,A)] : case_num(A,F1,F2,F32,one2) = F1 ).

% verit_eq_simplify(16)
tff(fact_5302_and__not__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa(num,num,bit1,M),aa(num,num,bit1,N)) = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M,N)) ).

% and_not_num.simps(9)
tff(fact_5303_and__not__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa(num,num,bit0,M),aa(num,num,bit1,N)) = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M,N)) ).

% and_not_num.simps(6)
tff(fact_5304_map__option__case,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Y: option(B)] : aa(option(B),option(A),map_option(B,A,F3),Y) = case_option(option(A),B,none(A),aTP_Lamp_mc(fun(B,A),fun(B,option(A)),F3),Y) ).

% map_option_case
tff(fact_5305_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ma(A,A)),X),aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set.fold_insort_key.remove
tff(fact_5306_and__not__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option(num)] :
      ( ( bit_and_not_num(X,Xa) = Y )
     => ( pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),product_Pair(num,num,X),Xa)))
       => ( ( ( X = one2 )
           => ( ( Xa = one2 )
             => ( ( Y = none(num) )
               => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),product_Pair(num,num,one2),one2))) ) ) )
         => ( ( ( X = one2 )
             => ! [N2: num] :
                  ( ( Xa = aa(num,num,bit0,N2) )
                 => ( ( Y = aa(num,option(num),some(num),one2) )
                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit0,N2)))) ) ) )
           => ( ( ( X = one2 )
               => ! [N2: num] :
                    ( ( Xa = aa(num,num,bit1,N2) )
                   => ( ( Y = none(num) )
                     => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit1,N2)))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ( ( Xa = one2 )
                     => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M5)) )
                       => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M5)),one2))) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N2: num] :
                          ( ( Xa = aa(num,num,bit0,N2) )
                         => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N2)) )
                           => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M5)),aa(num,num,bit0,N2)))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa(num,num,bit0,M5) )
                       => ! [N2: num] :
                            ( ( Xa = aa(num,num,bit1,N2) )
                           => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N2)) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M5)),aa(num,num,bit1,N2)))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ( ( Xa = one2 )
                           => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M5)) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M5)),one2))) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N2: num] :
                                ( ( Xa = aa(num,num,bit0,N2) )
                               => ( ( Y = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_lt(num,option(num)),bit_and_not_num(M5,N2)) )
                                 => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M5)),aa(num,num,bit0,N2)))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa(num,num,bit1,M5) )
                             => ! [N2: num] :
                                  ( ( Xa = aa(num,num,bit1,N2) )
                                 => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),bit_and_not_num(M5,N2)) )
                                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_and_not_num_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M5)),aa(num,num,bit1,N2)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
tff(fact_5307_and__num_Oelims,axiom,
    ! [X: num,Xa: num,Y: option(num)] :
      ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,X),Xa) = Y )
     => ( ( ( X = one2 )
         => ( ( Xa = one2 )
           => ( Y != aa(num,option(num),some(num),one2) ) ) )
       => ( ( ( X = one2 )
           => ( ? [N2: num] : Xa = aa(num,num,bit0,N2)
             => ( Y != none(num) ) ) )
         => ( ( ( X = one2 )
             => ( ? [N2: num] : Xa = aa(num,num,bit1,N2)
               => ( Y != aa(num,option(num),some(num),one2) ) ) )
           => ( ( ? [M5: num] : X = aa(num,num,bit0,M5)
               => ( ( Xa = one2 )
                 => ( Y != none(num) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ! [N2: num] :
                        ( ( Xa = aa(num,num,bit0,N2) )
                       => ( Y != aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M5),N2)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N2: num] :
                          ( ( Xa = aa(num,num,bit1,N2) )
                         => ( Y != aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M5),N2)) ) ) )
                 => ( ( ? [M5: num] : X = aa(num,num,bit1,M5)
                     => ( ( Xa = one2 )
                       => ( Y != aa(num,option(num),some(num),one2) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ! [N2: num] :
                              ( ( Xa = aa(num,num,bit0,N2) )
                             => ( Y != aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M5),N2)) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N2: num] :
                                ( ( Xa = aa(num,num,bit1,N2) )
                               => ( Y != case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_lt(num,option(num)),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M5),N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
tff(fact_5308_xor__num_Oelims,axiom,
    ! [X: num,Xa: num,Y: option(num)] :
      ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,X),Xa) = Y )
     => ( ( ( X = one2 )
         => ( ( Xa = one2 )
           => ( Y != none(num) ) ) )
       => ( ( ( X = one2 )
           => ! [N2: num] :
                ( ( Xa = aa(num,num,bit0,N2) )
               => ( Y != aa(num,option(num),some(num),aa(num,num,bit1,N2)) ) ) )
         => ( ( ( X = one2 )
             => ! [N2: num] :
                  ( ( Xa = aa(num,num,bit1,N2) )
                 => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,N2)) ) ) )
           => ( ! [M5: num] :
                  ( ( X = aa(num,num,bit0,M5) )
                 => ( ( Xa = one2 )
                   => ( Y != aa(num,option(num),some(num),aa(num,num,bit1,M5)) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ! [N2: num] :
                        ( ( Xa = aa(num,num,bit0,N2) )
                       => ( Y != aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M5),N2)) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N2: num] :
                          ( ( Xa = aa(num,num,bit1,N2) )
                         => ( Y != aa(num,option(num),some(num),case_option(num,num,one2,bit1,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M5),N2))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa(num,num,bit1,M5) )
                       => ( ( Xa = one2 )
                         => ( Y != aa(num,option(num),some(num),aa(num,num,bit0,M5)) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ! [N2: num] :
                              ( ( Xa = aa(num,num,bit0,N2) )
                             => ( Y != aa(num,option(num),some(num),case_option(num,num,one2,bit1,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M5),N2))) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N2: num] :
                                ( ( Xa = aa(num,num,bit1,N2) )
                               => ( Y != aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M5),N2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
tff(fact_5309_and__num_Osimps_I1_J,axiom,
    aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,one2),one2) = aa(num,option(num),some(num),one2) ).

% and_num.simps(1)
tff(fact_5310_xor__num_Osimps_I1_J,axiom,
    aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,one2),one2) = none(num) ).

% xor_num.simps(1)
tff(fact_5311_and__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,aa(num,num,bit0,M)),aa(num,num,bit0,N)) = aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M),N)) ).

% and_num.simps(5)
tff(fact_5312_xor__num_Osimps_I5_J,axiom,
    ! [M: num,N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,aa(num,num,bit0,M)),aa(num,num,bit0,N)) = aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M),N)) ).

% xor_num.simps(5)
tff(fact_5313_and__num_Osimps_I7_J,axiom,
    ! [M: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,aa(num,num,bit1,M)),one2) = aa(num,option(num),some(num),one2) ).

% and_num.simps(7)
tff(fact_5314_and__num_Osimps_I3_J,axiom,
    ! [N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,one2),aa(num,num,bit1,N)) = aa(num,option(num),some(num),one2) ).

% and_num.simps(3)
tff(fact_5315_and__num_Osimps_I4_J,axiom,
    ! [M: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,aa(num,num,bit0,M)),one2) = none(num) ).

% and_num.simps(4)
tff(fact_5316_and__num_Osimps_I2_J,axiom,
    ! [N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,one2),aa(num,num,bit0,N)) = none(num) ).

% and_num.simps(2)
tff(fact_5317_and__num__eq__Some__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num,Q3: num] :
          ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M),N) = aa(num,option(num),some(num),Q3) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),Q3) ) ) ) ).

% and_num_eq_Some_iff
tff(fact_5318_xor__num__eq__Some__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num,Q3: num] :
          ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M),N) = aa(num,option(num),some(num),Q3) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),Q3) ) ) ) ).

% xor_num_eq_Some_iff
tff(fact_5319_xor__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,aa(num,num,bit1,M)),aa(num,num,bit1,N)) = aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M),N)) ).

% xor_num.simps(9)
tff(fact_5320_and__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,aa(num,num,bit0,M)),aa(num,num,bit1,N)) = aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M),N)) ).

% and_num.simps(6)
tff(fact_5321_and__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,aa(num,num,bit1,M)),aa(num,num,bit0,N)) = aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M),N)) ).

% and_num.simps(8)
tff(fact_5322_xor__num_Osimps_I7_J,axiom,
    ! [M: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,aa(num,num,bit1,M)),one2) = aa(num,option(num),some(num),aa(num,num,bit0,M)) ).

% xor_num.simps(7)
tff(fact_5323_xor__num_Osimps_I4_J,axiom,
    ! [M: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,aa(num,num,bit0,M)),one2) = aa(num,option(num),some(num),aa(num,num,bit1,M)) ).

% xor_num.simps(4)
tff(fact_5324_xor__num_Osimps_I3_J,axiom,
    ! [N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,one2),aa(num,num,bit1,N)) = aa(num,option(num),some(num),aa(num,num,bit0,N)) ).

% xor_num.simps(3)
tff(fact_5325_xor__num_Osimps_I2_J,axiom,
    ! [N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,one2),aa(num,num,bit0,N)) = aa(num,option(num),some(num),aa(num,num,bit1,N)) ).

% xor_num.simps(2)
tff(fact_5326_and__num__eq__None__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num] :
          ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M),N) = none(num) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).

% and_num_eq_None_iff
tff(fact_5327_xor__num__eq__None__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num] :
          ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M),N) = none(num) )
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).

% xor_num_eq_None_iff
tff(fact_5328_numeral__and__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = case_option(A,num,zero_zero(A),numeral_numeral(A),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M),N)) ) ).

% numeral_and_num
tff(fact_5329_numeral__xor__num,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = case_option(A,num,zero_zero(A),numeral_numeral(A),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M),N)) ) ).

% numeral_xor_num
tff(fact_5330_and__num_Osimps_I9_J,axiom,
    ! [M: num,N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,aa(num,num,bit1,M)),aa(num,num,bit1,N)) = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_lt(num,option(num)),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M),N)) ).

% and_num.simps(9)
tff(fact_5331_xor__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,aa(num,num,bit1,M)),aa(num,num,bit0,N)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M),N))) ).

% xor_num.simps(8)
tff(fact_5332_xor__num_Osimps_I6_J,axiom,
    ! [M: num,N: num] : aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,aa(num,num,bit0,M)),aa(num,num,bit1,N)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M),N))) ).

% xor_num.simps(6)
tff(fact_5333_and__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option(num)] :
      ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,X),Xa) = Y )
     => ( pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),product_Pair(num,num,X),Xa)))
       => ( ( ( X = one2 )
           => ( ( Xa = one2 )
             => ( ( Y = aa(num,option(num),some(num),one2) )
               => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),product_Pair(num,num,one2),one2))) ) ) )
         => ( ( ( X = one2 )
             => ! [N2: num] :
                  ( ( Xa = aa(num,num,bit0,N2) )
                 => ( ( Y = none(num) )
                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit0,N2)))) ) ) )
           => ( ( ( X = one2 )
               => ! [N2: num] :
                    ( ( Xa = aa(num,num,bit1,N2) )
                   => ( ( Y = aa(num,option(num),some(num),one2) )
                     => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit1,N2)))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ( ( Xa = one2 )
                     => ( ( Y = none(num) )
                       => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M5)),one2))) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N2: num] :
                          ( ( Xa = aa(num,num,bit0,N2) )
                         => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M5),N2)) )
                           => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M5)),aa(num,num,bit0,N2)))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa(num,num,bit0,M5) )
                       => ! [N2: num] :
                            ( ( Xa = aa(num,num,bit1,N2) )
                           => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M5),N2)) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M5)),aa(num,num,bit1,N2)))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ( ( Xa = one2 )
                           => ( ( Y = aa(num,option(num),some(num),one2) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M5)),one2))) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N2: num] :
                                ( ( Xa = aa(num,num,bit0,N2) )
                               => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M5),N2)) )
                                 => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M5)),aa(num,num,bit0,N2)))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa(num,num,bit1,M5) )
                             => ! [N2: num] :
                                  ( ( Xa = aa(num,num,bit1,N2) )
                                 => ( ( Y = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_lt(num,option(num)),aa(num,option(num),aa(num,fun(num,option(num)),bit_un7362597486090784418nd_num,M5),N2)) )
                                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un4731106466462545111um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M5)),aa(num,num,bit1,N2)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
tff(fact_5334_xor__num_Opelims,axiom,
    ! [X: num,Xa: num,Y: option(num)] :
      ( ( aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,X),Xa) = Y )
     => ( pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),product_Pair(num,num,X),Xa)))
       => ( ( ( X = one2 )
           => ( ( Xa = one2 )
             => ( ( Y = none(num) )
               => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),product_Pair(num,num,one2),one2))) ) ) )
         => ( ( ( X = one2 )
             => ! [N2: num] :
                  ( ( Xa = aa(num,num,bit0,N2) )
                 => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit1,N2)) )
                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit0,N2)))) ) ) )
           => ( ( ( X = one2 )
               => ! [N2: num] :
                    ( ( Xa = aa(num,num,bit1,N2) )
                   => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,N2)) )
                     => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),product_Pair(num,num,one2),aa(num,num,bit1,N2)))) ) ) )
             => ( ! [M5: num] :
                    ( ( X = aa(num,num,bit0,M5) )
                   => ( ( Xa = one2 )
                     => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit1,M5)) )
                       => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M5)),one2))) ) ) )
               => ( ! [M5: num] :
                      ( ( X = aa(num,num,bit0,M5) )
                     => ! [N2: num] :
                          ( ( Xa = aa(num,num,bit0,N2) )
                         => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M5),N2)) )
                           => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M5)),aa(num,num,bit0,N2)))) ) ) )
                 => ( ! [M5: num] :
                        ( ( X = aa(num,num,bit0,M5) )
                       => ! [N2: num] :
                            ( ( Xa = aa(num,num,bit1,N2) )
                           => ( ( Y = aa(num,option(num),some(num),case_option(num,num,one2,bit1,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M5),N2))) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit0,M5)),aa(num,num,bit1,N2)))) ) ) )
                   => ( ! [M5: num] :
                          ( ( X = aa(num,num,bit1,M5) )
                         => ( ( Xa = one2 )
                           => ( ( Y = aa(num,option(num),some(num),aa(num,num,bit0,M5)) )
                             => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M5)),one2))) ) ) )
                     => ( ! [M5: num] :
                            ( ( X = aa(num,num,bit1,M5) )
                           => ! [N2: num] :
                                ( ( Xa = aa(num,num,bit0,N2) )
                               => ( ( Y = aa(num,option(num),some(num),case_option(num,num,one2,bit1,aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M5),N2))) )
                                 => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M5)),aa(num,num,bit0,N2)))) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X = aa(num,num,bit1,M5) )
                             => ! [N2: num] :
                                  ( ( Xa = aa(num,num,bit1,N2) )
                                 => ( ( Y = aa(option(num),option(num),map_option(num,num,bit0),aa(num,option(num),aa(num,fun(num,option(num)),bit_un2480387367778600638or_num,M5),N2)) )
                                   => ~ pp(aa(product_prod(num,num),bool,accp(product_prod(num,num),bit_un2901131394128224187um_rel),aa(num,product_prod(num,num),product_Pair(num,num,aa(num,num,bit1,M5)),aa(num,num,bit1,N2)))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
tff(fact_5335_xor__num__dict,axiom,
    bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).

% xor_num_dict
tff(fact_5336_and__num__rel__dict,axiom,
    bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).

% and_num_rel_dict
tff(fact_5337_xor__num__rel__dict,axiom,
    bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).

% xor_num_rel_dict
tff(fact_5338_and__num__dict,axiom,
    bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).

% and_num_dict
tff(fact_5339_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(nat,nat,minus_minus(nat,J),I2)))
     => ( aa(nat,nat,nth(nat,aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,I2,J))),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_5340_pow_Osimps_I3_J,axiom,
    ! [X: num,Y: num] : pow(X,aa(num,num,bit1,Y)) = aa(num,num,aa(num,fun(num,num),times_times(num),sqr(pow(X,Y))),X) ).

% pow.simps(3)
tff(fact_5341_in__set__product__lists__length,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),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_5342_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : pp(aa(set(nat),bool,finite_finite(nat),set_or3652927894154168847AtMost(nat,L,U))) ).

% finite_greaterThanAtMost
tff(fact_5343_greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,L: A,U: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),set_or3652927894154168847AtMost(A,L,U)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),I2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),I2),U)) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_5344_greaterThanAtMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),K))
         => ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanAtMost_empty
tff(fact_5345_greaterThanAtMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) ) ) ).

% greaterThanAtMost_empty_iff
tff(fact_5346_greaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K,L) )
        <=> ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),L)) ) ) ).

% greaterThanAtMost_empty_iff2
tff(fact_5347_infinite__Ioc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),set_or3652927894154168847AtMost(A,A2,B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2)) ) ) ).

% infinite_Ioc_iff
tff(fact_5348_cSup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,Y,X)) = X ) ) ) ).

% cSup_greaterThanAtMost
tff(fact_5349_Sup__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Sup_Sup(A),set_or3652927894154168847AtMost(A,X,Y)) = Y ) ) ) ).

% Sup_greaterThanAtMost
tff(fact_5350_cInf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & dense_linorder(A) )
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,Y,X)) = Y ) ) ) ).

% cInf_greaterThanAtMost
tff(fact_5351_Inf__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( aa(set(A),A,complete_Inf_Inf(A),set_or3652927894154168847AtMost(A,X,Y)) = X ) ) ) ).

% Inf_greaterThanAtMost
tff(fact_5352_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : aa(set(nat),nat,finite_card(nat),set_or3652927894154168847AtMost(nat,L,U)) = aa(nat,nat,minus_minus(nat,U),L) ).

% card_greaterThanAtMost
tff(fact_5353_Ioc__inj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( set_or3652927894154168847AtMost(A,A2,B2) = set_or3652927894154168847AtMost(A,C2,D2) )
        <=> ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D2),C2)) )
            | ( ( A2 = C2 )
              & ( B2 = D2 ) ) ) ) ) ).

% Ioc_inj
tff(fact_5354_sqr_Osimps_I2_J,axiom,
    ! [N: num] : sqr(aa(num,num,bit0,N)) = aa(num,num,bit0,aa(num,num,bit0,sqr(N))) ).

% sqr.simps(2)
tff(fact_5355_sqr_Osimps_I1_J,axiom,
    sqr(one2) = one2 ).

% sqr.simps(1)
tff(fact_5356_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_5357_Ioc__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            | ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% Ioc_subset_iff
tff(fact_5358_infinite__Ioc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ~ pp(aa(set(A),bool,finite_finite(A),set_or3652927894154168847AtMost(A,A2,B2))) ) ) ).

% infinite_Ioc
tff(fact_5359_sqr__conv__mult,axiom,
    ! [X: num] : sqr(X) = aa(num,num,aa(num,fun(num,num),times_times(num),X),X) ).

% sqr_conv_mult
tff(fact_5360_numeral__sqr,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num] : aa(num,A,numeral_numeral(A),sqr(K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),K)) ) ).

% numeral_sqr
tff(fact_5361_sum_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or3652927894154168847AtMost(nat,M,N))) ) ) ) ).

% sum.head
tff(fact_5362_prod_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or3652927894154168847AtMost(nat,M,N))) ) ) ) ).

% prod.head
tff(fact_5363_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_5364_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),D2)) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_5365_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2)) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_5366_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] : set_or3652927894154168847AtMost(A,A2,B2) = aa(set(A),set(A),minus_minus(set(A),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))) ) ).

% greaterThanAtMost_eq_atLeastAtMost_diff
tff(fact_5367_distinct__product__lists,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ! [X3: list(A)] :
          ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),X3),aa(list(list(A)),set(list(A)),set2(list(A)),Xss)))
         => distinct(A,X3) )
     => distinct(list(A),product_lists(A,Xss)) ) ).

% distinct_product_lists
tff(fact_5368_pow_Osimps_I2_J,axiom,
    ! [X: num,Y: num] : pow(X,aa(num,num,bit0,Y)) = sqr(pow(X,Y)) ).

% pow.simps(2)
tff(fact_5369_sqr_Osimps_I3_J,axiom,
    ! [N: num] : sqr(aa(num,num,bit1,N)) = aa(num,num,bit1,aa(num,num,bit0,aa(num,num,aa(num,fun(num,num),plus_plus(num),sqr(N)),N))) ).

% sqr.simps(3)
tff(fact_5370_sorted__list__of__set__def,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linord4507533701916653071of_set(A) = linord144544945434240204of_set(A,A,aTP_Lamp_ma(A,A)) ) ) ).

% sorted_list_of_set_def
tff(fact_5371_integer__of__num__triv_I2_J,axiom,
    code_integer_of_num(aa(num,num,bit0,one2)) = aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)) ).

% integer_of_num_triv(2)
tff(fact_5372_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,bool),aTP_Lamp_md(list(A),fun(set(nat),fun(A,bool)),Xs),I5)) ).

% set_nths
tff(fact_5373_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : pp(aa(set(int),bool,finite_finite(int),set_or3652927894154168847AtMost(int,L,U))) ).

% finite_greaterThanAtMost_int
tff(fact_5374_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : aa(set(int),nat,finite_card(int),set_or3652927894154168847AtMost(int,L,U)) = nat2(aa(int,int,minus_minus(int,U),L)) ).

% card_greaterThanAtMost_int
tff(fact_5375_in__set__nthsD,axiom,
    ! [A: $tType,X: A,Xs: list(A),I5: set(nat)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),nths(A,Xs,I5))))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs))) ) ).

% in_set_nthsD
tff(fact_5376_notin__set__nthsI,axiom,
    ! [A: $tType,X: A,Xs: list(A),I5: set(nat)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),nths(A,Xs,I5)))) ) ).

% notin_set_nthsI
tff(fact_5377_distinct__nthsI,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( distinct(A,Xs)
     => distinct(A,nths(A,Xs,I5)) ) ).

% distinct_nthsI
tff(fact_5378_set__nths__subset,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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_5379_nths__all,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( ! [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),I4),I5)) )
     => ( nths(A,Xs,I5) = Xs ) ) ).

% nths_all
tff(fact_5380_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_5381_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,bool),aTP_Lamp_me(list(A),fun(set(nat),fun(nat,bool)),Xs),I5))) ).

% length_nths
tff(fact_5382_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one2) = one_one(code_integer) ).

% integer_of_num_triv(1)
tff(fact_5383_integer__of__num_I2_J,axiom,
    ! [N: num] : code_integer_of_num(aa(num,num,bit0,N)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),code_integer_of_num(N)),code_integer_of_num(N)) ).

% integer_of_num(2)
tff(fact_5384_distinct__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(list(A),Xs)
     => ( ! [Ys3: list(A)] :
            ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Ys3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
           => distinct(A,Ys3) )
       => ( ! [Ys3: list(A),Zs2: list(A)] :
              ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Ys3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
             => ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs2),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
               => ( ( Ys3 != 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),Ys3)),aa(list(A),set(A),set2(A),Zs2)) = bot_bot(set(A)) ) ) ) )
         => distinct(A,concat(A,Xs)) ) ) ) ).

% distinct_concat
tff(fact_5385_Rats__eq__int__div__nat,axiom,
    field_char_0_Rats(real) = collect(real,aTP_Lamp_mf(real,bool)) ).

% Rats_eq_int_div_nat
tff(fact_5386_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Y: nat,X: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C2),Y))
       => ( aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_mg(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,minus_minus(nat,X),C2),aa(nat,nat,minus_minus(nat,Y),C2)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),C2),Y))
       => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
           => ( aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_mg(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),bot_bot(set(nat))) ) )
          & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X),Y))
           => ( aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_mg(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,X,Y)) = bot_bot(set(nat)) ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_5387_inf_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2)) ) ) ) ).

% inf.bounded_iff
tff(fact_5388_le__inf__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z)) ) ) ) ).

% le_inf_iff
tff(fact_5389_Int__subset__iff,axiom,
    ! [A: $tType,C5: set(A),A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C5),A3))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C5),B3)) ) ) ).

% Int_subset_iff
tff(fact_5390_bij__betw__Suc,axiom,
    ! [M6: set(nat),N3: set(nat)] :
      ( bij_betw(nat,nat,suc,M6,N3)
    <=> ( aa(set(nat),set(nat),image(nat,nat,suc),M6) = N3 ) ) ).

% bij_betw_Suc
tff(fact_5391_Rats__abs__iff,axiom,
    ! [X: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,abs_abs(real),X)),field_char_0_Rats(real)))
    <=> pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X),field_char_0_Rats(real))) ) ).

% Rats_abs_iff
tff(fact_5392_image__add__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A))),S2) = S2 ) ).

% image_add_0
tff(fact_5393_image__add__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I2: A,J: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),K)),set_or1337092689740270186AtMost(A,I2,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastAtMost
tff(fact_5394_image__diff__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D2: A,A2: A,B2: A] : aa(set(A),set(A),image(A,A,minus_minus(A,D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,D2),B2),aa(A,A,minus_minus(A,D2),A2)) ) ).

% image_diff_atLeastAtMost
tff(fact_5395_image__uminus__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A] : aa(set(A),set(A),image(A,A,uminus_uminus(A)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_atLeastAtMost
tff(fact_5396_image__add__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I2: A,J: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),K)),set_or7035219750837199246ssThan(A,I2,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastLessThan
tff(fact_5397_image__add__atMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [C2: A,A2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),C2)),aa(A,set(A),set_ord_atMost(A),A2)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)) ) ).

% image_add_atMost
tff(fact_5398_bij__betw__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A3: set(A),B3: set(A)] :
          ( bij_betw(A,A,aa(A,fun(A,A),plus_plus(A),A2),A3,B3)
        <=> ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),A3) = B3 ) ) ) ).

% bij_betw_add
tff(fact_5399_image__add__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [C2: A,A2: A,B2: A] : aa(set(A),set(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_5400_image__uminus__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A] : aa(set(A),set(A),image(A,A,uminus_uminus(A)),set_or5935395276787703475ssThan(A,X,Y)) = set_or5935395276787703475ssThan(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_greaterThanLessThan
tff(fact_5401_image__Suc__atLeastAtMost,axiom,
    ! [I2: nat,J: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_or1337092689740270186AtMost(nat,I2,J)) = set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,I2),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_5402_image__Suc__atLeastLessThan,axiom,
    ! [I2: nat,J: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_or7035219750837199246ssThan(nat,I2,J)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,I2),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_5403_bij__betw__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N3: set(nat),A3: set(A)] :
          ( bij_betw(nat,A,semiring_1_of_nat(A),N3,A3)
        <=> ( aa(set(nat),set(A),image(nat,A,semiring_1_of_nat(A)),N3) = A3 ) ) ) ).

% bij_betw_of_nat
tff(fact_5404_image__add__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I2: A,J: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_mh(A,fun(A,A),K)),set_or1337092689740270186AtMost(A,I2,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastAtMost'
tff(fact_5405_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D2: A,A2: A,B2: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_mi(A,fun(A,A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,A2),D2),aa(A,A,minus_minus(A,B2),D2)) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_5406_image__add__atLeastLessThan_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I2: A,J: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_mh(A,fun(A,A),K)),set_or7035219750837199246ssThan(A,I2,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I2),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastLessThan'
tff(fact_5407_image__minus__const__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A,B2: A] : aa(set(A),set(A),image(A,A,minus_minus(A,C2)),set_or3652927894154168847AtMost(A,A2,B2)) = set_or7035219750837199246ssThan(A,aa(A,A,minus_minus(A,C2),B2),aa(A,A,minus_minus(A,C2),A2)) ) ).

% image_minus_const_greaterThanAtMost
tff(fact_5408_image__diff__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A,B2: A] : aa(set(A),set(A),image(A,A,minus_minus(A,C2)),set_or7035219750837199246ssThan(A,A2,B2)) = set_or3652927894154168847AtMost(A,aa(A,A,minus_minus(A,C2),B2),aa(A,A,minus_minus(A,C2),A2)) ) ).

% image_diff_atLeastLessThan
tff(fact_5409_image__uminus__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A] : aa(set(A),set(A),image(A,A,uminus_uminus(A)),set_or3652927894154168847AtMost(A,X,Y)) = set_or7035219750837199246ssThan(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_greaterThanAtMost
tff(fact_5410_image__uminus__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A] : aa(set(A),set(A),image(A,A,uminus_uminus(A)),set_or7035219750837199246ssThan(A,X,Y)) = set_or3652927894154168847AtMost(A,aa(A,A,uminus_uminus(A),Y),aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_atLeastLessThan
tff(fact_5411_INF__eq__bot__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A3: set(B)] :
          ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3)) = bot_bot(A) )
        <=> ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),bot_bot(A)),X4))
             => ? [Xa4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Xa4),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,Xa4)),X4)) ) ) ) ) ).

% INF_eq_bot_iff
tff(fact_5412_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D2))
         => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),D2),B2)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_5413_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),D2))
         => ( aa(set(A),set(A),image(A,A,aTP_Lamp_mj(A,fun(A,A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,divide_divide(A,A2,D2),divide_divide(A,B2,D2)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_5414_SUP__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),B3: set(C),F3: fun(B,A),G: fun(C,A)] :
          ( ! [I4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),A3))
             => ? [X2: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X2),B3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I4)),aa(C,A,G,X2))) ) )
         => ( ! [J2: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),J2),B3))
               => ? [X2: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A3))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,G,J2)),aa(B,A,F3,X2))) ) )
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3)) = aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,G),B3)) ) ) ) ) ).

% SUP_eq
tff(fact_5415_translation__Int,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S: set(A),T2: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),T2)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),S)),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),T2)) ) ).

% translation_Int
tff(fact_5416_inf_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2)) ) ) ).

% inf.strict_coboundedI2
tff(fact_5417_inf_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2)) ) ) ).

% inf.strict_coboundedI1
tff(fact_5418_inf_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_5419_inf_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2)) ) ) ) ).

% inf.strict_boundedE
tff(fact_5420_inf_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb4
tff(fact_5421_inf_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb3
tff(fact_5422_less__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,X: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),X)) ) ) ).

% less_infI2
tff(fact_5423_less__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),X)) ) ) ).

% less_infI1
tff(fact_5424_Rats__dense__in__real,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
     => ? [X3: real] :
          ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),field_char_0_Rats(real)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),X3))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),Y)) ) ) ).

% Rats_dense_in_real
tff(fact_5425_Rats__no__bot__less,axiom,
    ! [X: real] :
    ? [X3: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),field_char_0_Rats(real)))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),X)) ) ).

% Rats_no_bot_less
tff(fact_5426_Rats__divide,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),field_char_0_Rats(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field_char_0_Rats(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),divide_divide(A,A2,B2)),field_char_0_Rats(A))) ) ) ) ).

% Rats_divide
tff(fact_5427_Rats__power,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,N: nat] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),field_char_0_Rats(A)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,power_power(A,A2),N)),field_char_0_Rats(A))) ) ) ).

% Rats_power
tff(fact_5428_Rats__number__of,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [W: num] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(num,A,numeral_numeral(A),W)),field_char_0_Rats(A))) ) ).

% Rats_number_of
tff(fact_5429_Rats__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),one_one(A)),field_char_0_Rats(A))) ) ).

% Rats_1
tff(fact_5430_Rats__add,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),field_char_0_Rats(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),field_char_0_Rats(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),field_char_0_Rats(A))) ) ) ) ).

% Rats_add
tff(fact_5431_Setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A3: set(B)] : collect(A,aa(set(B),fun(A,bool),aTP_Lamp_mk(fun(B,A),fun(set(B),fun(A,bool)),F3),A3)) = aa(set(B),set(A),image(B,A,F3),A3) ).

% Setcompr_eq_image
tff(fact_5432_setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),P: fun(B,bool)] : collect(A,aa(fun(B,bool),fun(A,bool),aTP_Lamp_ml(fun(B,A),fun(fun(B,bool),fun(A,bool)),F3),P)) = aa(set(B),set(A),image(B,A,F3),collect(B,P)) ).

% setcompr_eq_image
tff(fact_5433_all__subset__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A3: set(B),P: fun(set(A),bool)] :
      ( ! [B9: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B9),aa(set(B),set(A),image(B,A,F3),A3)))
         => pp(aa(set(A),bool,P,B9)) )
    <=> ! [B9: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B9),A3))
         => pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F3),B9))) ) ) ).

% all_subset_image
tff(fact_5434_Int__mono,axiom,
    ! [A: $tType,A3: set(A),C5: set(A),B3: set(A),D4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),D4))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),C5),D4))) ) ) ).

% Int_mono
tff(fact_5435_Int__lower1,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),A3)) ).

% Int_lower1
tff(fact_5436_Int__lower2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),B3)) ).

% Int_lower2
tff(fact_5437_Int__absorb1,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = B3 ) ) ).

% Int_absorb1
tff(fact_5438_Int__absorb2,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3) = A3 ) ) ).

% Int_absorb2
tff(fact_5439_Int__greatest,axiom,
    ! [A: $tType,C5: set(A),A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C5),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C5),B3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ).

% Int_greatest
tff(fact_5440_Int__Collect__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),P: fun(A,bool),Q: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => ( pp(aa(A,bool,P,X3))
             => pp(aa(A,bool,Q,X3)) ) )
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),collect(A,P))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),B3),collect(A,Q)))) ) ) ).

% Int_Collect_mono
tff(fact_5441_image__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(A),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),A3)),aa(set(A),set(B),image(A,B,F3),B3))) ) ).

% image_mono
tff(fact_5442_image__subsetI,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F3: fun(A,B),B3: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F3,X3)),B3)) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),A3)),B3)) ) ).

% image_subsetI
tff(fact_5443_subset__imageE,axiom,
    ! [A: $tType,B: $tType,B3: set(A),F3: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(B),set(A),image(B,A,F3),A3)))
     => ~ ! [C6: set(B)] :
            ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C6),A3))
           => ( B3 != aa(set(B),set(A),image(B,A,F3),C6) ) ) ) ).

% subset_imageE
tff(fact_5444_image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A3: set(B),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F3),A3)),B3))
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,F3,X4)),B3)) ) ) ).

% image_subset_iff
tff(fact_5445_subset__image__iff,axiom,
    ! [A: $tType,B: $tType,B3: set(A),F3: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(B),set(A),image(B,A,F3),A3)))
    <=> ? [AA: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),AA),A3))
          & ( B3 = aa(set(B),set(A),image(B,A,F3),AA) ) ) ) ).

% subset_image_iff
tff(fact_5446_image__Int__subset,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A3: set(B),B3: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(B),set(A),image(B,A,F3),A3)),aa(set(B),set(A),image(B,A,F3),B3)))) ).

% image_Int_subset
tff(fact_5447_Rats__no__top__le,axiom,
    ! [X: real] :
    ? [X3: real] :
      ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),field_char_0_Rats(real)))
      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),X3)) ) ).

% Rats_no_top_le
tff(fact_5448_inf__sup__ord_I2_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y)) ) ).

% inf_sup_ord(2)
tff(fact_5449_inf__sup__ord_I1_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),X)) ) ).

% inf_sup_ord(1)
tff(fact_5450_inf__le1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),X)) ) ).

% inf_le1
tff(fact_5451_inf__le2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Y)) ) ).

% inf_le2
tff(fact_5452_le__infE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2)) ) ) ) ).

% le_infE
tff(fact_5453_le__infI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2))) ) ) ) ).

% le_infI
tff(fact_5454_inf__mono,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),aa(A,A,aa(A,fun(A,A),inf_inf(A),C2),D2))) ) ) ) ).

% inf_mono
tff(fact_5455_le__infI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),X)) ) ) ).

% le_infI1
tff(fact_5456_le__infI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,X: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),X)) ) ) ).

% le_infI2
tff(fact_5457_inf_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( A2 = aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) ) ) ) ).

% inf.orderE
tff(fact_5458_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) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% inf.orderI
tff(fact_5459_inf__unique,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [F3: fun(A,fun(A,A)),X: A,Y: A] :
          ( ! [X3: A,Y4: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X3),Y4)),X3))
         => ( ! [X3: A,Y4: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,X3),Y4)),Y4))
           => ( ! [X3: A,Y4: A,Z2: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y4))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z2))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),aa(A,A,aa(A,fun(A,A),F3,Y4),Z2))) ) )
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = aa(A,A,aa(A,fun(A,A),F3,X),Y) ) ) ) ) ) ).

% inf_unique
tff(fact_5460_le__iff__inf,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
        <=> ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = X ) ) ) ).

% le_iff_inf
tff(fact_5461_inf_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = A2 ) ) ) ).

% inf.absorb1
tff(fact_5462_inf_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2) = B2 ) ) ) ).

% inf.absorb2
tff(fact_5463_inf__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = X ) ) ) ).

% inf_absorb1
tff(fact_5464_inf__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = Y ) ) ) ).

% inf_absorb2
tff(fact_5465_inf_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2)) ) ) ) ).

% inf.boundedE
tff(fact_5466_inf_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),inf_inf(A),B2),C2))) ) ) ) ).

% inf.boundedI
tff(fact_5467_inf__greatest,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),inf_inf(A),Y),Z))) ) ) ) ).

% inf_greatest
tff(fact_5468_inf_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_5469_inf_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),A2)) ) ).

% inf.cobounded1
tff(fact_5470_inf_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),B2)) ) ).

% inf.cobounded2
tff(fact_5471_inf_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_5472_inf_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_5473_inf_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2)) ) ) ).

% inf.coboundedI1
tff(fact_5474_inf_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),B2)),C2)) ) ) ).

% inf.coboundedI2
tff(fact_5475_image__Pow__mono,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A3: set(B),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F3),A3)),B3))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image(set(B),set(A),image(B,A,F3)),pow2(B,A3))),pow2(A,B3))) ) ).

% image_Pow_mono
tff(fact_5476_image__Collect__subsetI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F3: fun(A,B),B3: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(A,bool,P,X3))
         => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F3,X3)),B3)) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),collect(A,P))),B3)) ) ).

% image_Collect_subsetI
tff(fact_5477_Union__Int__subset,axiom,
    ! [A: $tType,A3: set(set(A)),B3: set(set(A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),A3),B3))),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),A3)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),B3)))) ).

% Union_Int_subset
tff(fact_5478_Sup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B3: set(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3)))) ) ).

% Sup_inter_less_eq
tff(fact_5479_INF__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),B3: set(C),G: fun(C,A),F3: fun(B,A)] :
          ( ! [I4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),A3))
             => ? [X2: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X2),B3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,G,X2)),aa(B,A,F3,I4))) ) )
         => ( ! [J2: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),J2),B3))
               => ? [X2: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A3))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X2)),aa(C,A,G,J2))) ) )
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3)) = aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image(C,A,G),B3)) ) ) ) ) ).

% INF_eq
tff(fact_5480_all__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A3: set(B),P: fun(set(A),bool)] :
      ( ! [B9: set(A)] :
          ( ( pp(aa(set(A),bool,finite_finite(A),B9))
            & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B9),aa(set(B),set(A),image(B,A,F3),A3))) )
         => pp(aa(set(A),bool,P,B9)) )
    <=> ! [B9: set(B)] :
          ( ( pp(aa(set(B),bool,finite_finite(B),B9))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B9),A3)) )
         => pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F3),B9))) ) ) ).

% all_finite_subset_image
tff(fact_5481_ex__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A3: set(B),P: fun(set(A),bool)] :
      ( ? [B9: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),B9))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B9),aa(set(B),set(A),image(B,A,F3),A3)))
          & pp(aa(set(A),bool,P,B9)) )
    <=> ? [B9: set(B)] :
          ( pp(aa(set(B),bool,finite_finite(B),B9))
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B9),A3))
          & pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F3),B9))) ) ) ).

% ex_finite_subset_image
tff(fact_5482_finite__subset__image,axiom,
    ! [A: $tType,B: $tType,B3: set(A),F3: fun(B,A),A3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite(A),B3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(B),set(A),image(B,A,F3),A3)))
       => ? [C6: set(B)] :
            ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C6),A3))
            & pp(aa(set(B),bool,finite_finite(B),C6))
            & ( B3 = aa(set(B),set(A),image(B,A,F3),C6) ) ) ) ) ).

% finite_subset_image
tff(fact_5483_finite__surj,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),aa(set(A),set(B),image(A,B,F3),A3)))
       => pp(aa(set(B),bool,finite_finite(B),B3)) ) ) ).

% finite_surj
tff(fact_5484_zero__notin__Suc__image,axiom,
    ! [A3: set(nat)] : ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),A3))) ).

% zero_notin_Suc_image
tff(fact_5485_translation__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S: set(A),T2: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),aa(set(A),set(A),minus_minus(set(A),S),T2)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),S)),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),T2)) ) ).

% translation_diff
tff(fact_5486_image__diff__subset,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A3: set(B),B3: set(B)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),aa(set(B),set(A),image(B,A,F3),A3)),aa(set(B),set(A),image(B,A,F3),B3))),aa(set(B),set(A),image(B,A,F3),aa(set(B),set(B),minus_minus(set(B),A3),B3)))) ).

% image_diff_subset
tff(fact_5487_bij__betw__byWitness,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F10: fun(B,A),F3: fun(A,B),A8: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => ( aa(B,A,F10,aa(A,B,F3,X3)) = X3 ) )
     => ( ! [X3: B] :
            ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A8))
           => ( aa(A,B,F3,aa(B,A,F10,X3)) = X3 ) )
       => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),A3)),A8))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F10),A8)),A3))
           => bij_betw(A,B,F3,A3,A8) ) ) ) ) ).

% bij_betw_byWitness
tff(fact_5488_bij__betw__subset,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A3: set(A),A8: set(B),B3: set(A),B12: set(B)] :
      ( bij_betw(A,B,F3,A3,A8)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
       => ( ( aa(set(A),set(B),image(A,B,F3),B3) = B12 )
         => bij_betw(A,B,F3,B3,B12) ) ) ) ).

% bij_betw_subset
tff(fact_5489_translation__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,T2: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),aa(set(A),set(A),uminus_uminus(set(A)),T2)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),T2)) ) ).

% translation_Compl
tff(fact_5490_ivl__disj__int__two_I3_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(3)
tff(fact_5491_ivl__disj__int__two_I6_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(6)
tff(fact_5492_SUP__upper2,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I2: B,A3: set(B),U: A,F3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,I2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3)))) ) ) ) ).

% SUP_upper2
tff(fact_5493_SUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A3: set(B),U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3))),U))
        <=> ! [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X4)),U)) ) ) ) ).

% SUP_le_iff
tff(fact_5494_SUP__upper,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I2: B,A3: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I2)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3)))) ) ) ).

% SUP_upper
tff(fact_5495_SUP__mono_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),G: fun(B,A),A3: set(B)] :
          ( ! [X3: B] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G,X3)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,G),A3)))) ) ) ).

% SUP_mono'
tff(fact_5496_SUP__least,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),F3: fun(B,A),U: A] :
          ( ! [I4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I4)),U)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3))),U)) ) ) ).

% SUP_least
tff(fact_5497_SUP__mono,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),B3: set(C),F3: fun(B,A),G: fun(C,A)] :
          ( ! [N2: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),N2),A3))
             => ? [X2: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X2),B3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,N2)),aa(C,A,G,X2))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,G),B3)))) ) ) ).

% SUP_mono
tff(fact_5498_SUP__eqI,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),F3: fun(B,A),X: A] :
          ( ! [I4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I4)),X)) )
         => ( ! [Y4: A] :
                ( ! [I: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I)),Y4)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) )
           => ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3)) = X ) ) ) ) ).

% SUP_eqI
tff(fact_5499_less__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A2: A,F3: fun(B,A),A3: set(B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3))))
        <=> ? [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F3,X4))) ) ) ) ).

% less_SUP_iff
tff(fact_5500_SUP__lessD,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),A3: set(B),Y: A,I2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3))),Y))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,I2)),Y)) ) ) ) ).

% SUP_lessD
tff(fact_5501_INF__greatest,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),U: A,F3: fun(B,A)] :
          ( ! [I4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,I4))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3)))) ) ) ).

% INF_greatest
tff(fact_5502_le__INF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [U: A,F3: fun(B,A),A3: set(B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3))))
        <=> ! [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,X4))) ) ) ) ).

% le_INF_iff
tff(fact_5503_INF__lower2,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I2: B,A3: set(B),F3: fun(B,A),U: A] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I2)),U))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3))),U)) ) ) ) ).

% INF_lower2
tff(fact_5504_INF__mono_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),G: fun(B,A),A3: set(B)] :
          ( ! [X3: B] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G,X3)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,G),A3)))) ) ) ).

% INF_mono'
tff(fact_5505_INF__lower,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I2: B,A3: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3))),aa(B,A,F3,I2))) ) ) ).

% INF_lower
tff(fact_5506_INF__mono,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: set(B),A3: set(C),F3: fun(C,A),G: fun(B,A)] :
          ( ! [M5: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),M5),B3))
             => ? [X2: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X2),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(C,A,F3,X2)),aa(B,A,G,M5))) ) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image(C,A,F3),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,G),B3)))) ) ) ).

% INF_mono
tff(fact_5507_INF__eqI,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),X: A,F3: fun(B,A)] :
          ( ! [I4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),A3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(B,A,F3,I4))) )
         => ( ! [Y4: A] :
                ( ! [I: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),aa(B,A,F3,I))) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) )
           => ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3)) = X ) ) ) ) ).

% INF_eqI
tff(fact_5508_INF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A3: set(B),A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3))),A2))
        <=> ? [X4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X4)),A2)) ) ) ) ).

% INF_less_iff
tff(fact_5509_less__INF__D,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Y: A,F3: fun(B,A),A3: set(B),I2: B] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3))))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),aa(B,A,F3,I2))) ) ) ) ).

% less_INF_D
tff(fact_5510_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A3: set(A),F3: fun(nat,A),N: nat] :
      ( ( A3 = aa(set(nat),set(A),image(nat,A,F3),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ea(nat,fun(nat,bool)),N))) )
     => pp(aa(set(A),bool,finite_finite(A),A3)) ) ).

% nat_seg_image_imp_finite
tff(fact_5511_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
    <=> ? [N5: nat,F6: fun(nat,A)] : A3 = aa(set(nat),set(A),image(nat,A,F6),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ea(nat,fun(nat,bool)),N5))) ) ).

% finite_conv_nat_seg_image
tff(fact_5512_Ints__subset__Rats,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),ring_1_Ints(A)),field_char_0_Rats(A))) ) ).

% Ints_subset_Rats
tff(fact_5513_le__SUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [X: A,F3: fun(B,A),A3: set(B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3))))
        <=> ! [Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X))
             => ? [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),aa(B,A,F3,X4))) ) ) ) ) ).

% le_SUP_iff
tff(fact_5514_inf__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y) = bot_bot(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,uminus_uminus(A),Y))) ) ) ).

% inf_shunt
tff(fact_5515_INF__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A3: set(B),X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3))),X))
        <=> ! [Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y5))
             => ? [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X4)),Y5)) ) ) ) ) ).

% INF_le_iff
tff(fact_5516_cSUP__least,axiom,
    ! [B: $tType,A: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(B),F3: fun(B,A),M6: A] :
          ( ( A3 != bot_bot(set(B)) )
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),M6)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3))),M6)) ) ) ) ).

% cSUP_least
tff(fact_5517_SUP__eq__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I5: set(B),C2: A,F3: fun(B,A)] :
          ( ( I5 != bot_bot(set(B)) )
         => ( ! [I4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),I5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(B,A,F3,I4))) )
           => ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),I5)) = C2 )
            <=> ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),I5))
                 => ( aa(B,A,F3,X4) = C2 ) ) ) ) ) ) ).

% SUP_eq_iff
tff(fact_5518_cINF__greatest,axiom,
    ! [A: $tType,B: $tType] :
      ( condit1219197933456340205attice(A)
     => ! [A3: set(B),M: A,F3: fun(B,A)] :
          ( ( A3 != bot_bot(set(B)) )
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),aa(B,A,F3,X3))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3)))) ) ) ) ).

% cINF_greatest
tff(fact_5519_INF__eq__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [I5: set(B),F3: fun(B,A),C2: A] :
          ( ( I5 != bot_bot(set(B)) )
         => ( ! [I4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),I5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I4)),C2)) )
           => ( ( aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),I5)) = C2 )
            <=> ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),I5))
                 => ( aa(B,A,F3,X4) = C2 ) ) ) ) ) ) ).

% INF_eq_iff
tff(fact_5520_card__image__le,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(A),set(B),image(A,B,F3),A3))),aa(set(A),nat,finite_card(A),A3))) ) ).

% card_image_le
tff(fact_5521_ivl__disj__int__two_I7_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(7)
tff(fact_5522_ivl__disj__int__one_I4_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or1337092689740270186AtMost(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(4)
tff(fact_5523_Ioc__disjoint,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D2)) = bot_bot(set(A)) )
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D2),C2))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D2),A2)) ) ) ) ).

% Ioc_disjoint
tff(fact_5524_ivl__disj__int__one_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or7035219750837199246ssThan(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(2)
tff(fact_5525_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)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),B3))) ) ).

% disjoint_eq_subset_Compl
tff(fact_5526_ivl__disj__int__two_I4_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(4)
tff(fact_5527_ivl__disj__int__two_I5_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(5)
tff(fact_5528_ivl__disj__int__two_I8_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(8)
tff(fact_5529_ivl__disj__int__two_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(1)
tff(fact_5530_ivl__disj__int__one_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or5935395276787703475ssThan(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(1)
tff(fact_5531_ivl__disj__int__one_I3_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or3652927894154168847AtMost(A,L,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(3)
tff(fact_5532_SUP__subset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),B3: set(B),F3: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),B3))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G,X3))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,G),B3)))) ) ) ) ).

% SUP_subset_mono
tff(fact_5533_INF__superset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [B3: set(B),A3: set(B),F3: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),A3))
         => ( ! [X3: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),B3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G,X3))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,G),B3)))) ) ) ) ).

% INF_superset_mono
tff(fact_5534_sum_Ogroup,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(B),T6: set(C),G: fun(B,C),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S2))
         => ( pp(aa(set(C),bool,finite_finite(C),T6))
           => ( pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),aa(set(B),set(C),image(B,C,G),S2)),T6))
             => ( aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_mn(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),S2),G),H)),T6) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),S2) ) ) ) ) ) ).

% sum.group
tff(fact_5535_ivl__disj__int__two_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,M: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_two(2)
tff(fact_5536_prod_Ogroup,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [S2: set(B),T6: set(C),G: fun(B,C),H: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),S2))
         => ( pp(aa(set(C),bool,finite_finite(C),T6))
           => ( pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),aa(set(B),set(C),image(B,C,G),S2)),T6))
             => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_mo(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),S2),G),H)),T6) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),S2) ) ) ) ) ) ).

% prod.group
tff(fact_5537_prod_Ointer__restrict,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),G: fun(B,A),B3: set(B)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(set(B),fun(B,A),aTP_Lamp_mp(fun(B,A),fun(set(B),fun(B,A)),G),B3)),A3) ) ) ) ).

% prod.inter_restrict
tff(fact_5538_INF__le__SUP,axiom,
    ! [A: $tType,B: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(B),F3: fun(B,A)] :
          ( ( A3 != bot_bot(set(B)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3)))) ) ) ).

% INF_le_SUP
tff(fact_5539_surj__card__le,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(B),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),aa(set(A),set(B),image(A,B,F3),A3)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),B3)),aa(set(A),nat,finite_card(A),A3))) ) ) ).

% surj_card_le
tff(fact_5540_scaleR__image__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,X: A,Y: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
         => ( aa(set(A),set(A),image(A,A,real_V8093663219630862766scaleR(A,C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),X),aa(A,A,real_V8093663219630862766scaleR(A,C2),Y)) ) ) ) ).

% scaleR_image_atLeastAtMost
tff(fact_5541_Iio__Int__singleton,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,K: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),K))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),K)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),K))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),K)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) ) ) ) ) ).

% Iio_Int_singleton
tff(fact_5542_sum_OInt__Diff,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),G: fun(B,A),B3: set(B)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3))),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),minus_minus(set(B),A3),B3))) ) ) ) ).

% sum.Int_Diff
tff(fact_5543_image__Suc__lessThan,axiom,
    ! [N: nat] : aa(set(nat),set(nat),image(nat,nat,suc),aa(nat,set(nat),set_ord_lessThan(nat),N)) = set_or1337092689740270186AtMost(nat,one_one(nat),N) ).

% image_Suc_lessThan
tff(fact_5544_image__Suc__atMost,axiom,
    ! [N: nat] : aa(set(nat),set(nat),image(nat,nat,suc),aa(nat,set(nat),set_ord_atMost(nat),N)) = set_or1337092689740270186AtMost(nat,one_one(nat),aa(nat,nat,suc,N)) ).

% image_Suc_atMost
tff(fact_5545_prod_Omono__neutral__cong,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [T6: set(B),S2: set(B),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),T6))
         => ( pp(aa(set(B),bool,finite_finite(B),S2))
           => ( ! [I4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),aa(set(B),set(B),minus_minus(set(B),T6),S2)))
                 => ( aa(B,A,H,I4) = one_one(A) ) )
             => ( ! [I4: B] :
                    ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),aa(set(B),set(B),minus_minus(set(B),S2),T6)))
                   => ( aa(B,A,G,I4) = one_one(A) ) )
               => ( ! [X3: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),S2),T6)))
                     => ( aa(B,A,G,X3) = aa(B,A,H,X3) ) )
                 => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),S2) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),H),T6) ) ) ) ) ) ) ) ).

% prod.mono_neutral_cong
tff(fact_5546_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_5547_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_5548_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),aa(nat,set(nat),set_ord_lessThan(nat),N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_5549_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),aa(nat,set(nat),set_ord_atMost(nat),N))) ).

% atMost_Suc_eq_insert_0
tff(fact_5550_Rats__eq__int__div__int,axiom,
    field_char_0_Rats(real) = collect(real,aTP_Lamp_mq(real,bool)) ).

% Rats_eq_int_div_int
tff(fact_5551_sum_OIf__cases,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),P: fun(B,bool),H: fun(B,A),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aa(fun(B,A),fun(fun(B,A),fun(B,A)),aTP_Lamp_mr(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),P),H),G)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,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,groups7311177749621191930dd_sum(B,A,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))))) ) ) ) ).

% sum.If_cases
tff(fact_5552_sum__div__partition,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: set(B),F3: fun(B,A),B2: A] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( divide_divide(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_ms(fun(B,A),fun(A,fun(B,A)),F3),B2)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),collect(B,aa(A,fun(B,bool),aTP_Lamp_mt(fun(B,A),fun(A,fun(B,bool)),F3),B2))))),divide_divide(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),collect(B,aa(A,fun(B,bool),aTP_Lamp_mu(fun(B,A),fun(A,fun(B,bool)),F3),B2)))),B2)) ) ) ) ).

% sum_div_partition
tff(fact_5553_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),X),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y),aa(A,A,aa(A,fun(A,A),times_times(A),C2),X)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = bot_bot(set(A)) ) ) ) ) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_5554_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X: A,Y: A,C2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
               => ( aa(set(A),set(A),image(A,A,aTP_Lamp_mv(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),C2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
               => ( aa(set(A),set(A),image(A,A,aTP_Lamp_mv(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2),aa(A,A,aa(A,fun(A,A),times_times(A),X),C2)) ) ) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => ( aa(set(A),set(A),image(A,A,aTP_Lamp_mv(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,X,Y)) = bot_bot(set(A)) ) ) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_5555_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,M: A,C2: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
           => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_mw(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A2,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_mw(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_mw(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_5556_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,M: A,C2: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
           => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_mx(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A2,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_mx(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_mx(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_5557_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,M: A,C2: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
           => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_my(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A2,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_my(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B2,M)),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_my(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,B2,M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,A2,M)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_5558_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,M: A,C2: A] :
          ( ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
           => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_mz(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = bot_bot(set(A)) ) )
          & ( ( set_or1337092689740270186AtMost(A,A2,B2) != bot_bot(set(A)) )
           => ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_mz(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,divide_divide(A,A2,M)),C2),aa(A,A,minus_minus(A,divide_divide(A,B2,M)),C2)) ) )
              & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),M))
               => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_mz(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,divide_divide(A,B2,M)),C2),aa(A,A,minus_minus(A,divide_divide(A,A2,M)),C2)) ) ) ) ) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_5559_sum__fun__comp,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_1(C)
     => ! [S2: set(A),R: set(B),G: fun(A,B),F3: fun(B,C)] :
          ( pp(aa(set(A),bool,finite_finite(A),S2))
         => ( pp(aa(set(B),bool,finite_finite(B),R))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G),S2)),R))
             => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_na(fun(A,B),fun(fun(B,C),fun(A,C)),G),F3)),S2) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,aa(fun(B,C),fun(B,C),aa(fun(A,B),fun(fun(B,C),fun(B,C)),aTP_Lamp_nc(set(A),fun(fun(A,B),fun(fun(B,C),fun(B,C))),S2),G),F3)),R) ) ) ) ) ) ).

% sum_fun_comp
tff(fact_5560_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),aa(set(nat),set(A),image(nat,A,aTP_Lamp_nd(A,fun(nat,A),B3)),collect(nat,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),inf_inf(A),A3),B3) ) ).

% INF_nat_binary
tff(fact_5561_finite__inf__Sup,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [A2: A,A3: set(A)] : aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(set(A),A,complete_Sup_Sup(A),A3)) = aa(set(A),A,complete_Sup_Sup(A),collect(A,aa(set(A),fun(A,bool),aTP_Lamp_ne(A,fun(set(A),fun(A,bool)),A2),A3))) ) ).

% finite_inf_Sup
tff(fact_5562_distinct__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( distinct(A,concat(A,Xs))
    <=> ( distinct(list(A),aa(list(list(A)),list(list(A)),removeAll(list(A),nil(A)),Xs))
        & ! [Ys4: list(A)] :
            ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Ys4),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
           => distinct(A,Ys4) )
        & ! [Ys4: list(A),Zs3: list(A)] :
            ( ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Ys4),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
              & pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs3),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
              & ( Ys4 != Zs3 ) )
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Ys4)),aa(list(A),set(A),set2(A),Zs3)) = bot_bot(set(A)) ) ) ) ) ).

% distinct_concat_iff
tff(fact_5563_list__update__nonempty,axiom,
    ! [A: $tType,Xs: list(A),K: nat,X: A] :
      ( ( list_update(A,Xs,K,X) = nil(A) )
    <=> ( Xs = nil(A) ) ) ).

% list_update_nonempty
tff(fact_5564_concat__replicate__trivial,axiom,
    ! [A: $tType,I2: nat] : concat(A,replicate(list(A),I2,nil(A))) = nil(A) ).

% concat_replicate_trivial
tff(fact_5565_nths__nil,axiom,
    ! [A: $tType,A3: set(nat)] : nths(A,nil(A),A3) = nil(A) ).

% nths_nil
tff(fact_5566_enumerate__simps_I1_J,axiom,
    ! [A: $tType,N: nat] : enumerate(A,N,nil(A)) = nil(product_prod(nat,A)) ).

% enumerate_simps(1)
tff(fact_5567_rotate1__is__Nil__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),list(A),rotate1(A),Xs) = nil(A) )
    <=> ( Xs = nil(A) ) ) ).

% rotate1_is_Nil_conv
tff(fact_5568_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_5569_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_5570_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_5571_replicate__empty,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( replicate(A,N,X) = nil(A) )
    <=> ( N = zero_zero(nat) ) ) ).

% replicate_empty
tff(fact_5572_empty__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( nil(A) = replicate(A,N,X) )
    <=> ( N = zero_zero(nat) ) ) ).

% empty_replicate
tff(fact_5573_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( aa(set(A),list(A),linord4507533701916653071of_set(A),bot_bot(set(A))) = nil(A) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_5574_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),A3))
         => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = nil(A) ) ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
tff(fact_5575_concat__eq__Nil__conv,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ( concat(A,Xss) = nil(A) )
    <=> ! [X4: list(A)] :
          ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),X4),aa(list(list(A)),set(list(A)),set2(list(A)),Xss)))
         => ( X4 = nil(A) ) ) ) ).

% concat_eq_Nil_conv
tff(fact_5576_Nil__eq__concat__conv,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ( nil(A) = concat(A,Xss) )
    <=> ! [X4: list(A)] :
          ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),X4),aa(list(list(A)),set(list(A)),set2(list(A)),Xss)))
         => ( X4 = nil(A) ) ) ) ).

% Nil_eq_concat_conv
tff(fact_5577_nths__empty,axiom,
    ! [A: $tType,Xs: list(A)] : nths(A,Xs,bot_bot(set(nat))) = nil(A) ).

% nths_empty
tff(fact_5578_length__greater__0__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs)))
    <=> ( Xs != nil(A) ) ) ).

% length_greater_0_conv
tff(fact_5579_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = nil(A) )
          <=> ( A3 = bot_bot(set(A)) ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_5580_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)),aa(set(list(A)),set(set(A)),image(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),Xs))) ).

% set_concat
tff(fact_5581_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),aa(set(real),set(real),image(real,real,uminus_uminus(real)),X6))) ).

% Inf_real_def
tff(fact_5582_UN__Pow__subset,axiom,
    ! [A: $tType,B: $tType,B3: fun(B,set(A)),A3: set(B)] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(set(A))),set(set(A)),complete_Sup_Sup(set(set(A))),aa(set(B),set(set(set(A))),image(B,set(set(A)),aTP_Lamp_nf(fun(B,set(A)),fun(B,set(set(A))),B3)),A3))),pow2(A,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),B3),A3))))) ).

% UN_Pow_subset
tff(fact_5583_insort__not__Nil,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),A2: B,Xs: list(B)] : aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),A2),Xs) != nil(B) ) ).

% insort_not_Nil
tff(fact_5584_removeAll_Osimps_I1_J,axiom,
    ! [A: $tType,X: A] : aa(list(A),list(A),removeAll(A,X),nil(A)) = nil(A) ).

% removeAll.simps(1)
tff(fact_5585_remove1_Osimps_I1_J,axiom,
    ! [A: $tType,X: A] : remove1(A,X,nil(A)) = nil(A) ).

% remove1.simps(1)
tff(fact_5586_rotate1_Osimps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),rotate1(A),nil(A)) = nil(A) ).

% rotate1.simps(1)
tff(fact_5587_product_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType,Uu: list(B)] : product(A,B,nil(A),Uu) = nil(product_prod(A,B)) ).

% product.simps(1)
tff(fact_5588_concat_Osimps_I1_J,axiom,
    ! [A: $tType] : concat(A,nil(list(A))) = nil(A) ).

% concat.simps(1)
tff(fact_5589_list__update__code_I1_J,axiom,
    ! [A: $tType,I2: nat,Y: A] : list_update(A,nil(A),I2,Y) = nil(A) ).

% list_update_code(1)
tff(fact_5590_list__update_Osimps_I1_J,axiom,
    ! [A: $tType,I2: nat,V: A] : list_update(A,nil(A),I2,V) = nil(A) ).

% list_update.simps(1)
tff(fact_5591_distinct_Osimps_I1_J,axiom,
    ! [A: $tType] : distinct(A,nil(A)) ).

% distinct.simps(1)
tff(fact_5592_None__notin__image__Some,axiom,
    ! [A: $tType,A3: set(A)] : ~ pp(aa(set(option(A)),bool,aa(option(A),fun(set(option(A)),bool),member(option(A)),none(A)),aa(set(A),set(option(A)),image(A,option(A),some(A)),A3))) ).

% None_notin_image_Some
tff(fact_5593_empty__set,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(list(A),set(A),set2(A),nil(A)) ).

% empty_set
tff(fact_5594_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_5595_replicate__0,axiom,
    ! [A: $tType,X: A] : replicate(A,zero_zero(nat),X) = nil(A) ).

% replicate_0
tff(fact_5596_finite__int__iff__bounded__le,axiom,
    ! [S2: set(int)] :
      ( pp(aa(set(int),bool,finite_finite(int),S2))
    <=> ? [K3: int] : pp(aa(set(int),bool,aa(set(int),fun(set(int),bool),ord_less_eq(set(int)),aa(set(int),set(int),image(int,int,abs_abs(int)),S2)),aa(int,set(int),set_ord_atMost(int),K3))) ) ).

% finite_int_iff_bounded_le
tff(fact_5597_finite__int__iff__bounded,axiom,
    ! [S2: set(int)] :
      ( pp(aa(set(int),bool,finite_finite(int),S2))
    <=> ? [K3: int] : pp(aa(set(int),bool,aa(set(int),fun(set(int),bool),ord_less_eq(set(int)),aa(set(int),set(int),image(int,int,abs_abs(int)),S2)),aa(int,set(int),set_ord_lessThan(int),K3))) ) ).

% finite_int_iff_bounded
tff(fact_5598_UN__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(A),F3: fun(A,set(B)),G: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F3,X3)),aa(A,set(B),G,X3))) )
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),F3),A3))),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),G),B3)))) ) ) ).

% UN_mono
tff(fact_5599_UN__least,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: fun(A,set(B)),C5: set(B)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B3,X3)),C5)) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B3),A3))),C5)) ) ).

% UN_least
tff(fact_5600_UN__upper,axiom,
    ! [B: $tType,A: $tType,A2: A,A3: set(A),B3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B3,A2)),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),B3),A3)))) ) ).

% UN_upper
tff(fact_5601_UN__subset__iff,axiom,
    ! [B: $tType,A: $tType,A3: fun(B,set(A)),I5: set(B),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))),B3))
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),I5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(B,set(A),A3,X4)),B3)) ) ) ).

% UN_subset_iff
tff(fact_5602_INT__lower,axiom,
    ! [B: $tType,A: $tType,A2: A,A3: set(A),B3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),B3),A3))),aa(A,set(B),B3,A2))) ) ).

% INT_lower
tff(fact_5603_INT__greatest,axiom,
    ! [B: $tType,A: $tType,A3: set(A),C5: set(B),B3: fun(A,set(B))] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
         => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C5),aa(A,set(B),B3,X3))) )
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C5),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),B3),A3)))) ) ).

% INT_greatest
tff(fact_5604_INT__anti__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(A),F3: fun(A,set(B)),G: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F3,X3)),aa(A,set(B),G,X3))) )
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),F3),B3))),aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(A),set(set(B)),image(A,set(B),G),A3)))) ) ) ).

% INT_anti_mono
tff(fact_5605_INT__subset__iff,axiom,
    ! [A: $tType,B: $tType,B3: set(A),A3: fun(B,set(A)),I5: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5))))
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),I5))
         => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(B,set(A),A3,X4))) ) ) ).

% INT_subset_iff
tff(fact_5606_list_Osize__gen_I1_J,axiom,
    ! [A: $tType,X: fun(A,nat)] : aa(list(A),nat,size_list(A,X),nil(A)) = zero_zero(nat) ).

% list.size_gen(1)
tff(fact_5607_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_5608_image__int__atLeastAtMost,axiom,
    ! [A2: nat,B2: nat] : aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),set_or1337092689740270186AtMost(nat,A2,B2)) = set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% image_int_atLeastAtMost
tff(fact_5609_image__int__atLeastLessThan,axiom,
    ! [A2: nat,B2: nat] : aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),set_or7035219750837199246ssThan(nat,A2,B2)) = set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),A2),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% image_int_atLeastLessThan
tff(fact_5610_finite__Inf__Sup,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [A3: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image(set(A),A,complete_Sup_Sup(A)),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image(set(A),A,complete_Inf_Inf(A)),collect(set(A),aTP_Lamp_ng(set(set(A)),fun(set(A),bool),A3)))))) ) ).

% finite_Inf_Sup
tff(fact_5611_bij__betw__UNION__chain,axiom,
    ! [B: $tType,C: $tType,A: $tType,I5: set(A),A3: fun(A,set(B)),F3: fun(B,C),A8: fun(A,set(C))] :
      ( ! [I4: A,J2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I4),I5))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),J2),I5))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A3,I4)),aa(A,set(B),A3,J2)))
              | pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A3,J2)),aa(A,set(B),A3,I4))) ) ) )
     => ( ! [I4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I4),I5))
           => bij_betw(B,C,F3,aa(A,set(B),A3,I4),aa(A,set(C),A8,I4)) )
       => bij_betw(B,C,F3,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5)),aa(set(set(C)),set(C),complete_Sup_Sup(set(C)),aa(set(A),set(set(C)),image(A,set(C),A8),I5))) ) ) ).

% bij_betw_UNION_chain
tff(fact_5612_Collect__split__mono__strong,axiom,
    ! [B: $tType,A: $tType,X6: set(A),A3: set(product_prod(A,B)),Y6: set(B),P: fun(A,fun(B,bool)),Q: fun(A,fun(B,bool))] :
      ( ( X6 = aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),A3) )
     => ( ( Y6 = aa(set(product_prod(A,B)),set(B),image(product_prod(A,B),B,product_snd(A,B)),A3) )
       => ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),X6))
             => ! [Xa3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Xa3),Y6))
                 => ( pp(aa(B,bool,aa(A,fun(B,bool),P,X3),Xa3))
                   => pp(aa(B,bool,aa(A,fun(B,bool),Q,X3),Xa3)) ) ) )
         => ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A3),collect(product_prod(A,B),product_case_prod(A,B,bool,P))))
           => pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A3),collect(product_prod(A,B),product_case_prod(A,B,bool,Q)))) ) ) ) ) ).

% Collect_split_mono_strong
tff(fact_5613_UN__le__add__shift__strict,axiom,
    ! [A: $tType,M6: fun(nat,set(A)),K: nat,N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_nh(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M6),K)),aa(nat,set(nat),set_ord_lessThan(nat),N))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),M6),set_or7035219750837199246ssThan(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)))) ).

% UN_le_add_shift_strict
tff(fact_5614_UN__le__add__shift,axiom,
    ! [A: $tType,M6: fun(nat,set(A)),K: nat,N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_nh(fun(nat,set(A)),fun(nat,fun(nat,set(A))),M6),K)),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),M6),set_or1337092689740270186AtMost(nat,K,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)))) ).

% UN_le_add_shift
tff(fact_5615_subset__subseqs,axiom,
    ! [A: $tType,X6: set(A),Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X6),aa(list(A),set(A),set2(A),Xs)))
     => pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X6),aa(set(list(A)),set(set(A)),image(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs))))) ) ).

% subset_subseqs
tff(fact_5616_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))),insert(set(A)),bot_bot(set(A))),bot_bot(set(set(A)))) ).

% Pow_set(1)
tff(fact_5617_subseqs__powset,axiom,
    ! [A: $tType,Xs: list(A)] : aa(set(list(A)),set(set(A)),image(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs))) = pow2(A,aa(list(A),set(A),set2(A),Xs)) ).

% subseqs_powset
tff(fact_5618_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : aa(set(int),set(int),image(int,int,aTP_Lamp_ni(int,fun(int,int),L)),set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,minus_minus(int,U),L))) = set_or7035219750837199246ssThan(int,L,U) ).

% image_add_int_atLeastLessThan
tff(fact_5619_Sup__Inf__le,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image(set(A),A,complete_Inf_Inf(A)),collect(set(A),aTP_Lamp_nj(set(set(A)),fun(set(A),bool),A3))))),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image(set(A),A,complete_Sup_Sup(A)),A3)))) ) ).

% Sup_Inf_le
tff(fact_5620_Inf__Sup__le,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [A3: set(set(A))] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(set(A)),set(A),image(set(A),A,complete_Sup_Sup(A)),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(set(A)),set(A),image(set(A),A,complete_Inf_Inf(A)),collect(set(A),aTP_Lamp_nk(set(set(A)),fun(set(A),bool),A3)))))) ) ).

% Inf_Sup_le
tff(fact_5621_card__UN__le,axiom,
    ! [B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B))] :
      ( pp(aa(set(A),bool,finite_finite(A),I5))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(B),nat,finite_card(B),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5)))),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aTP_Lamp_nl(fun(A,set(B)),fun(A,nat),A3)),I5))) ) ).

% card_UN_le
tff(fact_5622_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),U))
     => ( set_or7035219750837199246ssThan(int,zero_zero(int),U) = aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),aa(nat,set(nat),set_ord_lessThan(nat),nat2(U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_5623_UN__image__subset,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,set(A)),G: fun(C,set(B)),X: C,X6: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),F3),aa(C,set(B),G,X)))),X6))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(C,set(B),G,X)),collect(B,aa(set(A),fun(B,bool),aTP_Lamp_nm(fun(B,set(A)),fun(set(A),fun(B,bool)),F3),X6)))) ) ).

% UN_image_subset
tff(fact_5624_mlex__eq,axiom,
    ! [A: $tType,F3: fun(A,nat),R: set(product_prod(A,A))] : mlex_prod(A,F3,R) = collect(product_prod(A,A),product_case_prod(A,A,bool,aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aTP_Lamp_nn(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,bool))),F3),R))) ).

% mlex_eq
tff(fact_5625_INF__filter__not__bot,axiom,
    ! [I6: $tType,A: $tType,B3: set(I6),F5: fun(I6,filter(A))] :
      ( ! [X7: set(I6)] :
          ( pp(aa(set(I6),bool,aa(set(I6),fun(set(I6),bool),ord_less_eq(set(I6)),X7),B3))
         => ( pp(aa(set(I6),bool,finite_finite(I6),X7))
           => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(I6),set(filter(A)),image(I6,filter(A),F5),X7)) != bot_bot(filter(A)) ) ) )
     => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(I6),set(filter(A)),image(I6,filter(A),F5),B3)) != bot_bot(filter(A)) ) ) ).

% INF_filter_not_bot
tff(fact_5626_INF__filter__bot__base,axiom,
    ! [A: $tType,B: $tType,I5: set(A),F5: fun(A,filter(B))] :
      ( ! [I4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I4),I5))
         => ! [J2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),J2),I5))
             => ? [X2: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),I5))
                  & pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),aa(A,filter(B),F5,X2)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F5,I4)),aa(A,filter(B),F5,J2)))) ) ) )
     => ( ( aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),F5),I5)) = bot_bot(filter(B)) )
      <=> ? [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),I5))
            & ( aa(A,filter(B),F5,X4) = bot_bot(filter(B)) ) ) ) ) ).

% INF_filter_bot_base
tff(fact_5627_Inf__filter__not__bot,axiom,
    ! [A: $tType,B3: set(filter(A))] :
      ( ! [X7: set(filter(A))] :
          ( pp(aa(set(filter(A)),bool,aa(set(filter(A)),fun(set(filter(A)),bool),ord_less_eq(set(filter(A))),X7),B3))
         => ( pp(aa(set(filter(A)),bool,finite_finite(filter(A)),X7))
           => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X7) != bot_bot(filter(A)) ) ) )
     => ( aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B3) != bot_bot(filter(A)) ) ) ).

% Inf_filter_not_bot
tff(fact_5628_Inf__filter__def,axiom,
    ! [A: $tType,S2: set(filter(A))] : aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),S2) = aa(set(filter(A)),filter(A),complete_Sup_Sup(filter(A)),collect(filter(A),aTP_Lamp_no(set(filter(A)),fun(filter(A),bool),S2))) ).

% Inf_filter_def
tff(fact_5629_mlex__less,axiom,
    ! [A: $tType,F3: fun(A,nat),X: A,Y: A,R: set(product_prod(A,A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
     => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y)),mlex_prod(A,F3,R))) ) ).

% mlex_less
tff(fact_5630_mlex__iff,axiom,
    ! [A: $tType,X: A,Y: A,F3: fun(A,nat),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y)),mlex_prod(A,F3,R)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
        | ( ( aa(A,nat,F3,X) = aa(A,nat,F3,Y) )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y)),R)) ) ) ) ).

% mlex_iff
tff(fact_5631_mlex__leq,axiom,
    ! [A: $tType,F3: fun(A,nat),X: A,Y: A,R: set(product_prod(A,A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,F3,X)),aa(A,nat,F3,Y)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y)),R))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y)),mlex_prod(A,F3,R))) ) ) ).

% mlex_leq
tff(fact_5632_conj__subset__def,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,bool),Q: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_np(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q))))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),collect(A,P)))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),collect(A,Q))) ) ) ).

% conj_subset_def
tff(fact_5633_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)),aa(set(list(A)),set(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_5634_suminf__eq__SUP__real,axiom,
    ! [X6: fun(nat,real)] :
      ( summable(real,X6)
     => ( ! [I4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,X6,I4)))
       => ( suminf(real,X6) = aa(set(real),real,complete_Sup_Sup(real),aa(set(nat),set(real),image(nat,real,aTP_Lamp_nq(fun(nat,real),fun(nat,real),X6)),top_top(set(nat)))) ) ) ) ).

% suminf_eq_SUP_real
tff(fact_5635_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
    ! [A: $tType,F3: fun(nat,set(A)),S2: set(A)] :
      ( ! [I4: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),F3,I4)),S2))
     => ( pp(aa(set(A),bool,finite_finite(A),S2))
       => ( ? [N8: nat] :
              ( ! [N2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N2),N8))
                 => ! [M5: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M5),N8))
                     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N2))
                       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),aa(nat,set(A),F3,M5)),aa(nat,set(A),F3,N2))) ) ) )
              & ! [N2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
                 => ( aa(nat,set(A),F3,N8) = aa(nat,set(A),F3,N2) ) ) )
         => ( aa(nat,set(A),F3,aa(set(A),nat,finite_card(A),S2)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),F3),top_top(set(nat)))) ) ) ) ) ).

% finite_mono_strict_prefix_implies_finite_fixpoint
tff(fact_5636_atMost__UNIV__triv,axiom,
    ! [A: $tType] : aa(set(A),set(set(A)),set_ord_atMost(set(A)),top_top(set(A))) = top_top(set(set(A))) ).

% atMost_UNIV_triv
tff(fact_5637_remdups__eq__nil__right__iff,axiom,
    ! [A: $tType,X: list(A)] :
      ( ( nil(A) = remdups(A,X) )
    <=> ( X = nil(A) ) ) ).

% remdups_eq_nil_right_iff
tff(fact_5638_remdups__eq__nil__iff,axiom,
    ! [A: $tType,X: list(A)] :
      ( ( remdups(A,X) = nil(A) )
    <=> ( X = nil(A) ) ) ).

% remdups_eq_nil_iff
tff(fact_5639_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_5640_length__remdups__eq,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),remdups(A,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) )
    <=> ( remdups(A,Xs) = Xs ) ) ).

% length_remdups_eq
tff(fact_5641_remdups__id__iff__distinct,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( remdups(A,Xs) = Xs )
    <=> distinct(A,Xs) ) ).

% remdups_id_iff_distinct
tff(fact_5642_distinct__remdups,axiom,
    ! [A: $tType,Xs: list(A)] : distinct(A,remdups(A,Xs)) ).

% distinct_remdups
tff(fact_5643_range__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),top_top(set(A))) = top_top(set(A)) ) ).

% range_add
tff(fact_5644_surj__plus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_plus
tff(fact_5645_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] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),top_top(A)))
             => ? [Xa4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Xa4)) ) ) ) ) ).

% Sup_eq_top_iff
tff(fact_5646_surj__fn,axiom,
    ! [A: $tType,F3: fun(A,A),N: nat] :
      ( ( aa(set(A),set(A),image(A,A,F3),top_top(set(A))) = top_top(set(A)) )
     => ( aa(set(A),set(A),image(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3)),top_top(set(A))) = top_top(set(A)) ) ) ).

% surj_fn
tff(fact_5647_length__remdups__leq,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),remdups(A,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_remdups_leq
tff(fact_5648_SUP__eq__top__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [F3: fun(B,A),A3: set(B)] :
          ( ( aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3)) = top_top(A) )
        <=> ! [X4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),top_top(A)))
             => ? [Xa4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Xa4),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),aa(B,A,F3,Xa4))) ) ) ) ) ).

% SUP_eq_top_iff
tff(fact_5649_Inf__atMostLessThan,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),top_top(A)),X))
         => ( aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_lessThan(A),X)) = bot_bot(A) ) ) ) ).

% Inf_atMostLessThan
tff(fact_5650_sums__SUP,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] : sums(A,F3,aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image(nat,A,aTP_Lamp_nr(fun(nat,A),fun(nat,A),F3)),top_top(set(nat))))) ) ).

% sums_SUP
tff(fact_5651_UNIV__option__conv,axiom,
    ! [A: $tType] : top_top(set(option(A))) = aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),aa(set(A),set(option(A)),image(A,option(A),some(A)),top_top(set(A)))) ).

% UNIV_option_conv
tff(fact_5652_UN__atMost__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image(nat,set(nat),set_ord_atMost(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_atMost_UNIV
tff(fact_5653_UN__lessThan__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image(nat,set(nat),set_ord_lessThan(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_lessThan_UNIV
tff(fact_5654_remdups_Osimps_I1_J,axiom,
    ! [A: $tType] : remdups(A,nil(A)) = nil(A) ).

% remdups.simps(1)
tff(fact_5655_atLeastAtMost__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( bounded_lattice(A)
     => ! [X: A,Y: A] :
          ( ( set_or1337092689740270186AtMost(A,X,Y) = top_top(set(A)) )
        <=> ( ( X = bot_bot(A) )
            & ( Y = top_top(A) ) ) ) ) ).

% atLeastAtMost_eq_UNIV_iff
tff(fact_5656_not__UNIV__eq__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [H2: A] : top_top(set(A)) != aa(A,set(A),set_ord_atMost(A),H2) ) ).

% not_UNIV_eq_Iic
tff(fact_5657_atMost__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [X: A] :
          ( ( aa(A,set(A),set_ord_atMost(A),X) = top_top(set(A)) )
        <=> ( X = top_top(A) ) ) ) ).

% atMost_eq_UNIV_iff
tff(fact_5658_subset__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),top_top(set(A)))) ).

% subset_UNIV
tff(fact_5659_top_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),top_top(A)),A2))
         => ( A2 = top_top(A) ) ) ) ).

% top.extremum_uniqueI
tff(fact_5660_top_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),top_top(A)),A2))
        <=> ( A2 = top_top(A) ) ) ) ).

% top.extremum_unique
tff(fact_5661_top__greatest,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),top_top(A))) ) ).

% top_greatest
tff(fact_5662_less__filter__def,axiom,
    ! [A: $tType,F5: filter(A),F11: filter(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less(filter(A)),F5),F11))
    <=> ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F5),F11))
        & ~ pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F11),F5)) ) ) ).

% less_filter_def
tff(fact_5663_not__UNIV__eq__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L3: A,H2: A] : top_top(set(A)) != set_or1337092689740270186AtMost(A,L3,H2) ) ).

% not_UNIV_eq_Icc
tff(fact_5664_remdups__remdups,axiom,
    ! [A: $tType,Xs: list(A)] : remdups(A,remdups(A,Xs)) = remdups(A,Xs) ).

% remdups_remdups
tff(fact_5665_top_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] : ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),top_top(A)),A2)) ) ).

% top.extremum_strict
tff(fact_5666_top_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( ( A2 != top_top(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),top_top(A))) ) ) ).

% top.not_eq_extremum
tff(fact_5667_distinct__remdups__id,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( remdups(A,Xs) = Xs ) ) ).

% distinct_remdups_id
tff(fact_5668_finite__fun__UNIVD1,axiom,
    ! [B: $tType,A: $tType] :
      ( pp(aa(set(fun(A,B)),bool,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)) )
       => pp(aa(set(A),bool,finite_finite(A),top_top(set(A)))) ) ) ).

% finite_fun_UNIVD1
tff(fact_5669_range__subsetD,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),B3: set(A),I2: B] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F3),top_top(set(B)))),B3))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,F3,I2)),B3)) ) ).

% range_subsetD
tff(fact_5670_not__UNIV__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),set_or1337092689740270186AtMost(A,L,H))) ) ).

% not_UNIV_le_Icc
tff(fact_5671_not__UNIV__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [H: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),aa(A,set(A),set_ord_atMost(A),H))) ) ).

% not_UNIV_le_Iic
tff(fact_5672_bij__fn,axiom,
    ! [A: $tType,F3: fun(A,A),N: nat] :
      ( bij_betw(A,A,F3,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)),N),F3),top_top(set(A)),top_top(set(A))) ) ).

% bij_fn
tff(fact_5673_full__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A)] : collect(A,aTP_Lamp_ns(fun(B,A),fun(A,bool),F3)) = aa(set(B),set(A),image(B,A,F3),top_top(set(B))) ).

% full_SetCompr_eq
tff(fact_5674_remove1__remdups,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( remove1(A,X,remdups(A,Xs)) = remdups(A,remove1(A,X,Xs)) ) ) ).

% remove1_remdups
tff(fact_5675_surj__Compl__image__subset,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A3: set(B)] :
      ( ( aa(set(B),set(A),image(B,A,F3),top_top(set(B))) = top_top(set(A)) )
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(B),set(A),image(B,A,F3),A3))),aa(set(B),set(A),image(B,A,F3),aa(set(B),set(B),uminus_uminus(set(B)),A3)))) ) ).

% surj_Compl_image_subset
tff(fact_5676_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_5677_Collect__ex__eq,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool))] : collect(A,aTP_Lamp_nt(fun(A,fun(B,bool)),fun(A,bool),P)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_nv(fun(A,fun(B,bool)),fun(B,set(A)),P)),top_top(set(B)))) ).

% Collect_ex_eq
tff(fact_5678_notin__range__Some,axiom,
    ! [A: $tType,X: option(A)] :
      ( ~ pp(aa(set(option(A)),bool,aa(option(A),fun(set(option(A)),bool),member(option(A)),X),aa(set(A),set(option(A)),image(A,option(A),some(A)),top_top(set(A)))))
    <=> ( X = none(A) ) ) ).

% notin_range_Some
tff(fact_5679_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_5680_finite__UNIV__card__ge__0,axiom,
    ! [A: $tType] :
      ( pp(aa(set(A),bool,finite_finite(A),top_top(set(A))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),top_top(set(A))))) ) ).

% finite_UNIV_card_ge_0
tff(fact_5681_UN__UN__finite__eq,axiom,
    ! [A: $tType,A3: fun(nat,set(A))] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),aTP_Lamp_nw(fun(nat,set(A)),fun(nat,set(A)),A3)),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),top_top(set(nat)))) ).

% UN_UN_finite_eq
tff(fact_5682_card__range__greater__zero,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A)] :
      ( pp(aa(set(A),bool,finite_finite(A),aa(set(B),set(A),image(B,A,F3),top_top(set(B)))))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image(B,A,F3),top_top(set(B)))))) ) ).

% card_range_greater_zero
tff(fact_5683_UN__finite__subset,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),C5: set(A)] :
      ( ! [N2: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)))),C5))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),top_top(set(nat))))),C5)) ) ).

% UN_finite_subset
tff(fact_5684_UN__finite2__eq,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),B3: fun(nat,set(A)),K: nat] :
      ( ! [N2: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K))))
     => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),top_top(set(nat)))) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B3),top_top(set(nat)))) ) ) ).

% UN_finite2_eq
tff(fact_5685_suminf__eq__SUP,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(nat,A)] : suminf(A,F3) = aa(set(A),A,complete_Sup_Sup(A),aa(set(nat),set(A),image(nat,A,aTP_Lamp_nr(fun(nat,A),fun(nat,A),F3)),top_top(set(nat)))) ) ).

% suminf_eq_SUP
tff(fact_5686_range__mod,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_nx(nat,fun(nat,nat),N)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ) ) ).

% range_mod
tff(fact_5687_UN__finite2__subset,axiom,
    ! [A: $tType,A3: fun(nat,set(A)),B3: fun(nat,set(A)),K: nat] :
      ( ! [N2: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N2)))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),K))))))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),A3),top_top(set(nat))))),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),B3),top_top(set(nat)))))) ) ).

% UN_finite2_subset
tff(fact_5688_UNIV__nat__eq,axiom,
    top_top(set(nat)) = aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),top_top(set(nat)))) ).

% UNIV_nat_eq
tff(fact_5689_card__disjoint__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( aa(set(list(A)),nat,finite_card(list(A)),shuffles(A,Xs,Ys)) = 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)),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% card_disjoint_shuffles
tff(fact_5690_image__Fpow__mono,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),A3: set(B),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F3),A3)),B3))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(B)),set(set(A)),image(set(B),set(A),image(B,A,F3)),finite_Fpow(B,A3))),finite_Fpow(A,B3))) ) ).

% image_Fpow_mono
tff(fact_5691_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_5692_Nil__in__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),nil(A)),shuffles(A,Xs,Ys)))
    <=> ( ( Xs = nil(A) )
        & ( Ys = nil(A) ) ) ) ).

% Nil_in_shuffles
tff(fact_5693_finite__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : pp(aa(set(list(A)),bool,finite_finite(list(A)),shuffles(A,Xs,Ys))) ).

% finite_shuffles
tff(fact_5694_card__UNIV__bool,axiom,
    aa(set(bool),nat,finite_card(bool),top_top(set(bool))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)) ).

% card_UNIV_bool
tff(fact_5695_infinite__UNIV__listI,axiom,
    ! [A: $tType] : ~ pp(aa(set(list(A)),bool,finite_finite(list(A)),top_top(set(list(A))))) ).

% infinite_UNIV_listI
tff(fact_5696_shuffles__commutes,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : shuffles(A,Xs,Ys) = shuffles(A,Ys,Xs) ).

% shuffles_commutes
tff(fact_5697_shuffles_Osimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A)] : shuffles(A,Xs,nil(A)) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Xs),bot_bot(set(list(A)))) ).

% shuffles.simps(2)
tff(fact_5698_shuffles_Osimps_I1_J,axiom,
    ! [A: $tType,Ys: list(A)] : shuffles(A,nil(A),Ys) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Ys),bot_bot(set(list(A)))) ).

% shuffles.simps(1)
tff(fact_5699_Nil__in__shufflesI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = nil(A) )
     => ( ( Ys = nil(A) )
       => pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),nil(A)),shuffles(A,Xs,Ys))) ) ) ).

% Nil_in_shufflesI
tff(fact_5700_length__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
     => ( aa(list(A),nat,size_size(list(A)),Zs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ) ) ).

% length_shuffles
tff(fact_5701_Fpow__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),finite_Fpow(A,A3)),finite_Fpow(A,B3))) ) ).

% Fpow_mono
tff(fact_5702_Fpow__subset__Pow,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),finite_Fpow(A,A3)),pow2(A,A3))) ).

% Fpow_subset_Pow
tff(fact_5703_Fpow__def,axiom,
    ! [A: $tType,A3: set(A)] : finite_Fpow(A,A3) = collect(set(A),aTP_Lamp_ny(set(A),fun(set(A),bool),A3)) ).

% Fpow_def
tff(fact_5704_distinct__disjoint__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( distinct(A,Xs)
     => ( distinct(A,Ys)
       => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
         => ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
           => distinct(A,Zs) ) ) ) ) ).

% distinct_disjoint_shuffles
tff(fact_5705_root__def,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(real,real,root(N),X) = zero_zero(real) ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(real,real,root(N),X) = the_inv_into(real,real,top_top(set(real)),aTP_Lamp_nz(nat,fun(real,real),N),X) ) ) ) ).

% root_def
tff(fact_5706_card__UNIV__char,axiom,
    aa(set(char),nat,finite_card(char),top_top(set(char))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))))))) ).

% card_UNIV_char
tff(fact_5707_DERIV__even__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
         => has_field_derivative(real,root(N),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),N))),aa(nat,real,power_power(real,aa(real,real,root(N),X)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_even_real_root
tff(fact_5708_DERIV__at__within__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Y: A,Z: A,X: A,S2: set(A)] :
          ( has_field_derivative(A,F3,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),X),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),Z)),S2)))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_oa(fun(A,A),fun(A,fun(A,A)),F3),Z),Y,topolo174197925503356063within(A,X,S2)) ) ) ).

% DERIV_at_within_shift
tff(fact_5709_MVT2,axiom,
    ! [A2: real,B2: real,F3: fun(real,real),F10: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
             => has_field_derivative(real,F3,aa(real,real,F10,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
       => ? [Z2: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Z2))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z2),B2))
            & ( aa(real,real,minus_minus(real,aa(real,real,F3,B2)),aa(real,real,F3,A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,B2),A2)),aa(real,real,F10,Z2)) ) ) ) ) ).

% MVT2
tff(fact_5710_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
             => ? [Y3: real] :
                  ( has_field_derivative(real,F3,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Y3)) ) ) )
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,F3,A2)),aa(real,real,F3,B2))) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_5711_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
             => ? [Y3: real] :
                  ( has_field_derivative(real,F3,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y3),zero_zero(real))) ) ) )
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,F3,B2)),aa(real,real,F3,A2))) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_5712_DERIV__neg__imp__decreasing,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
             => ? [Y3: real] :
                  ( has_field_derivative(real,F3,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y3),zero_zero(real))) ) ) )
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F3,B2)),aa(real,real,F3,A2))) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_5713_DERIV__pos__imp__increasing,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
             => ? [Y3: real] :
                  ( has_field_derivative(real,F3,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y3)) ) ) )
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F3,A2)),aa(real,real,F3,B2))) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_5714_deriv__nonneg__imp__mono,axiom,
    ! [A2: real,B2: real,G: fun(real,real),G3: fun(real,real)] :
      ( ! [X3: real] :
          ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or1337092689740270186AtMost(real,A2,B2)))
         => has_field_derivative(real,G,aa(real,real,G3,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) )
     => ( ! [X3: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or1337092689740270186AtMost(real,A2,B2)))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,G3,X3))) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,G,A2)),aa(real,real,G,B2))) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_5715_DERIV__isconst3,axiom,
    ! [A2: real,B2: real,X: real,Y: real,F3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X),set_or5935395276787703475ssThan(real,A2,B2)))
       => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),Y),set_or5935395276787703475ssThan(real,A2,B2)))
         => ( ! [X3: real] :
                ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or5935395276787703475ssThan(real,A2,B2)))
               => has_field_derivative(real,F3,zero_zero(real),topolo174197925503356063within(real,X3,top_top(set(real)))) )
           => ( aa(real,real,F3,X) = aa(real,real,F3,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_5716_DERIV__local__const,axiom,
    ! [F3: fun(real,real),L: real,X: real,D2: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
       => ( ! [Y4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y4))),D2))
             => ( aa(real,real,F3,X) = aa(real,real,F3,Y4) ) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_const
tff(fact_5717_DERIV__neg__dec__left,axiom,
    ! [F3: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H4))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H4),D3))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F3,X)),aa(real,real,F3,aa(real,real,minus_minus(real,X),H4)))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_5718_DERIV__pos__inc__left,axiom,
    ! [F3: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H4))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H4),D3))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F3,aa(real,real,minus_minus(real,X),H4))),aa(real,real,F3,X))) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_5719_DERIV__neg__dec__right,axiom,
    ! [F3: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H4))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H4),D3))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4))),aa(real,real,F3,X))) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_5720_DERIV__pos__inc__right,axiom,
    ! [F3: fun(real,real),L: real,X: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H4))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H4),D3))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F3,X)),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4)))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_5721_DERIV__ln,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => has_field_derivative(real,ln_ln(real),aa(real,real,inverse_inverse(real),X),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_ln
tff(fact_5722_DERIV__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Y: A,X: A,Z: A] :
          ( has_field_derivative(A,F3,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z),top_top(set(A))))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_ob(fun(A,A),fun(A,fun(A,A)),F3),Z),Y,topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_shift
tff(fact_5723_has__real__derivative__neg__dec__right,axiom,
    ! [F3: fun(real,real),L: real,X: real,S2: set(real)] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,X,S2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H4))
               => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4)),S2))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H4),D3))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4))),aa(real,real,F3,X))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_5724_has__real__derivative__pos__inc__right,axiom,
    ! [F3: fun(real,real),L: real,X: real,S2: set(real)] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,X,S2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H4))
               => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4)),S2))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H4),D3))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F3,X)),aa(real,real,F3,aa(real,real,aa(real,fun(real,real),plus_plus(real),X),H4)))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_5725_has__real__derivative__pos__inc__left,axiom,
    ! [F3: fun(real,real),L: real,X: real,S2: set(real)] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,X,S2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H4))
               => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,minus_minus(real,X),H4)),S2))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H4),D3))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F3,aa(real,real,minus_minus(real,X),H4))),aa(real,real,F3,X))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_5726_has__real__derivative__neg__dec__left,axiom,
    ! [F3: fun(real,real),L: real,X: real,S2: set(real)] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,X,S2))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
       => ? [D3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D3))
            & ! [H4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H4))
               => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,minus_minus(real,X),H4)),S2))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H4),D3))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F3,X)),aa(real,real,F3,aa(real,real,minus_minus(real,X),H4)))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_5727_DERIV__mult_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,X: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,X,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_oc(fun(A,A),fun(fun(A,A),fun(A,A)),F3),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,F3,X)),E4)),aa(A,A,aa(A,fun(A,A),times_times(A),D4),aa(A,A,G,X))),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_mult'
tff(fact_5728_DERIV__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Da: A,X: A,S: set(A),G: fun(A,A),Db: A] :
          ( has_field_derivative(A,F3,Da,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,X,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_oc(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Da),aa(A,A,G,X))),aa(A,A,aa(A,fun(A,A),times_times(A),Db),aa(A,A,F3,X))),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_mult
tff(fact_5729_DERIV__ident,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F5: filter(A)] : has_field_derivative(A,aTP_Lamp_od(A,A),one_one(A),F5) ) ).

% DERIV_ident
tff(fact_5730_DERIV__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,X: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,X,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_oe(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),D4),E4),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_add
tff(fact_5731_field__differentiable__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),F10: A,F5: filter(A),G: fun(A,A),G3: A] :
          ( has_field_derivative(A,F3,F10,F5)
         => ( has_field_derivative(A,G,G3,F5)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_oe(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),F10),G3),F5) ) ) ) ).

% field_differentiable_add
tff(fact_5732_DERIV__cdivide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,X: A,S: set(A),C2: A] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_of(fun(A,A),fun(A,fun(A,A)),F3),C2),divide_divide(A,D4,C2),topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_cdivide
tff(fact_5733_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,X: A,S: set(A),G: fun(A,A),E4: A] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,E4,topolo174197925503356063within(A,X,S))
           => ( ( aa(A,A,G,X) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_og(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G),divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),D4),aa(A,A,G,X))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F3,X)),E4)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G,X)),aa(A,A,G,X))),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% DERIV_divide
tff(fact_5734_has__field__derivative__subset,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Y: A,X: A,S: set(A),T2: set(A)] :
          ( has_field_derivative(A,F3,Y,topolo174197925503356063within(A,X,S))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),S))
           => has_field_derivative(A,F3,Y,topolo174197925503356063within(A,X,T2)) ) ) ) ).

% has_field_derivative_subset
tff(fact_5735_DERIV__subset,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),F10: A,X: A,S: set(A),T2: set(A)] :
          ( has_field_derivative(A,F3,F10,topolo174197925503356063within(A,X,S))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),S))
           => has_field_derivative(A,F3,F10,topolo174197925503356063within(A,X,T2)) ) ) ) ).

% DERIV_subset
tff(fact_5736_at__le,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),T2: set(A),X: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),T2))
         => pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),topolo174197925503356063within(A,X,S)),topolo174197925503356063within(A,X,T2))) ) ) ).

% at_le
tff(fact_5737_DERIV__const__average,axiom,
    ! [A2: real,B2: real,V: fun(real,real),K: real] :
      ( ( A2 != B2 )
     => ( ! [X3: real] : has_field_derivative(real,V,K,topolo174197925503356063within(real,X3,top_top(set(real))))
       => ( aa(real,real,V,divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))) = divide_divide(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,V,A2)),aa(real,real,V,B2)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) ) ) ).

% DERIV_const_average
tff(fact_5738_DERIV__local__max,axiom,
    ! [F3: fun(real,real),L: real,X: real,D2: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
       => ( ! [Y4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y4))),D2))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,F3,Y4)),aa(real,real,F3,X))) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_max
tff(fact_5739_DERIV__local__min,axiom,
    ! [F3: fun(real,real),L: real,X: real,D2: real] :
      ( has_field_derivative(real,F3,L,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
       => ( ! [Y4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X),Y4))),D2))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,F3,X)),aa(real,real,F3,Y4))) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_min
tff(fact_5740_DERIV__ln__divide,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => has_field_derivative(real,ln_ln(real),divide_divide(real,one_one(real),X),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_ln_divide
tff(fact_5741_DERIV__cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K: A,Xa: A] : has_field_derivative(A,aTP_Lamp_oh(A,fun(A,A),K),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xa),K))),topolo174197925503356063within(A,Xa,top_top(set(A)))) ) ).

% DERIV_cos_add
tff(fact_5742_DERIV__fun__pow,axiom,
    ! [G: fun(real,real),M: real,X: real,N: nat] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,X,top_top(set(real))))
     => has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_oi(fun(real,real),fun(nat,fun(real,real)),G),N),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),N)),aa(nat,real,power_power(real,aa(real,real,G,X)),aa(nat,nat,minus_minus(nat,N),one_one(nat))))),M),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_fun_pow
tff(fact_5743_DERIV__power__Suc,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,X: A,S: set(A),N: nat] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_oj(fun(A,A),fun(nat,fun(A,A)),F3),N),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),N))),aa(A,A,aa(A,fun(A,A),times_times(A),D4),aa(nat,A,power_power(A,aa(A,A,F3,X)),N))),topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_power_Suc
tff(fact_5744_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [X: A,S: set(A)] :
          ( ( X != zero_zero(A) )
         => has_field_derivative(A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(nat,A,power_power(A,aa(A,A,inverse_inverse(A),X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_inverse
tff(fact_5745_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,X: A,S: set(A),N: nat] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,X,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_ok(fun(A,A),fun(nat,fun(A,A)),F3),N),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),times_times(A),D4),aa(nat,A,power_power(A,aa(A,A,F3,X)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(A,X,S)) ) ) ).

% DERIV_power
tff(fact_5746_DERIV__pow,axiom,
    ! [N: nat,X: real,S: set(real)] : has_field_derivative(real,aTP_Lamp_ol(nat,fun(real,real),N),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,power_power(real,X),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(real,X,S)) ).

% DERIV_pow
tff(fact_5747_termdiffs__strong__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [Y4: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(A,fun(nat,A)),C2),Y4))
         => has_field_derivative(A,aTP_Lamp_om(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_bu(fun(nat,A),fun(A,fun(nat,A)),C2),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% termdiffs_strong_converges_everywhere
tff(fact_5748_at__within__Icc__at,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B2))
           => ( topolo174197925503356063within(A,X,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,X,top_top(set(A))) ) ) ) ) ).

% at_within_Icc_at
tff(fact_5749_has__real__derivative__powr,axiom,
    ! [Z: real,R3: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Z))
     => has_field_derivative(real,aTP_Lamp_on(real,fun(real,real),R3),aa(real,real,aa(real,fun(real,real),times_times(real),R3),powr(real,Z,aa(real,real,minus_minus(real,R3),one_one(real)))),topolo174197925503356063within(real,Z,top_top(set(real)))) ) ).

% has_real_derivative_powr
tff(fact_5750_at__within__Icc__at__left,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( topolo174197925503356063within(A,B2,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,B2,aa(A,set(A),set_ord_lessThan(A),B2)) ) ) ) ).

% at_within_Icc_at_left
tff(fact_5751_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D2: A,X: A,S: set(A),G: fun(A,A),E2: A] :
          ( has_field_derivative(A,F3,D2,topolo174197925503356063within(A,X,S))
         => ( has_field_derivative(A,G,E2,topolo174197925503356063within(A,X,S))
           => ( ( aa(A,A,G,X) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_og(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G),divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),D2),aa(A,A,G,X))),aa(A,A,aa(A,fun(A,A),times_times(A),E2),aa(A,A,F3,X))),aa(nat,A,power_power(A,aa(A,A,G,X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% DERIV_quotient
tff(fact_5752_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D2: A,X: A,S: set(A)] :
          ( has_field_derivative(A,F3,D2,topolo174197925503356063within(A,X,S))
         => ( ( aa(A,A,F3,X) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_oo(fun(A,A),fun(A,A),F3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D2),aa(A,A,inverse_inverse(A),aa(nat,A,power_power(A,aa(A,A,F3,X)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))))),topolo174197925503356063within(A,X,S)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_5753_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K5: real,C2: fun(nat,A),F3: fun(A,A),F10: A,Z: A] :
          ( ! [Z2: A] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z2)),K5))
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(A,fun(nat,A)),C2),Z2),aa(A,A,F3,Z2)) )
         => ( has_field_derivative(A,F3,F10,topolo174197925503356063within(A,Z,top_top(set(A))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K5))
             => sums(A,aa(A,fun(nat,A),aTP_Lamp_bu(fun(nat,A),fun(A,fun(nat,A)),C2),Z),F10) ) ) ) ) ).

% termdiffs_sums_strong
tff(fact_5754_DERIV__log,axiom,
    ! [X: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => has_field_derivative(real,log(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)),X)),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_log
tff(fact_5755_DERIV__fun__powr,axiom,
    ! [G: fun(real,real),M: real,X: real,R3: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,G,X)))
       => has_field_derivative(real,aa(real,fun(real,real),aTP_Lamp_op(fun(real,real),fun(real,fun(real,real)),G),R3),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),R3),powr(real,aa(real,real,G,X),aa(real,real,minus_minus(real,R3),aa(nat,real,semiring_1_of_nat(real),one_one(nat)))))),M),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_fun_powr
tff(fact_5756_DERIV__powr,axiom,
    ! [G: fun(real,real),M: real,X: real,F3: fun(real,real),R3: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,X,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,G,X)))
       => ( has_field_derivative(real,F3,R3,topolo174197925503356063within(real,X,top_top(set(real))))
         => has_field_derivative(real,aa(fun(real,real),fun(real,real),aTP_Lamp_oq(fun(real,real),fun(fun(real,real),fun(real,real)),G),F3),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(real,real,G,X),aa(real,real,F3,X))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),R3),aa(real,real,ln_ln(real),aa(real,real,G,X)))),divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),M),aa(real,real,F3,X)),aa(real,real,G,X)))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_powr
tff(fact_5757_DERIV__real__sqrt,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
     => has_field_derivative(real,sqrt,divide_divide(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_real_sqrt
tff(fact_5758_DERIV__series_H,axiom,
    ! [F3: fun(real,fun(nat,real)),F10: fun(real,fun(nat,real)),X0: real,A2: real,B2: real,L5: fun(nat,real)] :
      ( ! [N2: nat] : has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_or(fun(real,fun(nat,real)),fun(nat,fun(real,real)),F3),N2),aa(nat,real,aa(real,fun(nat,real),F10,X0),N2),topolo174197925503356063within(real,X0,top_top(set(real))))
     => ( ! [X3: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or5935395276787703475ssThan(real,A2,B2)))
           => summable(real,aa(real,fun(nat,real),F3,X3)) )
       => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X0),set_or5935395276787703475ssThan(real,A2,B2)))
         => ( summable(real,aa(real,fun(nat,real),F10,X0))
           => ( summable(real,L5)
             => ( ! [N2: nat,X3: real,Y4: real] :
                    ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or5935395276787703475ssThan(real,A2,B2)))
                   => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),Y4),set_or5935395276787703475ssThan(real,A2,B2)))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),F3,X3),N2)),aa(nat,real,aa(real,fun(nat,real),F3,Y4),N2)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,L5,N2)),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,X3),Y4))))) ) )
               => has_field_derivative(real,aTP_Lamp_os(fun(real,fun(nat,real)),fun(real,real),F3),suminf(real,aa(real,fun(nat,real),F10,X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_5759_DERIV__arctan,axiom,
    ! [X: real] : has_field_derivative(real,arctan,aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),topolo174197925503356063within(real,X,top_top(set(real)))) ).

% DERIV_arctan
tff(fact_5760_termdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_bu(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
           => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ot(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K5)))
               => has_field_derivative(A,aTP_Lamp_om(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_bu(fun(nat,A),fun(A,fun(nat,A)),C2),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ) ) ).

% termdiffs
tff(fact_5761_termdiffs__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K5)))
           => has_field_derivative(A,aTP_Lamp_om(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_bu(fun(nat,A),fun(A,fun(nat,A)),C2),X)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% termdiffs_strong
tff(fact_5762_termdiffs__strong_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K5: real,C2: fun(nat,A),Z: A] :
          ( ! [Z2: A] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z2)),K5))
             => summable(A,aa(A,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(A,fun(nat,A)),C2),Z2)) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,Z)),K5))
           => has_field_derivative(A,aTP_Lamp_om(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_bu(fun(nat,A),fun(A,fun(nat,A)),C2),Z)),topolo174197925503356063within(A,Z,top_top(set(A)))) ) ) ) ).

% termdiffs_strong'
tff(fact_5763_DERIV__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( cos(A,X) != zero_zero(A) )
         => has_field_derivative(A,tan(A),aa(A,A,inverse_inverse(A),aa(nat,A,power_power(A,cos(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_tan
tff(fact_5764_DERIV__real__sqrt__generic,axiom,
    ! [X: real,D4: real] :
      ( ( X != zero_zero(real) )
     => ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
         => ( D4 = divide_divide(real,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X)),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) )
       => ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
           => ( D4 = divide_divide(real,aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,X))),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))) ) )
         => has_field_derivative(real,sqrt,D4,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_5765_arsinh__real__has__field__derivative,axiom,
    ! [X: 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,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real)))),topolo174197925503356063within(real,X,A3)) ).

% arsinh_real_has_field_derivative
tff(fact_5766_DERIV__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( ( sin(A,X) != zero_zero(A) )
         => has_field_derivative(A,cot(A),aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),aa(nat,A,power_power(A,sin(A,X)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ).

% DERIV_cot
tff(fact_5767_has__field__derivative__tanh,axiom,
    ! [A9: $tType] :
      ( ( real_Vector_banach(A9)
        & real_V3459762299906320749_field(A9) )
     => ! [G: fun(A9,A9),X: A9,Db: A9,S: set(A9)] :
          ( ( cosh(A9,aa(A9,A9,G,X)) != zero_zero(A9) )
         => ( has_field_derivative(A9,G,Db,topolo174197925503356063within(A9,X,S))
           => has_field_derivative(A9,aTP_Lamp_ou(fun(A9,A9),fun(A9,A9),G),aa(A9,A9,aa(A9,fun(A9,A9),times_times(A9),aa(A9,A9,minus_minus(A9,one_one(A9)),aa(nat,A9,power_power(A9,aa(A9,A9,tanh(A9),aa(A9,A9,G,X))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),Db),topolo174197925503356063within(A9,X,S)) ) ) ) ).

% has_field_derivative_tanh
tff(fact_5768_arcosh__real__has__field__derivative,axiom,
    ! [X: real,A3: set(real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => has_field_derivative(real,arcosh(real),divide_divide(real,one_one(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),one_one(real)))),topolo174197925503356063within(real,X,A3)) ) ).

% arcosh_real_has_field_derivative
tff(fact_5769_artanh__real__has__field__derivative,axiom,
    ! [X: real,A3: set(real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => has_field_derivative(real,artanh(real),divide_divide(real,one_one(real),aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),topolo174197925503356063within(real,X,A3)) ) ).

% artanh_real_has_field_derivative
tff(fact_5770_DERIV__power__series_H,axiom,
    ! [R: real,F3: fun(nat,real),X0: real] :
      ( ! [X3: real] :
          ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R)))
         => summable(real,aa(real,fun(nat,real),aTP_Lamp_ov(fun(nat,real),fun(real,fun(nat,real)),F3),X3)) )
     => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X0),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R)))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R))
         => has_field_derivative(real,aTP_Lamp_ox(fun(nat,real),fun(real,real),F3),suminf(real,aa(real,fun(nat,real),aTP_Lamp_ov(fun(nat,real),fun(real,fun(nat,real)),F3),X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ).

% DERIV_power_series'
tff(fact_5771_DERIV__real__root,axiom,
    ! [N: nat,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
       => has_field_derivative(real,root(N),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),N)),aa(nat,real,power_power(real,aa(real,real,root(N),X)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_real_root
tff(fact_5772_DERIV__arccos,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => has_field_derivative(real,arccos,aa(real,real,inverse_inverse(real),aa(real,real,uminus_uminus(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_arccos
tff(fact_5773_DERIV__arcsin,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => has_field_derivative(real,arcsin,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,X),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_arcsin
tff(fact_5774_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun(nat,fun(real,real)),F3: fun(real,real),X: real,N: nat] :
      ( ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
        & ! [M5: nat,X3: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),X3),topolo174197925503356063within(real,X3,top_top(set(real)))) )
     => ? [T4: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X)))
          & ( aa(real,real,F3,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_oy(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T4),semiring_char_0_fact(real,N))),aa(nat,real,power_power(real,X),N))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_5775_Maclaurin__all__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F3: fun(real,real),X: real,N: nat] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
     => ( ! [M5: nat,X3: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),X3),topolo174197925503356063within(real,X3,top_top(set(real))))
       => ? [T4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X)))
            & ( aa(real,real,F3,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_oy(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T4),semiring_char_0_fact(real,N))),aa(nat,real,power_power(real,X),N))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_5776_DERIV__odd__real__root,axiom,
    ! [N: nat,X: real] :
      ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
     => ( ( X != zero_zero(real) )
       => has_field_derivative(real,root(N),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),N)),aa(nat,real,power_power(real,aa(real,real,root(N),X)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ).

% DERIV_odd_real_root
tff(fact_5777_Maclaurin__minus,axiom,
    ! [H: real,N: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H),zero_zero(real)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
         => ( ! [M5: nat,T4: real] :
                ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),H),T4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),zero_zero(real))) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
           => ? [T4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),H),T4))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),zero_zero(real)))
                & ( aa(real,real,F3,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_oz(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T4),semiring_char_0_fact(real,N))),aa(nat,real,power_power(real,H),N))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_5778_Maclaurin2,axiom,
    ! [H: real,Diff: fun(nat,fun(real,real)),F3: fun(real,real),N: nat] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
       => ( ! [M5: nat,T4: real] :
              ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T4))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),H)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
         => ? [T4: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T4))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),H))
              & ( aa(real,real,F3,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_oz(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T4),semiring_char_0_fact(real,N))),aa(nat,real,power_power(real,H),N))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_5779_Maclaurin,axiom,
    ! [H: real,N: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),H))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
         => ( ! [M5: nat,T4: real] :
                ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),H)) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
           => ? [T4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),T4))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),H))
                & ( aa(real,real,F3,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_oz(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T4),semiring_char_0_fact(real,N))),aa(nat,real,power_power(real,H),N))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_5780_Maclaurin__all__lt,axiom,
    ! [Diff: fun(nat,fun(real,real)),F3: fun(real,real),N: nat,X: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => ( ( X != zero_zero(real) )
         => ( ! [M5: nat,X3: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),X3),topolo174197925503356063within(real,X3,top_top(set(real))))
           => ? [T4: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,abs_abs(real),T4)))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X)))
                & ( aa(real,real,F3,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_oy(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T4),semiring_char_0_fact(real,N))),aa(nat,real,power_power(real,X),N))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_5781_Maclaurin__bi__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),F3: fun(real,real),N: nat,X: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
     => ( ! [M5: nat,T4: real] :
            ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X))) )
           => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
       => ? [T4: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),X)))
            & ( aa(real,real,F3,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_oy(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T4),semiring_char_0_fact(real,N))),aa(nat,real,power_power(real,X),N))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_5782_Taylor__down,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real),A2: real,B2: real,C2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
       => ( ! [M5: nat,T4: real] :
              ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),T4))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),B2)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),C2))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),C2),B2))
             => ? [T4: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),T4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),C2))
                  & ( aa(real,real,F3,A2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_pa(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),A2),C2)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T4),semiring_char_0_fact(real,N))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,A2),C2)),N))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_5783_Taylor__up,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real),A2: real,B2: real,C2: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
       => ( ! [M5: nat,T4: real] :
              ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),T4))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),B2)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),C2))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),B2))
             => ? [T4: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),T4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),B2))
                  & ( aa(real,real,F3,B2) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_pa(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),B2),C2)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T4),semiring_char_0_fact(real,N))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,B2),C2)),N))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_5784_Taylor,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),F3: fun(real,real),A2: real,B2: real,C2: real,X: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = F3 )
       => ( ! [M5: nat,T4: real] :
              ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),T4))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),B2)) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),C2))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),C2),B2))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),B2))
                 => ( ( X != C2 )
                   => ? [T4: real] :
                        ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),C2))
                         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),T4))
                            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),C2)) ) )
                        & ( ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),C2))
                         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),T4))
                            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),T4),X)) ) )
                        & ( aa(real,real,F3,X) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_pb(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),C2),X)),aa(nat,set(nat),set_ord_lessThan(nat),N))),aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(real,real,aa(nat,fun(real,real),Diff,N),T4),semiring_char_0_fact(real,N))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,X),C2)),N))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_5785_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: fun(nat,fun(real,real)),K: nat,B3: real] :
      ( ! [M5: nat,T4: real] :
          ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T4))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T4),H)) )
         => has_field_derivative(real,aa(nat,fun(real,real),Diff,M5),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M5)),T4),topolo174197925503356063within(real,T4,top_top(set(real)))) )
     => ( ( N = aa(nat,nat,suc,K) )
       => ! [M2: nat,T8: real] :
            ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),T8))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),T8),H)) )
           => has_field_derivative(real,aa(nat,fun(real,real),aa(real,fun(nat,fun(real,real)),aa(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real))),aTP_Lamp_pd(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),N),Diff),B3),M2),aa(real,real,minus_minus(real,aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T8)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_pe(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Diff),M2),T8)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,M2))))),aa(real,real,aa(real,fun(real,real),times_times(real),B3),divide_divide(real,aa(nat,real,power_power(real,T8),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,M2))),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,M2))))))),topolo174197925503356063within(real,T8,top_top(set(real)))) ) ) ) ).

% Maclaurin_lemma2
tff(fact_5786_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => has_field_derivative(real,aTP_Lamp_pf(real,real),suminf(real,aTP_Lamp_pg(real,fun(nat,real),X)),topolo174197925503356063within(real,X,top_top(set(real)))) ) ).

% DERIV_arctan_series
tff(fact_5787_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D4: real] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( ( X != zero_zero(real) )
       => ( ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),X))
             => ( D4 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,power_power(real,aa(real,real,root(N),X)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))) ) ) )
         => ( ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),zero_zero(real)))
               => ( D4 = aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,power_power(real,aa(real,real,root(N),X)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat))))))) ) ) )
           => ( ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
               => ( D4 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,power_power(real,aa(real,real,root(N),X)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))) ) )
             => has_field_derivative(real,root(N),D4,topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_5788_has__derivative__arcsin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G3: fun(A,real),S: set(A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,X)))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(A,real,G,X)),one_one(real)))
           => ( has_derivative(A,real,G,G3,topolo174197925503356063within(A,X,S))
             => has_derivative(A,real,aTP_Lamp_ph(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_pi(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G3),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% has_derivative_arcsin
tff(fact_5789_has__derivative__arccos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G3: fun(A,real),S: set(A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,X)))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(A,real,G,X)),one_one(real)))
           => ( has_derivative(A,real,G,G3,topolo174197925503356063within(A,X,S))
             => has_derivative(A,real,aTP_Lamp_pj(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_pk(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G3),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% has_derivative_arccos
tff(fact_5790_UNIV__char__of__nat,axiom,
    top_top(set(char)) = aa(set(nat),set(char),image(nat,char,unique5772411509450598832har_of(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))) ).

% UNIV_char_of_nat
tff(fact_5791_char__of__quasi__inj,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: A,N: A] :
          ( ( aa(A,char,unique5772411509450598832har_of(A),M) = aa(A,char,unique5772411509450598832har_of(A),N) )
        <=> ( modulo_modulo(A,M,aa(num,A,numeral_numeral(A),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))))))) = modulo_modulo(A,N,aa(num,A,numeral_numeral(A),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))))))) ) ) ) ).

% char_of_quasi_inj
tff(fact_5792_char__of__mod__256,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: A] : aa(A,char,unique5772411509450598832har_of(A),modulo_modulo(A,N,aa(num,A,numeral_numeral(A),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))) = aa(A,char,unique5772411509450598832har_of(A),N) ) ).

% char_of_mod_256
tff(fact_5793_has__derivative__scaleR,axiom,
    ! [C: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(D)
        & real_V822414075346904944vector(C) )
     => ! [F3: fun(D,real),F10: fun(D,real),X: D,S: set(D),G: fun(D,C),G3: fun(D,C)] :
          ( has_derivative(D,real,F3,F10,topolo174197925503356063within(D,X,S))
         => ( has_derivative(D,C,G,G3,topolo174197925503356063within(D,X,S))
           => has_derivative(D,C,aa(fun(D,C),fun(D,C),aTP_Lamp_pl(fun(D,real),fun(fun(D,C),fun(D,C)),F3),G),aa(fun(D,C),fun(D,C),aa(fun(D,C),fun(fun(D,C),fun(D,C)),aa(D,fun(fun(D,C),fun(fun(D,C),fun(D,C))),aa(fun(D,real),fun(D,fun(fun(D,C),fun(fun(D,C),fun(D,C)))),aTP_Lamp_pm(fun(D,real),fun(fun(D,real),fun(D,fun(fun(D,C),fun(fun(D,C),fun(D,C))))),F3),F10),X),G),G3),topolo174197925503356063within(D,X,S)) ) ) ) ).

% has_derivative_scaleR
tff(fact_5794_has__derivative__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F10: fun(A,B),X: A,S: set(A),T2: set(A)] :
          ( has_derivative(A,B,F3,F10,topolo174197925503356063within(A,X,S))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),S))
           => has_derivative(A,B,F3,F10,topolo174197925503356063within(A,X,T2)) ) ) ) ).

% has_derivative_subset
tff(fact_5795_has__derivative__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F10: fun(A,B),F5: filter(A),G: fun(A,B),G3: fun(A,B)] :
          ( has_derivative(A,B,F3,F10,F5)
         => ( has_derivative(A,B,G,G3,F5)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pn(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G),aa(fun(A,B),fun(A,B),aTP_Lamp_pn(fun(A,B),fun(fun(A,B),fun(A,B)),F10),G3),F5) ) ) ) ).

% has_derivative_add
tff(fact_5796_has__derivative__mult,axiom,
    ! [A: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(D)
        & real_V4412858255891104859lgebra(A) )
     => ! [F3: fun(D,A),F10: fun(D,A),X: D,S: set(D),G: fun(D,A),G3: fun(D,A)] :
          ( has_derivative(D,A,F3,F10,topolo174197925503356063within(D,X,S))
         => ( has_derivative(D,A,G,G3,topolo174197925503356063within(D,X,S))
           => has_derivative(D,A,aa(fun(D,A),fun(D,A),aTP_Lamp_po(fun(D,A),fun(fun(D,A),fun(D,A)),F3),G),aa(fun(D,A),fun(D,A),aa(fun(D,A),fun(fun(D,A),fun(D,A)),aa(D,fun(fun(D,A),fun(fun(D,A),fun(D,A))),aa(fun(D,A),fun(D,fun(fun(D,A),fun(fun(D,A),fun(D,A)))),aTP_Lamp_pp(fun(D,A),fun(fun(D,A),fun(D,fun(fun(D,A),fun(fun(D,A),fun(D,A))))),F3),F10),X),G),G3),topolo174197925503356063within(D,X,S)) ) ) ) ).

% has_derivative_mult
tff(fact_5797_has__derivative__in__compose2,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [T2: set(A),G: fun(A,B),G3: fun(A,fun(A,B)),F3: fun(C,A),S: set(C),X: C,F10: fun(C,A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),T2))
             => has_derivative(A,B,G,aa(A,fun(A,B),G3,X3),topolo174197925503356063within(A,X3,T2)) )
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,F3),S)),T2))
           => ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X),S))
             => ( has_derivative(C,A,F3,F10,topolo174197925503356063within(C,X,S))
               => has_derivative(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_pq(fun(A,B),fun(fun(C,A),fun(C,B)),G),F3),aa(fun(C,A),fun(C,B),aa(C,fun(fun(C,A),fun(C,B)),aa(fun(C,A),fun(C,fun(fun(C,A),fun(C,B))),aTP_Lamp_pr(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),G3),F3),X),F10),topolo174197925503356063within(C,X,S)) ) ) ) ) ) ).

% has_derivative_in_compose2
tff(fact_5798_has__derivative__divide_H,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V3459762299906320749_field(A) )
     => ! [F3: fun(C,A),F10: fun(C,A),X: C,S2: set(C),G: fun(C,A),G3: fun(C,A)] :
          ( has_derivative(C,A,F3,F10,topolo174197925503356063within(C,X,S2))
         => ( has_derivative(C,A,G,G3,topolo174197925503356063within(C,X,S2))
           => ( ( aa(C,A,G,X) != zero_zero(A) )
             => has_derivative(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_ps(fun(C,A),fun(fun(C,A),fun(C,A)),F3),G),aa(fun(C,A),fun(C,A),aa(fun(C,A),fun(fun(C,A),fun(C,A)),aa(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))),aa(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A)))),aTP_Lamp_pt(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),F3),F10),X),G),G3),topolo174197925503356063within(C,X,S2)) ) ) ) ) ).

% has_derivative_divide'
tff(fact_5799_has__derivative__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: fun(A,B),F10: fun(A,B),X: A,S2: set(A),N: nat] :
          ( has_derivative(A,B,F3,F10,topolo174197925503356063within(A,X,S2))
         => has_derivative(A,B,aa(nat,fun(A,B),aTP_Lamp_pu(fun(A,B),fun(nat,fun(A,B)),F3),N),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_pv(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),F3),F10),X),N),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_derivative_power
tff(fact_5800_has__derivative__ln,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G3: fun(A,real),S: set(A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G,X)))
         => ( has_derivative(A,real,G,G3,topolo174197925503356063within(A,X,S))
           => has_derivative(A,real,aTP_Lamp_pw(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_px(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G3),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_ln
tff(fact_5801_has__derivative__divide,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [F3: fun(C,A),F10: fun(C,A),X: C,S2: set(C),G: fun(C,A),G3: fun(C,A)] :
          ( has_derivative(C,A,F3,F10,topolo174197925503356063within(C,X,S2))
         => ( has_derivative(C,A,G,G3,topolo174197925503356063within(C,X,S2))
           => ( ( aa(C,A,G,X) != zero_zero(A) )
             => has_derivative(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_py(fun(C,A),fun(fun(C,A),fun(C,A)),F3),G),aa(fun(C,A),fun(C,A),aa(fun(C,A),fun(fun(C,A),fun(C,A)),aa(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))),aa(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A)))),aTP_Lamp_pz(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),F3),F10),X),G),G3),topolo174197925503356063within(C,X,S2)) ) ) ) ) ).

% has_derivative_divide
tff(fact_5802_has__derivative__powr,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G3: fun(A,real),X: A,X6: set(A),F3: fun(A,real),F10: fun(A,real)] :
          ( has_derivative(A,real,G,G3,topolo174197925503356063within(A,X,X6))
         => ( has_derivative(A,real,F3,F10,topolo174197925503356063within(A,X,X6))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G,X)))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),X6))
               => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_qa(fun(A,real),fun(fun(A,real),fun(A,real)),G),F3),aa(fun(A,real),fun(A,real),aa(fun(A,real),fun(fun(A,real),fun(A,real)),aa(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))),aa(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real)))),aTP_Lamp_qb(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),G),G3),X),F3),F10),topolo174197925503356063within(A,X,X6)) ) ) ) ) ) ).

% has_derivative_powr
tff(fact_5803_char__of__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,M: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))),N))
         => ( aa(A,char,unique5772411509450598832har_of(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),M)) = aa(A,char,unique5772411509450598832har_of(A),M) ) ) ) ).

% char_of_take_bit_eq
tff(fact_5804_has__derivative__real__sqrt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G3: fun(A,real),S: set(A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G,X)))
         => ( has_derivative(A,real,G,G3,topolo174197925503356063within(A,X,S))
           => has_derivative(A,real,aTP_Lamp_qc(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_qd(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G3),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_real_sqrt
tff(fact_5805_has__derivative__arctan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G3: fun(A,real),X: A,S: set(A)] :
          ( has_derivative(A,real,G,G3,topolo174197925503356063within(A,X,S))
         => has_derivative(A,real,aTP_Lamp_qe(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_qf(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G3),X),topolo174197925503356063within(A,X,S)) ) ) ).

% has_derivative_arctan
tff(fact_5806_has__derivative__tan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),X: A,G3: fun(A,real),S: set(A)] :
          ( ( cos(real,aa(A,real,G,X)) != zero_zero(real) )
         => ( has_derivative(A,real,G,G3,topolo174197925503356063within(A,X,S))
           => has_derivative(A,real,aTP_Lamp_qg(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_qh(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),X),G3),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_tan
tff(fact_5807_of__char__of,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [A2: A] : aa(char,A,comm_s6883823935334413003f_char(A),aa(A,char,unique5772411509450598832har_of(A),A2)) = modulo_modulo(A,A2,aa(num,A,numeral_numeral(A),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))))))) ) ).

% of_char_of
tff(fact_5808_char__of__def,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: A] : aa(A,char,unique5772411509450598832har_of(A),N) = char2(aa(bool,bool,fNot,dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),N)),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),one_one(nat)),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,one2)))),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit0,one2)))),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit1,one2)))),aa(nat,bool,bit_se5641148757651400278ts_bit(A,N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2))))) ) ).

% char_of_def
tff(fact_5809_listset_Osimps_I1_J,axiom,
    ! [A: $tType] : listset(A,nil(set(A))) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),nil(A)),bot_bot(set(list(A)))) ).

% listset.simps(1)
tff(fact_5810_of__char__mod__256,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [C2: char] : modulo_modulo(A,aa(char,A,comm_s6883823935334413003f_char(A),C2),aa(num,A,numeral_numeral(A),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))))))) = aa(char,A,comm_s6883823935334413003f_char(A),C2) ) ).

% of_char_mod_256
tff(fact_5811_nat__of__char__less__256,axiom,
    ! [C2: char] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(char,nat,comm_s6883823935334413003f_char(nat),C2)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))) ).

% nat_of_char_less_256
tff(fact_5812_range__nat__of__char,axiom,
    aa(set(char),set(nat),image(char,nat,comm_s6883823935334413003f_char(nat)),top_top(set(char))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2)))))))))) ).

% range_nat_of_char
tff(fact_5813_char__of__eq__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: A,C2: char] :
          ( ( aa(A,char,unique5772411509450598832har_of(A),N) = C2 )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))),N) = aa(char,A,comm_s6883823935334413003f_char(A),C2) ) ) ) ).

% char_of_eq_iff
tff(fact_5814_integer__of__char__code,axiom,
    ! [B0: bool,B1: bool,B22: bool,B32: bool,B42: bool,B52: bool,B62: bool,B72: bool] : integer_of_char(char2(B0,B1,B22,B32,B42,B52,B62,B72)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B72)),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B62))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B52))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B42))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B32))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B22))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B1))),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))),aa(bool,code_integer,zero_neq_one_of_bool(code_integer),B0)) ).

% integer_of_char_code
tff(fact_5815_of__char__Char,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [B0: bool,B1: bool,B22: bool,B32: bool,B42: bool,B52: bool,B62: bool,B72: bool] : aa(char,A,comm_s6883823935334413003f_char(A),char2(B0,B1,B22,B32,B42,B52,B62,B72)) = groups4207007520872428315er_sum(bool,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(list(bool),list(bool),aa(bool,fun(list(bool),list(bool)),cons(bool),B0),aa(list(bool),list(bool),aa(bool,fun(list(bool),list(bool)),cons(bool),B1),aa(list(bool),list(bool),aa(bool,fun(list(bool),list(bool)),cons(bool),B22),aa(list(bool),list(bool),aa(bool,fun(list(bool),list(bool)),cons(bool),B32),aa(list(bool),list(bool),aa(bool,fun(list(bool),list(bool)),cons(bool),B42),aa(list(bool),list(bool),aa(bool,fun(list(bool),list(bool)),cons(bool),B52),aa(list(bool),list(bool),aa(bool,fun(list(bool),list(bool)),cons(bool),B62),aa(list(bool),list(bool),aa(bool,fun(list(bool),list(bool)),cons(bool),B72),nil(bool)))))))))) ) ).

% of_char_Char
tff(fact_5816_shuffles_Opsimps_I1_J,axiom,
    ! [A: $tType,Ys: list(A)] :
      ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),nil(A)),Ys)))
     => ( shuffles(A,nil(A),Ys) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Ys),bot_bot(set(list(A)))) ) ) ).

% shuffles.psimps(1)
tff(fact_5817_list_Oinject,axiom,
    ! [A: $tType,X21: A,X222: list(A),Y21: A,Y22: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y21),Y22) )
    <=> ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
tff(fact_5818_list_Osimps_I15_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X21),aa(list(A),set(A),set2(A),X222)) ).

% list.simps(15)
tff(fact_5819_nth__Cons__Suc,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(nat,nat,suc,N)) = aa(nat,A,nth(A,Xs),N) ).

% nth_Cons_Suc
tff(fact_5820_nth__Cons__0,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),zero_zero(nat)) = X ).

% nth_Cons_0
tff(fact_5821_nths__singleton,axiom,
    ! [A: $tType,A3: set(nat),X: A] :
      ( ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3))
       => ( nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),A3) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) )
      & ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3))
       => ( nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),A3) = nil(A) ) ) ) ).

% nths_singleton
tff(fact_5822_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A2: A,X: B,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F3,A2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F3,X)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),groups4207007520872428315er_sum(B,A,F3,A2,Xs))) ) ).

% horner_sum_simps(2)
tff(fact_5823_enumerate__simps_I2_J,axiom,
    ! [B: $tType,N: nat,X: B,Xs: list(B)] : enumerate(B,N,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(list(product_prod(nat,B)),list(product_prod(nat,B)),aa(product_prod(nat,B),fun(list(product_prod(nat,B)),list(product_prod(nat,B))),cons(product_prod(nat,B)),aa(B,product_prod(nat,B),product_Pair(nat,B,N),X)),enumerate(B,aa(nat,nat,suc,N),Xs)) ).

% enumerate_simps(2)
tff(fact_5824_nth__Cons__numeral,axiom,
    ! [A: $tType,X: A,Xs: list(A),V: num] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V)),one_one(nat))) ).

% nth_Cons_numeral
tff(fact_5825_nth__Cons__pos,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,N),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_5826_splice_Ocases,axiom,
    ! [A: $tType,X: product_prod(list(A),list(A))] :
      ( ! [Ys3: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),nil(A)),Ys3)
     => ~ ! [X3: A,Xs2: list(A),Ys3: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Ys3) ) ).

% splice.cases
tff(fact_5827_shuffles_Ocases,axiom,
    ! [A: $tType,X: product_prod(list(A),list(A))] :
      ( ! [Ys3: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),nil(A)),Ys3)
     => ( ! [Xs2: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs2),nil(A))
       => ~ ! [X3: A,Xs2: list(A),Y4: A,Ys3: list(A)] : X != aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3)) ) ) ).

% shuffles.cases
tff(fact_5828_sorted__wrt_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,bool)),list(A))] :
      ( ! [P5: fun(A,fun(A,bool))] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),product_Pair(fun(A,fun(A,bool)),list(A),P5),nil(A))
     => ~ ! [P5: fun(A,fun(A,bool)),X3: A,Ys3: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),product_Pair(fun(A,fun(A,bool)),list(A),P5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3)) ) ).

% sorted_wrt.cases
tff(fact_5829_arg__min__list_Ocases,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [X: product_prod(fun(A,B),list(A))] :
          ( ! [F4: fun(A,B),X3: A] : X != aa(list(A),product_prod(fun(A,B),list(A)),product_Pair(fun(A,B),list(A),F4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))
         => ( ! [F4: fun(A,B),X3: A,Y4: A,Zs2: list(A)] : X != aa(list(A),product_prod(fun(A,B),list(A)),product_Pair(fun(A,B),list(A),F4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2)))
           => ~ ! [A5: fun(A,B)] : X != aa(list(A),product_prod(fun(A,B),list(A)),product_Pair(fun(A,B),list(A),A5),nil(A)) ) ) ) ).

% arg_min_list.cases
tff(fact_5830_successively_Ocases,axiom,
    ! [A: $tType,X: product_prod(fun(A,fun(A,bool)),list(A))] :
      ( ! [P5: fun(A,fun(A,bool))] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),product_Pair(fun(A,fun(A,bool)),list(A),P5),nil(A))
     => ( ! [P5: fun(A,fun(A,bool)),X3: A] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),product_Pair(fun(A,fun(A,bool)),list(A),P5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))
       => ~ ! [P5: fun(A,fun(A,bool)),X3: A,Y4: A,Xs2: list(A)] : X != aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),product_Pair(fun(A,fun(A,bool)),list(A),P5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs2))) ) ) ).

% successively.cases
tff(fact_5831_map__tailrec__rev_Ocases,axiom,
    ! [A: $tType,B: $tType,X: product_prod(fun(A,B),product_prod(list(A),list(B)))] :
      ( ! [F4: fun(A,B),Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),product_Pair(fun(A,B),product_prod(list(A),list(B)),F4),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),nil(A)),Bs2))
     => ~ ! [F4: fun(A,B),A5: A,As: list(A),Bs2: list(B)] : X != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),product_Pair(fun(A,B),product_prod(list(A),list(B)),F4),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),As)),Bs2)) ) ).

% map_tailrec_rev.cases
tff(fact_5832_shuffles_Opinduct,axiom,
    ! [A: $tType,A0: list(A),A1: list(A),P: fun(list(A),fun(list(A),bool))] :
      ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),A0),A1)))
     => ( ! [Ys3: list(A)] :
            ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),nil(A)),Ys3)))
           => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,nil(A)),Ys3)) )
       => ( ! [Xs2: list(A)] :
              ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs2),nil(A))))
             => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs2),nil(A))) )
         => ( ! [X3: A,Xs2: list(A),Y4: A,Ys3: list(A)] :
                ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3))))
               => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3)))
                 => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Ys3))
                   => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3))) ) ) )
           => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,A0),A1)) ) ) ) ) ).

% shuffles.pinduct
tff(fact_5833_list__nonempty__induct,axiom,
    ! [A: $tType,Xs: list(A),P: fun(list(A),bool)] :
      ( ( Xs != nil(A) )
     => ( ! [X3: A] : pp(aa(list(A),bool,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))))
       => ( ! [X3: A,Xs2: list(A)] :
              ( ( Xs2 != nil(A) )
             => ( pp(aa(list(A),bool,P,Xs2))
               => pp(aa(list(A),bool,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))) ) )
         => pp(aa(list(A),bool,P,Xs)) ) ) ) ).

% list_nonempty_induct
tff(fact_5834_list__induct2_H,axiom,
    ! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),bool)),Xs: list(A),Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),nil(B)))
     => ( ! [X3: A,Xs2: list(A)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),nil(B)))
       => ( ! [Y4: B,Ys3: list(B)] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys3)))
         => ( ! [X3: A,Xs2: list(A),Y4: B,Ys3: list(B)] :
                ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),Ys3))
               => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys3))) )
           => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs),Ys)) ) ) ) ) ).

% list_induct2'
tff(fact_5835_neq__Nil__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
    <=> ? [Y5: A,Ys4: list(A)] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) ) ).

% neq_Nil_conv
tff(fact_5836_remdups__adj_Ocases,axiom,
    ! [A: $tType,X: list(A)] :
      ( ( X != nil(A) )
     => ( ! [X3: A] : X != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))
       => ~ ! [X3: A,Y4: A,Xs2: list(A)] : X != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_5837_transpose_Ocases,axiom,
    ! [A: $tType,X: list(list(A))] :
      ( ( X != nil(list(A)) )
     => ( ! [Xss2: list(list(A))] : X != aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2)
       => ~ ! [X3: A,Xs2: list(A),Xss2: list(list(A))] : X != aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2) ) ) ).

% transpose.cases
tff(fact_5838_min__list_Ocases,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: list(A)] :
          ( ! [X3: A,Xs2: list(A)] : X != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)
         => ( X = nil(A) ) ) ) ).

% min_list.cases
tff(fact_5839_list_Oexhaust,axiom,
    ! [A: $tType,Y: list(A)] :
      ( ( Y != nil(A) )
     => ~ ! [X212: A,X223: list(A)] : Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X212),X223) ) ).

% list.exhaust
tff(fact_5840_list_OdiscI,axiom,
    ! [A: $tType,List: list(A),X21: A,X222: list(A)] :
      ( ( List = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222) )
     => ( List != nil(A) ) ) ).

% list.discI
tff(fact_5841_list_Odistinct_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : nil(A) != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222) ).

% list.distinct(1)
tff(fact_5842_list__induct2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(list(A),fun(list(B),bool))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,nil(A)),nil(B)))
       => ( ! [X3: A,Xs2: list(A),Y4: B,Ys3: list(B)] :
              ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys3) )
             => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs2),Ys3))
               => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys3))) ) )
         => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),P,Xs),Ys)) ) ) ) ).

% list_induct2
tff(fact_5843_list__induct3,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C),P: fun(list(A),fun(list(B),fun(list(C),bool)))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(C),nat,size_size(list(C)),Zs) )
       => ( pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P,nil(A)),nil(B)),nil(C)))
         => ( ! [X3: A,Xs2: list(A),Y4: B,Ys3: list(B),Z2: C,Zs2: list(C)] :
                ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys3) )
               => ( ( aa(list(B),nat,size_size(list(B)),Ys3) = aa(list(C),nat,size_size(list(C)),Zs2) )
                 => ( pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P,Xs2),Ys3),Zs2))
                   => pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys3)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Z2),Zs2))) ) ) )
           => pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),aa(list(A),fun(list(B),fun(list(C),bool)),P,Xs),Ys),Zs)) ) ) ) ) ).

% list_induct3
tff(fact_5844_list__induct4,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list(A),Ys: list(B),Zs: list(C),Ws: list(D),P: fun(list(A),fun(list(B),fun(list(C),fun(list(D),bool))))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(C),nat,size_size(list(C)),Zs) )
       => ( ( aa(list(C),nat,size_size(list(C)),Zs) = aa(list(D),nat,size_size(list(D)),Ws) )
         => ( pp(aa(list(D),bool,aa(list(C),fun(list(D),bool),aa(list(B),fun(list(C),fun(list(D),bool)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),bool))),P,nil(A)),nil(B)),nil(C)),nil(D)))
           => ( ! [X3: A,Xs2: list(A),Y4: B,Ys3: list(B),Z2: C,Zs2: list(C),W2: D,Ws2: list(D)] :
                  ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys3) )
                 => ( ( aa(list(B),nat,size_size(list(B)),Ys3) = aa(list(C),nat,size_size(list(C)),Zs2) )
                   => ( ( aa(list(C),nat,size_size(list(C)),Zs2) = aa(list(D),nat,size_size(list(D)),Ws2) )
                     => ( pp(aa(list(D),bool,aa(list(C),fun(list(D),bool),aa(list(B),fun(list(C),fun(list(D),bool)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),bool))),P,Xs2),Ys3),Zs2),Ws2))
                       => pp(aa(list(D),bool,aa(list(C),fun(list(D),bool),aa(list(B),fun(list(C),fun(list(D),bool)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),bool))),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys3)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Z2),Zs2)),aa(list(D),list(D),aa(D,fun(list(D),list(D)),cons(D),W2),Ws2))) ) ) ) )
             => pp(aa(list(D),bool,aa(list(C),fun(list(D),bool),aa(list(B),fun(list(C),fun(list(D),bool)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),bool))),P,Xs),Ys),Zs),Ws)) ) ) ) ) ) ).

% list_induct4
tff(fact_5845_impossible__Cons,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)))
     => ( Xs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys) ) ) ).

% impossible_Cons
tff(fact_5846_removeAll_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Y: A,Xs: list(A)] :
      ( ( ( X = Y )
       => ( aa(list(A),list(A),removeAll(A,X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs)) = aa(list(A),list(A),removeAll(A,X),Xs) ) )
      & ( ( X != Y )
       => ( aa(list(A),list(A),removeAll(A,X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),aa(list(A),list(A),removeAll(A,X),Xs)) ) ) ) ).

% removeAll.simps(2)
tff(fact_5847_distinct__length__2__or__more,axiom,
    ! [A: $tType,A2: A,B2: A,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Xs)))
    <=> ( ( A2 != B2 )
        & distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),Xs))
        & distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Xs)) ) ) ).

% distinct_length_2_or_more
tff(fact_5848_list__update_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),I2: nat,V: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),I2,V) = case_nat(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Xs),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_qi(A,fun(list(A),fun(A,fun(nat,list(A)))),X),Xs),V),I2) ).

% list_update.simps(2)
tff(fact_5849_not__Cons__self2,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) != Xs ).

% not_Cons_self2
tff(fact_5850_set__ConsD,axiom,
    ! [A: $tType,Y: A,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))))
     => ( ( Y = X )
        | pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% set_ConsD
tff(fact_5851_list_Oset__cases,axiom,
    ! [A: $tType,E2: A,A2: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),E2),aa(list(A),set(A),set2(A),A2)))
     => ( ! [Z23: list(A)] : A2 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),Z23)
       => ~ ! [Z12: A,Z23: list(A)] :
              ( ( A2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z12),Z23) )
             => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),E2),aa(list(A),set(A),set2(A),Z23))) ) ) ) ).

% list.set_cases
tff(fact_5852_list_Oset__intros_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X21),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)))) ).

% list.set_intros(1)
tff(fact_5853_list_Oset__intros_I2_J,axiom,
    ! [A: $tType,Y: A,X222: list(A),X21: A] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),aa(list(A),set(A),set2(A),X222)))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)))) ) ).

% list.set_intros(2)
tff(fact_5854_remove1_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Y: A,Xs: list(A)] :
      ( ( ( X = Y )
       => ( remove1(A,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs)) = Xs ) )
      & ( ( X != Y )
       => ( remove1(A,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),remove1(A,X,Xs)) ) ) ) ).

% remove1.simps(2)
tff(fact_5855_set__subset__Cons,axiom,
    ! [A: $tType,Xs: list(A),X: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)))) ).

% set_subset_Cons
tff(fact_5856_length__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_Cons
tff(fact_5857_Suc__length__conv,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( aa(nat,nat,suc,N) = aa(list(A),nat,size_size(list(A)),Xs) )
    <=> ? [Y5: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = N ) ) ) ).

% Suc_length_conv
tff(fact_5858_length__Suc__conv,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
    <=> ? [Y5: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = N ) ) ) ).

% length_Suc_conv
tff(fact_5859_Cons__in__shuffles__rightI,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A),Z: A] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
     => pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z),Zs)),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z),Ys)))) ) ).

% Cons_in_shuffles_rightI
tff(fact_5860_Cons__in__shuffles__leftI,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A),Z: A] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
     => pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z),Zs)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z),Xs),Ys))) ) ).

% Cons_in_shuffles_leftI
tff(fact_5861_Cons__shuffles__subset2,axiom,
    ! [A: $tType,Y: A,Xs: list(A),Ys: list(A)] : pp(aa(set(list(A)),bool,aa(set(list(A)),fun(set(list(A)),bool),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),image(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),shuffles(A,Xs,Ys))),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))) ).

% Cons_shuffles_subset2
tff(fact_5862_Cons__shuffles__subset1,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A)] : pp(aa(set(list(A)),bool,aa(set(list(A)),fun(set(list(A)),bool),ord_less_eq(set(list(A))),aa(set(list(A)),set(list(A)),image(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),shuffles(A,Xs,Ys))),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys))) ).

% Cons_shuffles_subset1
tff(fact_5863_distinct__singleton,axiom,
    ! [A: $tType,X: A] : distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ).

% distinct_singleton
tff(fact_5864_distinct_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))
    <=> ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
        & distinct(A,Xs) ) ) ).

% distinct.simps(2)
tff(fact_5865_replicate__Suc,axiom,
    ! [A: $tType,N: nat,X: A] : replicate(A,aa(nat,nat,suc,N),X) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),replicate(A,N,X)) ).

% replicate_Suc
tff(fact_5866_list__update__code_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),I2: nat,Y: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(nat,nat,suc,I2),Y) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),list_update(A,Xs,I2,Y)) ).

% list_update_code(3)
tff(fact_5867_insort__key_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),X: B,Y: B,Ys: list(B)] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X)),aa(B,A,F3,Y)))
           => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X)),aa(B,A,F3,Y)))
           => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),Ys)) ) ) ) ) ).

% insort_key.simps(2)
tff(fact_5868_list__update__code_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),zero_zero(nat),Y) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs) ).

% list_update_code(2)
tff(fact_5869_shufflesE,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
     => ( ( ( Zs = Xs )
         => ( Ys != nil(A) ) )
       => ( ( ( Zs = Ys )
           => ( Xs != nil(A) ) )
         => ( ! [X3: A,Xs4: list(A)] :
                ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs4) )
               => ! [Z2: A,Zs4: list(A)] :
                    ( ( Zs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z2),Zs4) )
                   => ( ( X3 = Z2 )
                     => ~ pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs4),shuffles(A,Xs4,Ys))) ) ) )
           => ~ ! [Y4: A,Ys5: list(A)] :
                  ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys5) )
                 => ! [Z2: A,Zs4: list(A)] :
                      ( ( Zs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z2),Zs4) )
                     => ( ( Y4 = Z2 )
                       => ~ pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs4),shuffles(A,Xs,Ys5))) ) ) ) ) ) ) ) ).

% shufflesE
tff(fact_5870_insort__key_Osimps_I1_J,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),X: B] : aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),nil(B)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),nil(B)) ) ).

% insort_key.simps(1)
tff(fact_5871_remdups_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( remdups(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = remdups(A,Xs) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( remdups(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),remdups(A,Xs)) ) ) ) ).

% remdups.simps(2)
tff(fact_5872_Cons__in__subseqsD,axiom,
    ! [A: $tType,Y: A,Ys: list(A),Xs: list(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs))))
     => pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Ys),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs)))) ) ).

% Cons_in_subseqsD
tff(fact_5873_nth__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = case_nat(A,X,nth(A,Xs),N) ).

% nth_Cons
tff(fact_5874_Suc__le__length__iff,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,N)),aa(list(A),nat,size_size(list(A)),Xs)))
    <=> ? [X4: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Ys4) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Ys4))) ) ) ).

% Suc_le_length_iff
tff(fact_5875_insort__is__Cons,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Xs: list(B),F3: fun(B,A),A2: B] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(list(B),set(B),set2(B),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,A2)),aa(B,A,F3,X3))) )
         => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),A2),Xs) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A2),Xs) ) ) ) ).

% insort_is_Cons
tff(fact_5876_count__list_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Y: A,Xs: list(A)] :
      ( ( ( X = Y )
       => ( aa(A,nat,count_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Y) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,count_list(A,Xs),Y)),one_one(nat)) ) )
      & ( ( X != Y )
       => ( aa(A,nat,count_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Y) = aa(A,nat,count_list(A,Xs),Y) ) ) ) ).

% count_list.simps(2)
tff(fact_5877_list_Osize_I4_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),X222)),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size(4)
tff(fact_5878_nth__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( ( ( N = zero_zero(nat) )
       => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = X ) )
      & ( ( N != zero_zero(nat) )
       => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,N),one_one(nat))) ) ) ) ).

% nth_Cons'
tff(fact_5879_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,X: fun(A,nat),X21: A,X222: list(A)] : aa(list(A),nat,size_list(A,X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,X,X21)),aa(list(A),nat,size_list(A,X),X222))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_5880_nth__equal__first__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = X )
        <=> ( N = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_5881_nth__non__equal__first__eq,axiom,
    ! [A: $tType,X: A,Y: A,Xs: list(A),N: nat] :
      ( ( X != Y )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),N) = Y )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,N),one_one(nat))) = Y )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N)) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_5882_Cons__replicate__eq,axiom,
    ! [A: $tType,X: A,Xs: list(A),N: nat,Y: A] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = replicate(A,N,Y) )
    <=> ( ( X = Y )
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
        & ( Xs = replicate(A,aa(nat,nat,minus_minus(nat,N),one_one(nat)),X) ) ) ) ).

% Cons_replicate_eq
tff(fact_5883_shuffles_Opsimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),nil(A))))
     => ( shuffles(A,Xs,nil(A)) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Xs),bot_bot(set(list(A)))) ) ) ).

% shuffles.psimps(2)
tff(fact_5884_set__Cons__sing__Nil,axiom,
    ! [A: $tType,A3: set(A)] : set_Cons(A,A3,aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),nil(A)),bot_bot(set(list(A))))) = aa(set(A),set(list(A)),image(A,list(A),aTP_Lamp_qj(A,list(A))),A3) ).

% set_Cons_sing_Nil
tff(fact_5885_String_Ochar__of__ascii__of,axiom,
    ! [C2: char] : aa(char,nat,comm_s6883823935334413003f_char(nat),ascii_of(C2)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),aa(char,nat,comm_s6883823935334413003f_char(nat),C2)) ).

% String.char_of_ascii_of
tff(fact_5886_concat__inth,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = X ).

% concat_inth
tff(fact_5887_same__append__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs) )
    <=> ( Ys = Zs ) ) ).

% same_append_eq
tff(fact_5888_append__same__eq,axiom,
    ! [A: $tType,Ys: list(A),Xs: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Xs) )
    <=> ( Ys = Zs ) ) ).

% append_same_eq
tff(fact_5889_append__assoc,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),Zs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs)) ).

% append_assoc
tff(fact_5890_append_Oassoc,axiom,
    ! [A: $tType,A2: list(A),B2: list(A),C2: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A2),B2)),C2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),B2),C2)) ).

% append.assoc
tff(fact_5891_append__is__Nil__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = nil(A) )
    <=> ( ( Xs = nil(A) )
        & ( Ys = nil(A) ) ) ) ).

% append_is_Nil_conv
tff(fact_5892_Nil__is__append__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) )
    <=> ( ( Xs = nil(A) )
        & ( Ys = nil(A) ) ) ) ).

% Nil_is_append_conv
tff(fact_5893_self__append__conv2,axiom,
    ! [A: $tType,Y: list(A),Xs: list(A)] :
      ( ( Y = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Y) )
    <=> ( Xs = nil(A) ) ) ).

% self_append_conv2
tff(fact_5894_append__self__conv2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = Ys )
    <=> ( Xs = nil(A) ) ) ).

% append_self_conv2
tff(fact_5895_self__append__conv,axiom,
    ! [A: $tType,Y: list(A),Ys: list(A)] :
      ( ( Y = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Y),Ys) )
    <=> ( Ys = nil(A) ) ) ).

% self_append_conv
tff(fact_5896_append__self__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = Xs )
    <=> ( Ys = nil(A) ) ) ).

% append_self_conv
tff(fact_5897_append__Nil2,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),nil(A)) = Xs ).

% append_Nil2
tff(fact_5898_append_Oright__neutral,axiom,
    ! [A: $tType,A2: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A2),nil(A)) = A2 ).

% append.right_neutral
tff(fact_5899_append__eq__append__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Us: list(A),Vs: list(A)] :
      ( ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        | ( aa(list(A),nat,size_size(list(A)),Us) = aa(list(A),nat,size_size(list(A)),Vs) ) )
     => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs) )
      <=> ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_5900_concat__append,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] : concat(A,aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),concat(A,Xs)),concat(A,Ys)) ).

% concat_append
tff(fact_5901_removeAll__append,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),removeAll(A,X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),removeAll(A,X),Xs)),aa(list(A),list(A),removeAll(A,X),Ys)) ).

% removeAll_append
tff(fact_5902_append1__eq__conv,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys: list(A),Y: A] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))) )
    <=> ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
tff(fact_5903_length__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ).

% length_append
tff(fact_5904_size__list__append,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_list(A,F3),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_list(A,F3),Xs)),aa(list(A),nat,size_list(A,F3),Ys)) ).

% size_list_append
tff(fact_5905_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : aa(set(nat),list(nat),linord4507533701916653071of_set(nat),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,suc,K))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),aa(nat,set(nat),set_ord_lessThan(nat),K))),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),K),nil(nat))) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_5906_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : aa(set(nat),list(nat),linord4507533701916653071of_set(nat),aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,suc,K))) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),aa(nat,set(nat),set_ord_atMost(nat),K))),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,K)),nil(nat))) ).

% sorted_list_of_set_atMost_Suc
tff(fact_5907_nth__append__length,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = X ).

% nth_append_length
tff(fact_5908_nth__append__length__plus,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) = aa(nat,A,nth(A,Ys),N) ).

% nth_append_length_plus
tff(fact_5909_list__update__length,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys: list(A),Y: A] : list_update(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys)),aa(list(A),nat,size_size(list(A)),Xs),Y) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ).

% list_update_length
tff(fact_5910_distinct__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( distinct(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
    <=> ( distinct(A,Xs)
        & distinct(A,Ys)
        & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) ) ) ) ).

% distinct_append
tff(fact_5911_n__lists__Nil,axiom,
    ! [A: $tType,N: nat] :
      ( ( ( N = zero_zero(nat) )
       => ( n_lists(A,N,nil(A)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ) )
      & ( ( N != zero_zero(nat) )
       => ( n_lists(A,N,nil(A)) = nil(list(A)) ) ) ) ).

% n_lists_Nil
tff(fact_5912_concat__eq__appendD,axiom,
    ! [A: $tType,Xss: list(list(A)),Ys: list(A),Zs: list(A)] :
      ( ( concat(A,Xss) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
     => ( ( Xss != nil(list(A)) )
       => ? [Xss1: list(list(A)),Xs2: list(A),Xs4: list(A),Xss22: list(list(A))] :
            ( ( Xss = aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),Xss1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),Xs4)),Xss22)) )
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),concat(A,Xss1)),Xs2) )
            & ( Zs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs4),concat(A,Xss22)) ) ) ) ) ).

% concat_eq_appendD
tff(fact_5913_concat__eq__append__conv,axiom,
    ! [A: $tType,Xss: list(list(A)),Ys: list(A),Zs: list(A)] :
      ( ( concat(A,Xss) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
    <=> ( ( ( Xss = nil(list(A)) )
         => ( ( Ys = nil(A) )
            & ( Zs = nil(A) ) ) )
        & ( ( Xss != nil(list(A)) )
         => ? [Xss12: list(list(A)),Xs3: list(A),Xs5: list(A),Xss23: list(list(A))] :
              ( ( Xss = aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),Xss12),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs3),Xs5)),Xss23)) )
              & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),concat(A,Xss12)),Xs3) )
              & ( Zs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs5),concat(A,Xss23)) ) ) ) ) ) ).

% concat_eq_append_conv
tff(fact_5914_rev__nonempty__induct,axiom,
    ! [A: $tType,Xs: list(A),P: fun(list(A),bool)] :
      ( ( Xs != nil(A) )
     => ( ! [X3: A] : pp(aa(list(A),bool,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))))
       => ( ! [X3: A,Xs2: list(A)] :
              ( ( Xs2 != nil(A) )
             => ( pp(aa(list(A),bool,P,Xs2))
               => pp(aa(list(A),bool,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))))) ) )
         => pp(aa(list(A),bool,P,Xs)) ) ) ) ).

% rev_nonempty_induct
tff(fact_5915_append__eq__Cons__conv,axiom,
    ! [A: $tType,Ys: list(A),Zs: list(A),X: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
    <=> ( ( ( Ys = nil(A) )
          & ( Zs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) )
        | ? [Ys6: list(A)] :
            ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys6) )
            & ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys6),Zs) = Xs ) ) ) ) ).

% append_eq_Cons_conv
tff(fact_5916_Cons__eq__append__conv,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
    <=> ( ( ( Ys = nil(A) )
          & ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = Zs ) )
        | ? [Ys6: list(A)] :
            ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys6) = Ys )
            & ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys6),Zs) ) ) ) ) ).

% Cons_eq_append_conv
tff(fact_5917_eq__Nil__appendI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
     => ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),Ys) ) ) ).

% eq_Nil_appendI
tff(fact_5918_rev__exhaust,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ~ ! [Ys3: list(A),Y4: A] : Xs != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),nil(A))) ) ).

% rev_exhaust
tff(fact_5919_rev__induct,axiom,
    ! [A: $tType,P: fun(list(A),bool),Xs: list(A)] :
      ( pp(aa(list(A),bool,P,nil(A)))
     => ( ! [X3: A,Xs2: list(A)] :
            ( pp(aa(list(A),bool,P,Xs2))
           => pp(aa(list(A),bool,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))))) )
       => pp(aa(list(A),bool,P,Xs)) ) ) ).

% rev_induct
tff(fact_5920_append_Oleft__neutral,axiom,
    ! [A: $tType,A2: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),A2) = A2 ).

% append.left_neutral
tff(fact_5921_append__Nil,axiom,
    ! [A: $tType,Ys: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nil(A)),Ys) = Ys ).

% append_Nil
tff(fact_5922_listset_Osimps_I2_J,axiom,
    ! [A: $tType,A3: set(A),As2: list(set(A))] : listset(A,aa(list(set(A)),list(set(A)),aa(set(A),fun(list(set(A)),list(set(A))),cons(set(A)),A3),As2)) = set_Cons(A,A3,listset(A,As2)) ).

% listset.simps(2)
tff(fact_5923_append__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) ).

% append_Cons
tff(fact_5924_Cons__eq__appendI,axiom,
    ! [A: $tType,X: A,Xs1: list(A),Ys: list(A),Xs: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs1) = Ys )
     => ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),Zs) )
       => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) ) ) ) ).

% Cons_eq_appendI
tff(fact_5925_concat_Osimps_I2_J,axiom,
    ! [A: $tType,X: list(A),Xs: list(list(A))] : concat(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),X),Xs)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),concat(A,Xs)) ).

% concat.simps(2)
tff(fact_5926_split__list__first__prop__iff,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ? [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
          & pp(aa(A,bool,P,X4)) )
    <=> ? [Ys4: list(A),X4: A] :
          ( ? [Zs3: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Zs3))
          & pp(aa(A,bool,P,X4))
          & ! [Xa4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),aa(list(A),set(A),set2(A),Ys4)))
             => ~ pp(aa(A,bool,P,Xa4)) ) ) ) ).

% split_list_first_prop_iff
tff(fact_5927_split__list__last__prop__iff,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ? [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
          & pp(aa(A,bool,P,X4)) )
    <=> ? [Ys4: list(A),X4: A,Zs3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Zs3)) )
          & pp(aa(A,bool,P,X4))
          & ! [Xa4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),aa(list(A),set(A),set2(A),Zs3)))
             => ~ pp(aa(A,bool,P,Xa4)) ) ) ) ).

% split_list_last_prop_iff
tff(fact_5928_in__set__conv__decomp__first,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
    <=> ? [Ys4: list(A),Zs3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs3)) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Ys4))) ) ) ).

% in_set_conv_decomp_first
tff(fact_5929_in__set__conv__decomp__last,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
    <=> ? [Ys4: list(A),Zs3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs3)) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Zs3))) ) ) ).

% in_set_conv_decomp_last
tff(fact_5930_split__list__first__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ? [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
          & pp(aa(A,bool,P,X2)) )
     => ~ ! [Ys3: list(A),X3: A] :
            ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Zs2))
           => ( pp(aa(A,bool,P,X3))
             => ~ ! [Xa2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),aa(list(A),set(A),set2(A),Ys3)))
                   => ~ pp(aa(A,bool,P,Xa2)) ) ) ) ) ).

% split_list_first_propE
tff(fact_5931_split__list__last__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ? [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
          & pp(aa(A,bool,P,X2)) )
     => ~ ! [Ys3: list(A),X3: A,Zs2: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Zs2)) )
           => ( pp(aa(A,bool,P,X3))
             => ~ ! [Xa2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),aa(list(A),set(A),set2(A),Zs2)))
                   => ~ pp(aa(A,bool,P,Xa2)) ) ) ) ) ).

% split_list_last_propE
tff(fact_5932_split__list__first__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ? [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
          & pp(aa(A,bool,P,X2)) )
     => ? [Ys3: list(A),X3: A] :
          ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Zs2))
          & pp(aa(A,bool,P,X3))
          & ! [Xa2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),aa(list(A),set(A),set2(A),Ys3)))
             => ~ pp(aa(A,bool,P,Xa2)) ) ) ) ).

% split_list_first_prop
tff(fact_5933_split__list__last__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ? [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
          & pp(aa(A,bool,P,X2)) )
     => ? [Ys3: list(A),X3: A,Zs2: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Zs2)) )
          & pp(aa(A,bool,P,X3))
          & ! [Xa2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),aa(list(A),set(A),set2(A),Zs2)))
             => ~ pp(aa(A,bool,P,Xa2)) ) ) ) ).

% split_list_last_prop
tff(fact_5934_in__set__conv__decomp,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
    <=> ? [Ys4: list(A),Zs3: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs3)) ) ).

% in_set_conv_decomp
tff(fact_5935_append__Cons__eq__iff,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A),Xs6: list(A),Ys7: list(A)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Ys)))
       => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs6),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys7)) )
        <=> ( ( Xs = Xs6 )
            & ( Ys = Ys7 ) ) ) ) ) ).

% append_Cons_eq_iff
tff(fact_5936_split__list__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ? [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
          & pp(aa(A,bool,P,X2)) )
     => ~ ! [Ys3: list(A),X3: A] :
            ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Zs2))
           => ~ pp(aa(A,bool,P,X3)) ) ) ).

% split_list_propE
tff(fact_5937_split__list__first,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ? [Ys3: list(A),Zs2: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs2)) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Ys3))) ) ) ).

% split_list_first
tff(fact_5938_split__list__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ? [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
          & pp(aa(A,bool,P,X2)) )
     => ? [Ys3: list(A),X3: A] :
          ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Zs2))
          & pp(aa(A,bool,P,X3)) ) ) ).

% split_list_prop
tff(fact_5939_split__list__last,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ? [Ys3: list(A),Zs2: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs2)) )
          & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Zs2))) ) ) ).

% split_list_last
tff(fact_5940_split__list,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ? [Ys3: list(A),Zs2: list(A)] : Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs2)) ) ).

% split_list
tff(fact_5941_replicate__app__Cons__same,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,N,X)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,N,X)),Xs)) ).

% replicate_app_Cons_same
tff(fact_5942_replicate__add,axiom,
    ! [A: $tType,N: nat,M: nat,X: A] : replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M),X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,N,X)),replicate(A,M,X)) ).

% replicate_add
tff(fact_5943_append__replicate__commute,axiom,
    ! [A: $tType,N: nat,X: A,K: nat] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,N,X)),replicate(A,K,X)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,K,X)),replicate(A,N,X)) ).

% append_replicate_commute
tff(fact_5944_append__eq__append__conv2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A),Ts: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Ts) )
    <=> ? [Us2: list(A)] :
          ( ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Us2) )
            & ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),Ys) = Ts ) )
          | ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us2) = Zs )
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),Ts) ) ) ) ) ).

% append_eq_append_conv2
tff(fact_5945_append__eq__appendI,axiom,
    ! [A: $tType,Xs: list(A),Xs1: list(A),Zs: list(A),Ys: list(A),Us: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Xs1) = Zs )
     => ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs1),Us) )
       => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Zs),Us) ) ) ) ).

% append_eq_appendI
tff(fact_5946_enumerate__append__eq,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] : enumerate(A,N,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),aa(list(product_prod(nat,A)),fun(list(product_prod(nat,A)),list(product_prod(nat,A))),append(product_prod(nat,A)),enumerate(A,N,Xs)),enumerate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).

% enumerate_append_eq
tff(fact_5947_remdups__append2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : remdups(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),remdups(A,Ys))) = remdups(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) ).

% remdups_append2
tff(fact_5948_remove1__append,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( remove1(A,X,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remove1(A,X,Xs)),Ys) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( remove1(A,X,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),remove1(A,X,Ys)) ) ) ) ).

% remove1_append
tff(fact_5949_comm__append__are__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) )
     => ? [M5: nat,N2: nat,Zs2: list(A)] :
          ( ( concat(A,replicate(list(A),M5,Zs2)) = Xs )
          & ( concat(A,replicate(list(A),N2,Zs2)) = Ys ) ) ) ).

% comm_append_are_replicate
tff(fact_5950_same__length__different,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != Ys )
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
       => ? [Pre: list(A),X3: A,Xs4: list(A),Y4: A,Ys5: list(A)] :
            ( ( X3 != Y4 )
            & ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Pre),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))),Xs4)) )
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Pre),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),nil(A))),Ys5)) ) ) ) ) ).

% same_length_different
tff(fact_5951_not__distinct__decomp,axiom,
    ! [A: $tType,Ws: list(A)] :
      ( ~ distinct(A,Ws)
     => ? [Xs2: list(A),Ys3: list(A),Zs2: list(A),Y4: A] : Ws = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs2),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),nil(A))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys3),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),nil(A))),Zs2)))) ) ).

% not_distinct_decomp
tff(fact_5952_not__distinct__conv__prefix,axiom,
    ! [A: $tType,As3: list(A)] :
      ( ~ distinct(A,As3)
    <=> ? [Xs3: list(A),Y5: A,Ys4: list(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y5),aa(list(A),set(A),set2(A),Xs3)))
          & distinct(A,Xs3)
          & ( As3 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4)) ) ) ) ).

% not_distinct_conv_prefix
tff(fact_5953_replicate__append__same,axiom,
    ! [A: $tType,I2: nat,X: A] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),replicate(A,I2,X)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),replicate(A,I2,X)) ).

% replicate_append_same
tff(fact_5954_list__update__append1,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),Ys: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( list_update(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys),I2,X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),list_update(A,Xs,I2,X)),Ys) ) ) ).

% list_update_append1
tff(fact_5955_remove1__split,axiom,
    ! [A: $tType,A2: A,Xs: list(A),Ys: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(list(A),set(A),set2(A),Xs)))
     => ( ( remove1(A,A2,Xs) = Ys )
      <=> ? [Ls: list(A),Rs: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ls),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),Rs)) )
            & ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(list(A),set(A),set2(A),Ls)))
            & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ls),Rs) ) ) ) ) ).

% remove1_split
tff(fact_5956_rotate1_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(A),rotate1(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ).

% rotate1.simps(2)
tff(fact_5957_nths__append,axiom,
    ! [A: $tType,L: list(A),L3: list(A),A3: set(nat)] : nths(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),L3),A3) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),nths(A,L,A3)),nths(A,L3,collect(nat,aa(set(nat),fun(nat,bool),aTP_Lamp_qk(list(A),fun(set(nat),fun(nat,bool)),L),A3)))) ).

% nths_append
tff(fact_5958_subseqs_Osimps_I1_J,axiom,
    ! [A: $tType] : subseqs(A,nil(A)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ).

% subseqs.simps(1)
tff(fact_5959_product__lists_Osimps_I1_J,axiom,
    ! [A: $tType] : product_lists(A,nil(list(A))) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ).

% product_lists.simps(1)
tff(fact_5960_length__Suc__conv__rev,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
    <=> ? [Y5: A,Ys4: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys4) = N ) ) ) ).

% length_Suc_conv_rev
tff(fact_5961_length__append__singleton,axiom,
    ! [A: $tType,Xs: list(A),X: A] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_append_singleton
tff(fact_5962_nth__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),N) = aa(nat,A,nth(A,Xs),N) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),N) = aa(nat,A,nth(A,Ys),aa(nat,nat,minus_minus(nat,N),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ) ).

% nth_append
tff(fact_5963_list__update__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: list(A),X: A] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( list_update(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys),N,X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),list_update(A,Xs,N,X)),Ys) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( list_update(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys),N,X) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),list_update(A,Ys,aa(nat,nat,minus_minus(nat,N),aa(list(A),nat,size_size(list(A)),Xs)),X)) ) ) ) ).

% list_update_append
tff(fact_5964_n__lists_Osimps_I1_J,axiom,
    ! [A: $tType,Xs: list(A)] : n_lists(A,zero_zero(nat),Xs) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),nil(list(A))) ).

% n_lists.simps(1)
tff(fact_5965_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,suc,I2)),J))
     => ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,I2,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I2)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or3652927894154168847AtMost(nat,aa(nat,nat,suc,I2),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_5966_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),J))
     => ( aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,I2,J)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,I2)),aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or5935395276787703475ssThan(nat,aa(nat,nat,suc,I2),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_5967_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F3: fun(B,A),A2: A,Xs: list(B),Ys: list(B)] : groups4207007520872428315er_sum(B,A,F3,A2,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups4207007520872428315er_sum(B,A,F3,A2,Xs)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,A2),aa(list(B),nat,size_size(list(B)),Xs))),groups4207007520872428315er_sum(B,A,F3,A2,Ys))) ) ).

% horner_sum_append
tff(fact_5968_nths__Cons,axiom,
    ! [A: $tType,X: A,L: list(A),A3: set(nat)] : nths(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),L),A3) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),if(list(A),aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),zero_zero(nat)),A3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),nil(A))),nths(A,L,collect(nat,aTP_Lamp_ql(set(nat),fun(nat,bool),A3)))) ).

% nths_Cons
tff(fact_5969_set__Cons__def,axiom,
    ! [A: $tType,A3: set(A),XS: set(list(A))] : set_Cons(A,A3,XS) = collect(list(A),aa(set(list(A)),fun(list(A),bool),aTP_Lamp_qm(set(A),fun(set(list(A)),fun(list(A),bool)),A3),XS)) ).

% set_Cons_def
tff(fact_5970_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(A) )
       => ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Xs) )
         => ? [N2: nat,Zs2: list(A)] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),one_one(nat)),N2))
              & ( concat(A,replicate(list(A),N2,Zs2)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_5971_shuffles_Opelims,axiom,
    ! [A: $tType,X: list(A),Xa: list(A),Y: set(list(A))] :
      ( ( shuffles(A,X,Xa) = Y )
     => ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),X),Xa)))
       => ( ( ( X = nil(A) )
           => ( ( Y = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Xa),bot_bot(set(list(A)))) )
             => ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),nil(A)),Xa))) ) )
         => ( ( ( Xa = nil(A) )
             => ( ( Y = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),X),bot_bot(set(list(A)))) )
               => ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),X),nil(A)))) ) )
           => ~ ! [X3: A,Xs2: list(A)] :
                  ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
                 => ! [Y4: A,Ys3: list(A)] :
                      ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3) )
                     => ( ( Y = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3)),shuffles(A,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3)))),aa(set(list(A)),set(list(A)),image(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2),Ys3))) )
                       => ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3)))) ) ) ) ) ) ) ) ).

% shuffles.pelims
tff(fact_5972_lexn__conv,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),N: nat] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R3),N) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_qn(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),R3),N))) ).

% lexn_conv
tff(fact_5973_upto__aux__rec,axiom,
    ! [J: int,I2: int,Js: list(int)] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2))
       => ( upto_aux(I2,J,Js) = Js ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2))
       => ( upto_aux(I2,J,Js) = upto_aux(I2,aa(int,int,minus_minus(int,J),one_one(int)),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),Js)) ) ) ) ).

% upto_aux_rec
tff(fact_5974_le__sup__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y)),Z))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z)) ) ) ) ).

% le_sup_iff
tff(fact_5975_sup_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2)) ) ) ) ).

% sup.bounded_iff
tff(fact_5976_Un__subset__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),C5))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C5))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C5)) ) ) ).

% Un_subset_iff
tff(fact_5977_set__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) ).

% set_append
tff(fact_5978_set__union,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),set(A),set2(A),union(A,Xs,Ys)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) ).

% set_union
tff(fact_5979_ivl__disj__un__two__touch_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(4)
tff(fact_5980_ivl__disj__un__two_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(6)
tff(fact_5981_Un__Int__assoc__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)),C5) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C5)) )
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C5),A3)) ) ).

% Un_Int_assoc_eq
tff(fact_5982_Diff__partition,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),aa(set(A),set(A),minus_minus(set(A),B3),A3)) = B3 ) ) ).

% Diff_partition
tff(fact_5983_Diff__subset__conv,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),C5: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),B3)),C5))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),C5))) ) ).

% Diff_subset_conv
tff(fact_5984_ivl__disj__un__two_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(3)
tff(fact_5985_Un__Pow__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),sup_sup(set(set(A))),pow2(A,A3)),pow2(A,B3))),pow2(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)))) ).

% Un_Pow_subset
tff(fact_5986_sup_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ) ).

% sup.coboundedI2
tff(fact_5987_sup_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ) ).

% sup.coboundedI1
tff(fact_5988_sup_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_5989_sup_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_5990_sup_Ocobounded2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ).

% sup.cobounded2
tff(fact_5991_sup_Ocobounded1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ).

% sup.cobounded1
tff(fact_5992_sup_Oorder__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_5993_sup_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2)) ) ) ) ).

% sup.boundedI
tff(fact_5994_sup_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2)) ) ) ) ).

% sup.boundedE
tff(fact_5995_sup__absorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).

% sup_absorb2
tff(fact_5996_sup__absorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = X ) ) ) ).

% sup_absorb1
tff(fact_5997_sup_Oabsorb2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb2
tff(fact_5998_sup_Oabsorb1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb1
tff(fact_5999_sup__unique,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [F3: fun(A,fun(A,A)),X: A,Y: A] :
          ( ! [X3: A,Y4: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),aa(A,A,aa(A,fun(A,A),F3,X3),Y4)))
         => ( ! [X3: A,Y4: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),aa(A,A,aa(A,fun(A,A),F3,X3),Y4)))
           => ( ! [X3: A,Y4: A,Z2: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z2),X3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),F3,Y4),Z2)),X3)) ) )
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = aa(A,A,aa(A,fun(A,A),F3,X),Y) ) ) ) ) ) ).

% sup_unique
tff(fact_6000_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) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2)) ) ) ).

% sup.orderI
tff(fact_6001_sup_OorderE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( A2 = aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) ) ) ) ).

% sup.orderE
tff(fact_6002_le__iff__sup,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
        <=> ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = Y ) ) ) ).

% le_iff_sup
tff(fact_6003_sup__least,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)),X)) ) ) ) ).

% sup_least
tff(fact_6004_sup__mono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D2))) ) ) ) ).

% sup_mono
tff(fact_6005_sup_Omono,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,D2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),A2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),D2),B2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),C2),D2)),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ) ) ).

% sup.mono
tff(fact_6006_le__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ) ).

% le_supI2
tff(fact_6007_le__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ) ).

% le_supI1
tff(fact_6008_sup__ge2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Y: A,X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).

% sup_ge2
tff(fact_6009_sup__ge1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).

% sup_ge1
tff(fact_6010_le__supI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,X: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),X)) ) ) ) ).

% le_supI
tff(fact_6011_le__supE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2)),X))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X)) ) ) ) ).

% le_supE
tff(fact_6012_inf__sup__ord_I3_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).

% inf_sup_ord(3)
tff(fact_6013_inf__sup__ord_I4_J,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [Y: A,X: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y))) ) ).

% inf_sup_ord(4)
tff(fact_6014_subset__Un__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
    <=> ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = B3 ) ) ).

% subset_Un_eq
tff(fact_6015_subset__UnE,axiom,
    ! [A: $tType,C5: set(A),A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)))
     => ~ ! [A10: set(A)] :
            ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A10),A3))
           => ! [B13: set(A)] :
                ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B13),B3))
               => ( C5 != aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A10),B13) ) ) ) ) ).

% subset_UnE
tff(fact_6016_Un__absorb2,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = A3 ) ) ).

% Un_absorb2
tff(fact_6017_Un__absorb1,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3) = B3 ) ) ).

% Un_absorb1
tff(fact_6018_Un__upper2,axiom,
    ! [A: $tType,B3: set(A),A3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))) ).

% Un_upper2
tff(fact_6019_Un__upper1,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3))) ).

% Un_upper1
tff(fact_6020_Un__least,axiom,
    ! [A: $tType,A3: set(A),C5: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C5))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),C5)) ) ) ).

% Un_least
tff(fact_6021_Un__mono,axiom,
    ! [A: $tType,A3: set(A),C5: set(A),B3: set(A),D4: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C5))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),D4))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),C5),D4))) ) ) ).

% Un_mono
tff(fact_6022_distrib__sup__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),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),X),Y)),aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Z)))) ) ).

% distrib_sup_le
tff(fact_6023_distrib__inf__le,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [X: A,Y: A,Z: A] : pp(aa(A,bool,aa(A,fun(A,bool),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),X),Y)),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Z))),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z)))) ) ).

% distrib_inf_le
tff(fact_6024_less__supI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ) ).

% less_supI1
tff(fact_6025_less__supI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ) ).

% less_supI2
tff(fact_6026_sup_Oabsorb3,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = A2 ) ) ) ).

% sup.absorb3
tff(fact_6027_sup_Oabsorb4,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2) = B2 ) ) ) ).

% sup.absorb4
tff(fact_6028_sup_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),B2),C2)),A2))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A2)) ) ) ) ).

% sup.strict_boundedE
tff(fact_6029_sup_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_6030_sup_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),A2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ) ).

% sup.strict_coboundedI1
tff(fact_6031_sup_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [C2: A,B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),B2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),C2),aa(A,A,aa(A,fun(A,A),sup_sup(A),A2),B2))) ) ) ).

% sup.strict_coboundedI2
tff(fact_6032_set__shuffles,axiom,
    ! [A: $tType,Zs: list(A),Xs: list(A),Ys: list(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
     => ( aa(list(A),set(A),set2(A),Zs) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) ) ) ).

% set_shuffles
tff(fact_6033_lexn_Osimps_I1_J,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : aa(nat,set(product_prod(list(A),list(A))),lexn(A,R3),zero_zero(nat)) = bot_bot(set(product_prod(list(A),list(A)))) ).

% lexn.simps(1)
tff(fact_6034_sup__shunt,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),Y) = top_top(A) )
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),X)),Y)) ) ) ).

% sup_shunt
tff(fact_6035_shunt1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),Y)),Z))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,uminus_uminus(A),Y)),Z))) ) ) ).

% shunt1
tff(fact_6036_shunt2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(A,A,uminus_uminus(A),Y))),Z))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,aa(A,fun(A,A),sup_sup(A),Y),Z))) ) ) ).

% shunt2
tff(fact_6037_sup__neg__inf,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [P2: A,Q3: A,R3: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P2),aa(A,A,aa(A,fun(A,A),sup_sup(A),Q3),R3)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),P2),aa(A,A,uminus_uminus(A),Q3))),R3)) ) ) ).

% sup_neg_inf
tff(fact_6038_less__eq__Inf__inter,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A),B3: set(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)))) ) ).

% less_eq_Inf_inter
tff(fact_6039_ivl__disj__un__two_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(7)
tff(fact_6040_ivl__disj__un__one_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or7035219750837199246ssThan(A,L,U)) = aa(A,set(A),set_ord_lessThan(A),U) ) ) ) ).

% ivl_disj_un_one(2)
tff(fact_6041_card__Un__le,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),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_6042_ivl__disj__un__two_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or3652927894154168847AtMost(A,M,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(8)
tff(fact_6043_ivl__disj__un__one_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or3652927894154168847AtMost(A,L,U)) = aa(A,set(A),set_ord_atMost(A),U) ) ) ) ).

% ivl_disj_un_one(3)
tff(fact_6044_shuffles_Osimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A)] : shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))),aa(set(list(A)),set(list(A)),image(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys))) ).

% shuffles.simps(3)
tff(fact_6045_Inter__Un__subset,axiom,
    ! [A: $tType,A3: set(set(A)),B3: set(set(A))] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A3)),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B3))),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(set(A)),set(set(A)),aa(set(set(A)),fun(set(set(A)),set(set(A))),inf_inf(set(set(A))),A3),B3)))) ).

% Inter_Un_subset
tff(fact_6046_ivl__disj__un__two__touch_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(2)
tff(fact_6047_sum_Ounion__inter,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),B3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,finite_finite(B),B3))
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3))),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),B3)) ) ) ) ) ).

% sum.union_inter
tff(fact_6048_card__Un__Int,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,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_6049_ivl__disj__un__two__touch_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(3)
tff(fact_6050_ivl__disj__un__two_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(1)
tff(fact_6051_ivl__disj__un__one_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),L)),set_or1337092689740270186AtMost(A,L,U)) = aa(A,set(A),set_ord_atMost(A),U) ) ) ) ).

% ivl_disj_un_one(4)
tff(fact_6052_ivl__disj__un__singleton_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_lessThan(A),U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))) = aa(A,set(A),set_ord_atMost(A),U) ) ).

% ivl_disj_un_singleton(2)
tff(fact_6053_ivl__disj__un__two_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(2)
tff(fact_6054_ivl__disj__un__one_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(A,set(A),set_ord_atMost(A),L)),set_or5935395276787703475ssThan(A,L,U)) = aa(A,set(A),set_ord_lessThan(A),U) ) ) ) ).

% ivl_disj_un_one(1)
tff(fact_6055_ivl__disj__un__two__touch_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,M)),set_or7035219750837199246ssThan(A,M,U)) = set_or5935395276787703475ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two_touch(1)
tff(fact_6056_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),aa(set(nat),set(A),image(nat,A,aTP_Lamp_nd(A,fun(nat,A),B3)),collect(nat,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3) ) ).

% SUP_nat_binary
tff(fact_6057_lexn__length,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A)),N: nat] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),aa(nat,set(product_prod(list(A),list(A))),lexn(A,R3),N)))
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = N )
        & ( aa(list(A),nat,size_size(list(A)),Ys) = N ) ) ) ).

% lexn_length
tff(fact_6058_sup__bot_Osemilattice__neutr__order__axioms,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => semila1105856199041335345_order(A,sup_sup(A),bot_bot(A),aTP_Lamp_qo(A,fun(A,bool)),aTP_Lamp_qp(A,fun(A,bool))) ) ).

% sup_bot.semilattice_neutr_order_axioms
tff(fact_6059_sum_Ounion__inter__neutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),B3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,finite_finite(B),B3))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3)))
                 => ( aa(B,A,G,X3) = zero_zero(A) ) )
             => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),B3)) ) ) ) ) ) ).

% sum.union_inter_neutral
tff(fact_6060_sum__Un,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [A3: set(B),B3: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,finite_finite(B),B3))
           => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),A3)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),B3))),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3))) ) ) ) ) ).

% sum_Un
tff(fact_6061_sum_Ounion__disjoint,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),B3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,finite_finite(B),B3))
           => ( ( aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3) = bot_bot(set(B)) )
             => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),B3)) ) ) ) ) ) ).

% sum.union_disjoint
tff(fact_6062_prod_Ounion__inter__neutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),B3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,finite_finite(B),B3))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3)))
                 => ( aa(B,A,G,X3) = one_one(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),B3)) ) ) ) ) ) ).

% prod.union_inter_neutral
tff(fact_6063_ivl__disj__un__singleton_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(6)
tff(fact_6064_sum__Un2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),B3: set(A),F3: fun(A,B)] :
          ( pp(aa(set(A),bool,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,groups7311177749621191930dd_sum(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),aa(set(A),set(A),minus_minus(set(A),A3),B3))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),aa(set(A),set(A),minus_minus(set(A),B3),A3)))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ).

% sum_Un2
tff(fact_6065_sum_Ounion__diff2,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B),B3: set(B),G: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,finite_finite(B),B3))
           => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),minus_minus(set(B),A3),B3))),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),minus_minus(set(B),B3),A3)))),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3))) ) ) ) ) ).

% sum.union_diff2
tff(fact_6066_card__Un__disjoint,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,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_6067_ivl__disj__un__singleton_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),L),bot_bot(set(A)))),set_or3652927894154168847AtMost(A,L,U)) = set_or1337092689740270186AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(5)
tff(fact_6068_ivl__disj__un__two_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,M)),set_or5935395276787703475ssThan(A,M,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(4)
tff(fact_6069_ivl__disj__un__singleton_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),L),bot_bot(set(A)))),set_or5935395276787703475ssThan(A,L,U)) = set_or7035219750837199246ssThan(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(3)
tff(fact_6070_sum__Un__nat,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,finite_finite(A),B3))
       => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),A3)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),B3))),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))) ) ) ) ).

% sum_Un_nat
tff(fact_6071_ivl__disj__un__two_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,M: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),M))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),U))
           => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,M)),set_or1337092689740270186AtMost(A,M,U)) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ) ).

% ivl_disj_un_two(5)
tff(fact_6072_ivl__disj__un__singleton_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),U),bot_bot(set(A)))) = set_or3652927894154168847AtMost(A,L,U) ) ) ) ).

% ivl_disj_un_singleton(4)
tff(fact_6073_prod__Un,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [A3: set(B),B3: set(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( pp(aa(set(B),bool,finite_finite(B),B3))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3)))
                 => ( aa(B,A,F3,X3) != zero_zero(A) ) )
             => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),A3),B3)) = divide_divide(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),A3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),B3)),aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),F3),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),A3),B3))) ) ) ) ) ) ).

% prod_Un
tff(fact_6074_UN__le__eq__Un0,axiom,
    ! [A: $tType,M6: fun(nat,set(A)),N: nat] : aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),M6),aa(nat,set(nat),set_ord_atMost(nat),N))) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),M6),set_or1337092689740270186AtMost(nat,one_one(nat),N)))),aa(nat,set(A),M6,zero_zero(nat))) ).

% UN_le_eq_Un0
tff(fact_6075_shuffles_Oelims,axiom,
    ! [A: $tType,X: list(A),Xa: list(A),Y: set(list(A))] :
      ( ( shuffles(A,X,Xa) = Y )
     => ( ( ( X = nil(A) )
         => ( Y != aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),Xa),bot_bot(set(list(A)))) ) )
       => ( ( ( Xa = nil(A) )
           => ( Y != aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),X),bot_bot(set(list(A)))) ) )
         => ~ ! [X3: A,Xs2: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
               => ! [Y4: A,Ys3: list(A)] :
                    ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3) )
                   => ( Y != aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3)),shuffles(A,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Ys3)))),aa(set(list(A)),set(list(A)),image(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2),Ys3))) ) ) ) ) ) ) ).

% shuffles.elims
tff(fact_6076_shuffles_Opsimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A)] :
      ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),shuffles_rel(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))))
     => ( shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),sup_sup(set(list(A))),aa(set(list(A)),set(list(A)),image(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),shuffles(A,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)))),aa(set(list(A)),set(list(A)),image(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y)),shuffles(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys))) ) ) ).

% shuffles.psimps(3)
tff(fact_6077_lex__conv,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : lex(A,R3) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aTP_Lamp_qq(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R3))) ).

% lex_conv
tff(fact_6078_upto_Opelims,axiom,
    ! [X: int,Xa: int,Y: list(int)] :
      ( ( upto(X,Xa) = Y )
     => ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),product_Pair(int,int,X),Xa)))
       => ~ ( ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa))
               => ( Y = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa)) ) )
              & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa))
               => ( Y = nil(int) ) ) )
           => ~ pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),product_Pair(int,int,X),Xa))) ) ) ) ).

% upto.pelims
tff(fact_6079_upto__Nil,axiom,
    ! [I2: int,J: int] :
      ( ( upto(I2,J) = nil(int) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2)) ) ).

% upto_Nil
tff(fact_6080_upto__Nil2,axiom,
    ! [I2: int,J: int] :
      ( ( nil(int) = upto(I2,J) )
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2)) ) ).

% upto_Nil2
tff(fact_6081_upto__empty,axiom,
    ! [J: int,I2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),J),I2))
     => ( upto(I2,J) = nil(int) ) ) ).

% upto_empty
tff(fact_6082_upto__single,axiom,
    ! [I2: int] : upto(I2,I2) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I2),nil(int)) ).

% upto_single
tff(fact_6083_nth__upto,axiom,
    ! [I2: int,K: nat,J: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),aa(nat,int,semiring_1_of_nat(int),K))),J))
     => ( aa(nat,int,nth(int,upto(I2,J)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),aa(nat,int,semiring_1_of_nat(int),K)) ) ) ).

% nth_upto
tff(fact_6084_Cons__in__lex,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))),lex(A,R3)))
    <=> ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y)),R3))
          & ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) )
        | ( ( X = Y )
          & pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),lex(A,R3))) ) ) ) ).

% Cons_in_lex
tff(fact_6085_length__upto,axiom,
    ! [I2: int,J: int] : aa(list(int),nat,size_size(list(int)),upto(I2,J)) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,J),I2)),one_one(int))) ).

% length_upto
tff(fact_6086_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N)) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(num,int,numeral_numeral(int),N))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N)) = nil(int) ) ) ) ).

% upto_rec_numeral(1)
tff(fact_6087_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = nil(int) ) ) ) ).

% upto_rec_numeral(2)
tff(fact_6088_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(num,int,numeral_numeral(int),N))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) = nil(int) ) ) ) ).

% upto_rec_numeral(3)
tff(fact_6089_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))
       => ( upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = nil(int) ) ) ) ).

% upto_rec_numeral(4)
tff(fact_6090_upto__aux__def,axiom,
    ! [I2: int,J: int,Js: list(int)] : upto_aux(I2,J,Js) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,J)),Js) ).

% upto_aux_def
tff(fact_6091_sup__enat__def,axiom,
    sup_sup(extended_enat) = ord_max(extended_enat) ).

% sup_enat_def
tff(fact_6092_sup__nat__def,axiom,
    sup_sup(nat) = ord_max(nat) ).

% sup_nat_def
tff(fact_6093_upto__code,axiom,
    ! [I2: int,J: int] : upto(I2,J) = upto_aux(I2,J,nil(int)) ).

% upto_code
tff(fact_6094_atLeastAtMost__upto,axiom,
    ! [I2: int,J: int] : set_or1337092689740270186AtMost(int,I2,J) = aa(list(int),set(int),set2(int),upto(I2,J)) ).

% atLeastAtMost_upto
tff(fact_6095_distinct__upto,axiom,
    ! [I2: int,J: int] : distinct(int,upto(I2,J)) ).

% distinct_upto
tff(fact_6096_Nil2__notin__lex,axiom,
    ! [A: $tType,Xs: list(A),R3: set(product_prod(A,A))] : ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),nil(A))),lex(A,R3))) ).

% Nil2_notin_lex
tff(fact_6097_Nil__notin__lex,axiom,
    ! [A: $tType,Ys: list(A),R3: set(product_prod(A,A))] : ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),nil(A)),Ys)),lex(A,R3))) ).

% Nil_notin_lex
tff(fact_6098_lex__append__leftI,axiom,
    ! [A: $tType,Ys: list(A),Zs: list(A),R3: set(product_prod(A,A)),Xs: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Ys),Zs)),lex(A,R3)))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lex(A,R3))) ) ).

% lex_append_leftI
tff(fact_6099_atLeastLessThan__add__Un,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( set_or7035219750837199246ssThan(nat,I2,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,I2,J)),set_or7035219750837199246ssThan(nat,J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% atLeastLessThan_add_Un
tff(fact_6100_upto__split2,axiom,
    ! [I2: int,J: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),K))
       => ( upto(I2,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,J)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K)) ) ) ) ).

% upto_split2
tff(fact_6101_upto__split1,axiom,
    ! [I2: int,J: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),K))
       => ( upto(I2,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,aa(int,int,minus_minus(int,J),one_one(int)))),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_6102_atLeastLessThan__upto,axiom,
    ! [I2: int,J: int] : set_or7035219750837199246ssThan(int,I2,J) = aa(list(int),set(int),set2(int),upto(I2,aa(int,int,minus_minus(int,J),one_one(int)))) ).

% atLeastLessThan_upto
tff(fact_6103_lex__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : lex(A,R3) = aa(set(set(product_prod(list(A),list(A)))),set(product_prod(list(A),list(A))),complete_Sup_Sup(set(product_prod(list(A),list(A)))),aa(set(nat),set(set(product_prod(list(A),list(A)))),image(nat,set(product_prod(list(A),list(A))),lexn(A,R3)),top_top(set(nat)))) ).

% lex_def
tff(fact_6104_greaterThanAtMost__upto,axiom,
    ! [I2: int,J: int] : set_or3652927894154168847AtMost(int,I2,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)) ).

% greaterThanAtMost_upto
tff(fact_6105_lex__append__left__iff,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X3),X3)),R3))
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lex(A,R3)))
      <=> pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Ys),Zs)),lex(A,R3))) ) ) ).

% lex_append_left_iff
tff(fact_6106_lex__append__leftD,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X3),X3)),R3))
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lex(A,R3)))
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Ys),Zs)),lex(A,R3))) ) ) ).

% lex_append_leftD
tff(fact_6107_lex__append__rightI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A)),Vs: list(A),Us: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),lex(A,R3)))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Us) )
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs))),lex(A,R3))) ) ) ).

% lex_append_rightI
tff(fact_6108_upto_Osimps,axiom,
    ! [I2: int,J: int] :
      ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
       => ( upto(I2,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I2),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)) ) )
      & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
       => ( upto(I2,J) = nil(int) ) ) ) ).

% upto.simps
tff(fact_6109_upto_Oelims,axiom,
    ! [X: int,Xa: int,Y: list(int)] :
      ( ( upto(X,Xa) = Y )
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa))
         => ( Y = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),X),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),X),one_one(int)),Xa)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X),Xa))
         => ( Y = nil(int) ) ) ) ) ).

% upto.elims
tff(fact_6110_upto__rec1,axiom,
    ! [I2: int,J: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
     => ( upto(I2,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I2),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)) ) ) ).

% upto_rec1
tff(fact_6111_upto__rec2,axiom,
    ! [I2: int,J: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
     => ( upto(I2,J) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,aa(int,int,minus_minus(int,J),one_one(int)))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),nil(int))) ) ) ).

% upto_rec2
tff(fact_6112_greaterThanLessThan__upto,axiom,
    ! [I2: int,J: int] : set_or5935395276787703475ssThan(int,I2,J) = aa(list(int),set(int),set2(int),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),aa(int,int,minus_minus(int,J),one_one(int)))) ).

% greaterThanLessThan_upto
tff(fact_6113_Pow__set_I2_J,axiom,
    ! [B: $tType,X: B,Xs: list(B)] : pow2(B,aa(list(B),set(B),set2(B),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs))) = aa(set(set(B)),set(set(B)),aa(set(set(B)),fun(set(set(B)),set(set(B))),sup_sup(set(set(B))),pow2(B,aa(list(B),set(B),set2(B),Xs))),aa(set(set(B)),set(set(B)),image(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X)),pow2(B,aa(list(B),set(B),set2(B),Xs)))) ).

% Pow_set(2)
tff(fact_6114_upto__split3,axiom,
    ! [I2: int,J: int,K: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
     => ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),J),K))
       => ( upto(I2,K) = aa(list(int),list(int),aa(list(int),fun(list(int),list(int)),append(int),upto(I2,aa(int,int,minus_minus(int,J),one_one(int)))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K))) ) ) ) ).

% upto_split3
tff(fact_6115_upto_Opsimps,axiom,
    ! [I2: int,J: int] :
      ( pp(aa(product_prod(int,int),bool,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),product_Pair(int,int,I2),J)))
     => ( ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
         => ( upto(I2,J) = aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),I2),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int)),J)) ) )
        & ( ~ pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),I2),J))
         => ( upto(I2,J) = nil(int) ) ) ) ) ).

% upto.psimps
tff(fact_6116_lenlex__conv,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : lenlex(A,R3) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aTP_Lamp_qr(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R3))) ).

% lenlex_conv
tff(fact_6117_splice_Opinduct,axiom,
    ! [A: $tType,A0: list(A),A1: list(A),P: fun(list(A),fun(list(A),bool))] :
      ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),A0),A1)))
     => ( ! [Ys3: list(A)] :
            ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),nil(A)),Ys3)))
           => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,nil(A)),Ys3)) )
       => ( ! [X3: A,Xs2: list(A),Ys3: list(A)] :
              ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Ys3)))
             => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,Ys3),Xs2))
               => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Ys3)) ) )
         => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),P,A0),A1)) ) ) ) ).

% splice.pinduct
tff(fact_6118_Nil__lenlex__iff1,axiom,
    ! [A: $tType,Ns: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),nil(A)),Ns)),lenlex(A,R3)))
    <=> ( Ns != nil(A) ) ) ).

% Nil_lenlex_iff1
tff(fact_6119_lenlex__irreflexive,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X3),X3)),R3))
     => ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Xs)),lenlex(A,R3))) ) ).

% lenlex_irreflexive
tff(fact_6120_Nil__lenlex__iff2,axiom,
    ! [A: $tType,Ns: list(A),R3: set(product_prod(A,A))] : ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Ns),nil(A))),lenlex(A,R3))) ).

% Nil_lenlex_iff2
tff(fact_6121_lenlex__length,axiom,
    ! [A: $tType,Ms: list(A),Ns: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),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,R3)))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns))) ) ).

% lenlex_length
tff(fact_6122_lenlex__append1,axiom,
    ! [A: $tType,Us: list(A),Xs: list(A),R: set(product_prod(A,A)),Vs: list(A),Ys: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Us),Xs)),lenlex(A,R)))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Ys) )
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Vs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))),lenlex(A,R))) ) ) ).

% lenlex_append1
tff(fact_6123_Cons__lenlex__iff,axiom,
    ! [A: $tType,M: A,Ms: list(A),N: A,Ns: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),M),Ms)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N),Ns))),lenlex(A,R3)))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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) )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,M),N)),R3)) )
        | ( ( M = N )
          & pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),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,R3))) ) ) ) ).

% Cons_lenlex_iff
tff(fact_6124_splice_Opelims,axiom,
    ! [A: $tType,X: list(A),Xa: list(A),Y: list(A)] :
      ( ( splice(A,X,Xa) = Y )
     => ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),X),Xa)))
       => ( ( ( X = nil(A) )
           => ( ( Y = Xa )
             => ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),nil(A)),Xa))) ) )
         => ~ ! [X3: A,Xs2: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
               => ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),splice(A,Xa,Xs2)) )
                 => ~ pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xa))) ) ) ) ) ) ).

% splice.pelims
tff(fact_6125_termdiffs__aux,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_ot(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K5)))
           => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_qt(fun(nat,A),fun(A,fun(A,A)),C2),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% termdiffs_aux
tff(fact_6126_split__Nil__iff,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( splice(A,Xs,Ys) = nil(A) )
    <=> ( ( Xs = nil(A) )
        & ( Ys = nil(A) ) ) ) ).

% split_Nil_iff
tff(fact_6127_splice__Nil2,axiom,
    ! [A: $tType,Xs: list(A)] : splice(A,Xs,nil(A)) = Xs ).

% splice_Nil2
tff(fact_6128_splice__in__shuffles,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),splice(A,Xs,Ys)),shuffles(A,Xs,Ys))) ).

% splice_in_shuffles
tff(fact_6129_length__splice,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),splice(A,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ).

% length_splice
tff(fact_6130_splice__replicate,axiom,
    ! [A: $tType,M: nat,X: A,N: nat] : splice(A,replicate(A,M,X),replicate(A,N,X)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),X) ).

% splice_replicate
tff(fact_6131_power__tendsto__0__iff,axiom,
    ! [A: $tType,N: nat,F3: fun(A,real),F5: filter(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_qu(nat,fun(fun(A,real),fun(A,real)),N),F3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F5)
      <=> filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,zero_zero(real)),F5) ) ) ).

% power_tendsto_0_iff
tff(fact_6132_splice_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A)] : splice(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),splice(A,Ys,Xs)) ).

% splice.simps(2)
tff(fact_6133_tendsto__artanh,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F5: filter(A)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F5)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),A2))
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),one_one(real)))
         => filterlim(A,real,aTP_Lamp_qv(fun(A,real),fun(A,real),F3),topolo7230453075368039082e_nhds(real,aa(real,real,artanh(real),A2)),F5) ) ) ) ).

% tendsto_artanh
tff(fact_6134_LIM__offset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),L5: B,A2: A,K: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_qw(fun(A,B),fun(A,fun(A,B)),F3),K),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,aa(A,A,minus_minus(A,A2),K),top_top(set(A)))) ) ) ).

% LIM_offset
tff(fact_6135_LIM__offset__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),A2: A,L5: B] :
          ( filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_qx(fun(A,B),fun(A,fun(A,B)),F3),A2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_offset_zero_cancel
tff(fact_6136_LIM__offset__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_qx(fun(A,B),fun(A,fun(A,B)),F3),A2),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_offset_zero
tff(fact_6137_LIM__isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),A2: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,A2)),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_qx(fun(A,B),fun(A,fun(A,B)),F3),A2),topolo7230453075368039082e_nhds(B,aa(A,B,F3,A2)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_isCont_iff
tff(fact_6138_has__field__derivativeD,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,X: A,S2: set(A)] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,X,S2))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_qy(fun(A,A),fun(A,fun(A,A)),F3),X),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_field_derivativeD
tff(fact_6139_has__field__derivative__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,X: A,S2: set(A)] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,X,S2))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_qy(fun(A,A),fun(A,fun(A,A)),F3),X),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,X,S2)) ) ) ).

% has_field_derivative_iff
tff(fact_6140_tendsto__null__power,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [F3: fun(A,B),F5: filter(A),N: nat] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F5)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_qz(fun(A,B),fun(nat,fun(A,B)),F3),N),topolo7230453075368039082e_nhds(B,zero_zero(B)),F5) ) ) ) ).

% tendsto_null_power
tff(fact_6141_tendsto__arcosh,axiom,
    ! [B: $tType,F3: fun(B,real),A2: real,F5: filter(B)] :
      ( filterlim(B,real,F3,topolo7230453075368039082e_nhds(real,A2),F5)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),A2))
       => filterlim(B,real,aTP_Lamp_ra(fun(B,real),fun(B,real),F3),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F5) ) ) ).

% tendsto_arcosh
tff(fact_6142_tendsto__log,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F5: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F5)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F5)
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
         => ( ( A2 != one_one(real) )
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
             => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_rb(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G),topolo7230453075368039082e_nhds(real,aa(real,real,log(A2),B2)),F5) ) ) ) ) ) ).

% tendsto_log
tff(fact_6143_tendsto__one__prod_H,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [I5: set(B),F3: fun(A,fun(B,C)),F5: filter(A)] :
          ( ! [I4: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),I5))
             => filterlim(A,C,aa(B,fun(A,C),aTP_Lamp_rc(fun(A,fun(B,C)),fun(B,fun(A,C)),F3),I4),topolo7230453075368039082e_nhds(C,one_one(C)),F5) )
         => filterlim(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_rd(set(B),fun(fun(A,fun(B,C)),fun(A,C)),I5),F3),topolo7230453075368039082e_nhds(C,one_one(C)),F5) ) ) ).

% tendsto_one_prod'
tff(fact_6144_tendsto__divide__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(B,A),F5: filter(B),C2: A] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,zero_zero(A)),F5)
         => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_re(fun(B,A),fun(A,fun(B,A)),F3),C2),topolo7230453075368039082e_nhds(A,zero_zero(A)),F5) ) ) ).

% tendsto_divide_zero
tff(fact_6145_tendsto__divide,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(B,A),A2: A,F5: filter(B),G: fun(B,A),B2: A] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,A2),F5)
         => ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,B2),F5)
           => ( ( B2 != zero_zero(A) )
             => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_rf(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G),topolo7230453075368039082e_nhds(A,divide_divide(A,A2,B2)),F5) ) ) ) ) ).

% tendsto_divide
tff(fact_6146_tendsto__mult__one,axiom,
    ! [B: $tType,D: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F3: fun(D,B),F5: filter(D),G: fun(D,B)] :
          ( filterlim(D,B,F3,topolo7230453075368039082e_nhds(B,one_one(B)),F5)
         => ( filterlim(D,B,G,topolo7230453075368039082e_nhds(B,one_one(B)),F5)
           => filterlim(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_rg(fun(D,B),fun(fun(D,B),fun(D,B)),F3),G),topolo7230453075368039082e_nhds(B,one_one(B)),F5) ) ) ) ).

% tendsto_mult_one
tff(fact_6147_tendsto__add,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo6943815403480290642id_add(A)
     => ! [F3: fun(B,A),A2: A,F5: filter(B),G: fun(B,A),B2: A] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,A2),F5)
         => ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,B2),F5)
           => filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_rh(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),F5) ) ) ) ).

% tendsto_add
tff(fact_6148_tendsto__add__const__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [C2: A,F3: fun(B,A),D2: A,F5: filter(B)] :
          ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ri(A,fun(fun(B,A),fun(B,A)),C2),F3),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D2)),F5)
        <=> filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,D2),F5) ) ) ).

% tendsto_add_const_iff
tff(fact_6149_tendsto__add__zero,axiom,
    ! [B: $tType,D: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [F3: fun(D,B),F5: filter(D),G: fun(D,B)] :
          ( filterlim(D,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F5)
         => ( filterlim(D,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F5)
           => filterlim(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_rj(fun(D,B),fun(fun(D,B),fun(D,B)),F3),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F5) ) ) ) ).

% tendsto_add_zero
tff(fact_6150_tendsto__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [F3: fun(A,B),A2: B,F5: filter(A),N: nat] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,A2),F5)
         => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_rk(fun(A,B),fun(nat,fun(A,B)),F3),N),topolo7230453075368039082e_nhds(B,aa(nat,B,power_power(B,A2),N)),F5) ) ) ).

% tendsto_power
tff(fact_6151_tendsto__power__strong,axiom,
    ! [B: $tType,C: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [F3: fun(C,B),A2: B,F5: filter(C),G: fun(C,nat),B2: nat] :
          ( filterlim(C,B,F3,topolo7230453075368039082e_nhds(B,A2),F5)
         => ( filterlim(C,nat,G,topolo7230453075368039082e_nhds(nat,B2),F5)
           => filterlim(C,B,aa(fun(C,nat),fun(C,B),aTP_Lamp_rl(fun(C,B),fun(fun(C,nat),fun(C,B)),F3),G),topolo7230453075368039082e_nhds(B,aa(nat,B,power_power(B,A2),B2)),F5) ) ) ) ).

% tendsto_power_strong
tff(fact_6152_tendsto__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F5: filter(B),F11: filter(B),F3: fun(B,A),L: A] :
          ( pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),F5),F11))
         => ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F11)
           => filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F5) ) ) ) ).

% tendsto_mono
tff(fact_6153_filterlim__mono,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),F22: filter(B),F12: filter(A),F23: filter(B),F13: filter(A)] :
      ( filterlim(A,B,F3,F22,F12)
     => ( pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),F22),F23))
       => ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F13),F12))
         => filterlim(A,B,F3,F23,F13) ) ) ) ).

% filterlim_mono
tff(fact_6154_tendsto__within__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(A,B),L: filter(B),X: A,S2: set(A),T6: set(A)] :
          ( filterlim(A,B,F3,L,topolo174197925503356063within(A,X,S2))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T6),S2))
           => filterlim(A,B,F3,L,topolo174197925503356063within(A,X,T6)) ) ) ) ).

% tendsto_within_subset
tff(fact_6155_real__LIM__sandwich__zero,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(A,real),A2: A,G: fun(A,real)] :
          ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( ! [X3: A] :
                ( ( X3 != A2 )
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(A,real,G,X3))) )
           => ( ! [X3: A] :
                  ( ( X3 != A2 )
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(A,real,G,X3)),aa(A,real,F3,X3))) )
             => filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% real_LIM_sandwich_zero
tff(fact_6156_LIM__imp__LIM,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),L: B,A2: A,G: fun(A,C),M: C] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( ! [X3: A] :
                ( ( X3 != A2 )
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(C,aa(C,C,minus_minus(C,aa(A,C,G,X3)),M))),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F3,X3)),L)))) )
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,M),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ).

% LIM_imp_LIM
tff(fact_6157_splice_Osimps_I1_J,axiom,
    ! [A: $tType,Ys: list(A)] : splice(A,nil(A),Ys) = Ys ).

% splice.simps(1)
tff(fact_6158_LIM__offset__zero__iff,axiom,
    ! [C: $tType,D: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(D)
        & zero(C) )
     => ! [A2: A,F3: fun(A,D),L5: D] :
          ( nO_MATCH(C,A,zero_zero(C),A2)
         => ( filterlim(A,D,F3,topolo7230453075368039082e_nhds(D,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
          <=> filterlim(A,D,aa(fun(A,D),fun(A,D),aTP_Lamp_rm(A,fun(fun(A,D),fun(A,D)),A2),F3),topolo7230453075368039082e_nhds(D,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% LIM_offset_zero_iff
tff(fact_6159_LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),L5: B,A2: A,R3: real] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
           => ? [S3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S3))
                & ! [X2: A] :
                    ( ( ( X2 != A2 )
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X2),A2))),S3)) )
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F3,X2)),L5))),R3)) ) ) ) ) ) ).

% LIM_D
tff(fact_6160_LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [A2: A,F3: fun(A,B),L5: B] :
          ( ! [R2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
             => ? [S7: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S7))
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X3),A2))),S7)) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F3,X3)),L5))),R2)) ) ) )
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% LIM_I
tff(fact_6161_LIM__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [S6: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S6))
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X4),A2))),S6)) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,F3,X4)),L5))),R5)) ) ) ) ) ) ).

% LIM_eq
tff(fact_6162_LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [R: real,A2: A,F3: fun(A,B),G: fun(A,B),L: B] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R))
         => ( ! [X3: A] :
                ( ( X3 != A2 )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X3),A2))),R))
                 => ( aa(A,B,F3,X3) = aa(A,B,G,X3) ) ) )
           => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
             => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_equal2
tff(fact_6163_DERIV__LIM__iff,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [F3: fun(A,A),A2: A,D4: A] :
          ( filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_rn(fun(A,A),fun(A,fun(A,A)),F3),A2),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_ro(fun(A,A),fun(A,fun(A,A)),F3),A2),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% DERIV_LIM_iff
tff(fact_6164_LIM__fun__gt__zero,axiom,
    ! [F3: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F3,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),L))
       => ? [R2: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
            & ! [X2: real] :
                ( ( ( X2 != C2 )
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,C2),X2))),R2)) )
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,F3,X2))) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_6165_LIM__fun__not__zero,axiom,
    ! [F3: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F3,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( ( L != zero_zero(real) )
       => ? [R2: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
            & ! [X2: real] :
                ( ( ( X2 != C2 )
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,C2),X2))),R2)) )
               => ( aa(real,real,F3,X2) != zero_zero(real) ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_6166_LIM__fun__less__zero,axiom,
    ! [F3: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,F3,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),L),zero_zero(real)))
       => ? [R2: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
            & ! [X2: real] :
                ( ( ( X2 != C2 )
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,C2),X2))),R2)) )
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F3,X2)),zero_zero(real))) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_6167_LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F3: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F3,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] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D6))
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X3),A2))),D6)) )
                     => ( aa(A,B,F3,X3) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_rp(fun(A,B),fun(fun(B,C),fun(A,C)),F3),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% LIM_compose2
tff(fact_6168_splice_Oelims,axiom,
    ! [A: $tType,X: list(A),Xa: list(A),Y: list(A)] :
      ( ( splice(A,X,Xa) = Y )
     => ( ( ( X = nil(A) )
         => ( Y != Xa ) )
       => ~ ! [X3: A,Xs2: list(A)] :
              ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
             => ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),splice(A,Xa,Xs2)) ) ) ) ) ).

% splice.elims
tff(fact_6169_DERIV__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,X: A] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,X,top_top(set(A))))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_rq(fun(A,A),fun(A,fun(A,A)),F3),X),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_D
tff(fact_6170_DERIV__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,X: A] :
          ( has_field_derivative(A,F3,D4,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_rq(fun(A,A),fun(A,fun(A,A)),F3),X),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_def
tff(fact_6171_lim__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => filterlim(A,A,aTP_Lamp_rr(A,A),topolo7230453075368039082e_nhds(A,one_one(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% lim_exp_minus_1
tff(fact_6172_lemma__termdiff4,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [K: real,F3: fun(A,B),K5: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K))
         => ( ! [H3: A] :
                ( ( H3 != zero_zero(A) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,H3)),K))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,H3))),aa(real,real,aa(real,fun(real,real),times_times(real),K5),real_V7770717601297561774m_norm(A,H3)))) ) )
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% lemma_termdiff4
tff(fact_6173_field__has__derivative__at,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),D4: A,X: A] :
          ( has_derivative(A,A,F3,aa(A,fun(A,A),times_times(A),D4),topolo174197925503356063within(A,X,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_rq(fun(A,A),fun(A,fun(A,A)),F3),X),topolo7230453075368039082e_nhds(A,D4),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% field_has_derivative_at
tff(fact_6174_filterlim__at__to__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(A,B),F5: filter(B),A2: A] :
          ( filterlim(A,B,F3,F5,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_rs(fun(A,B),fun(A,fun(A,B)),F3),A2),F5,topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% filterlim_at_to_0
tff(fact_6175_powser__limit__0__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: real,A2: fun(nat,A),F3: fun(A,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S))
         => ( ! [X3: A] :
                ( ( X3 != zero_zero(A) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X3)),S))
                 => sums(A,aa(A,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(A,fun(nat,A)),A2),X3),aa(A,A,F3,X3)) ) )
           => filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,aa(nat,A,A2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0_strong
tff(fact_6176_powser__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: real,A2: fun(nat,A),F3: fun(A,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S))
         => ( ! [X3: A] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X3)),S))
               => sums(A,aa(A,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(A,fun(nat,A)),A2),X3),aa(A,A,F3,X3)) )
           => filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,aa(nat,A,A2,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0
tff(fact_6177_lemma__termdiff5,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [K: real,F3: fun(nat,real),G: fun(A,fun(nat,B))] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K))
         => ( summable(real,F3)
           => ( ! [H3: A,N2: nat] :
                  ( ( H3 != zero_zero(A) )
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,H3)),K))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(nat,B,aa(A,fun(nat,B),G,H3),N2))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,F3,N2)),real_V7770717601297561774m_norm(A,H3)))) ) )
             => filterlim(A,B,aTP_Lamp_rt(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_6178_LIM__cos__div__sin,axiom,
    filterlim(real,real,aTP_Lamp_ru(real,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),top_top(set(real)))) ).

% LIM_cos_div_sin
tff(fact_6179_splice_Opsimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A)] :
      ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys)))
     => ( splice(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),splice(A,Ys,Xs)) ) ) ).

% splice.psimps(2)
tff(fact_6180_splice_Opsimps_I1_J,axiom,
    ! [A: $tType,Ys: list(A)] :
      ( pp(aa(product_prod(list(A),list(A)),bool,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),nil(A)),Ys)))
     => ( splice(A,nil(A),Ys) = Ys ) ) ).

% splice.psimps(1)
tff(fact_6181_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)
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(nat,real,A2,zero_zero(nat))),zero_zero(real)))
         => ! [N7: nat] : pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),suminf(real,aTP_Lamp_rv(fun(nat,real),fun(nat,real),A2))),set_or1337092689740270186AtMost(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_rv(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),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),aa(num,num,bit0,one2))),N7)),one_one(nat)))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_rv(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N7)))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_6182_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)
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(nat,real,A2,zero_zero(nat))))
         => ! [N7: nat] : pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),suminf(real,aTP_Lamp_rv(fun(nat,real),fun(nat,real),A2))),set_or1337092689740270186AtMost(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_rv(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N7))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_rv(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),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),aa(num,num,bit0,one2))),N7)),one_one(nat))))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_6183_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_rw(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_6184_approx__from__below__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ? [U3: fun(nat,A)] :
              ( ! [N7: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,U3,N7)),X))
              & filterlim(nat,A,U3,topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ) ).

% approx_from_below_dense_linorder
tff(fact_6185_approx__from__above__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ? [U3: fun(nat,A)] :
              ( ! [N7: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(nat,A,U3,N7)))
              & filterlim(nat,A,U3,topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ) ).

% approx_from_above_dense_linorder
tff(fact_6186_filterlim__Suc,axiom,
    filterlim(nat,nat,suc,at_top(nat),at_top(nat)) ).

% filterlim_Suc
tff(fact_6187_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,F3: fun(nat,A),F5: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_rx(fun(nat,A),fun(nat,A),F3),F5,at_top(nat))
    <=> filterlim(nat,A,F3,F5,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_6188_LIMSEQ__imp__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A),L: A] :
          ( filterlim(nat,A,aTP_Lamp_ry(fun(nat,A),fun(nat,A),F3),topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_imp_Suc
tff(fact_6189_LIMSEQ__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A),L: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,aTP_Lamp_ry(fun(nat,A),fun(nat,A),F3),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_Suc
tff(fact_6190_LIMSEQ__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A),A2: A,K: nat] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
         => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_rz(fun(nat,A),fun(nat,fun(nat,A)),F3),K),topolo7230453075368039082e_nhds(A,A2),at_top(nat)) ) ) ).

% LIMSEQ_ignore_initial_segment
tff(fact_6191_LIMSEQ__offset,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A),K: nat,A2: A] :
          ( filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_rz(fun(nat,A),fun(nat,fun(nat,A)),F3),K),topolo7230453075368039082e_nhds(A,A2),at_top(nat))
         => filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,A2),at_top(nat)) ) ) ).

% LIMSEQ_offset
tff(fact_6192_lim__mono,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [N3: nat,X6: fun(nat,A),Y6: fun(nat,A),X: A,Y: A] :
          ( ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N3),N2))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N2)),aa(nat,A,Y6,N2))) )
         => ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
           => ( filterlim(nat,A,Y6,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ) ).

% lim_mono
tff(fact_6193_LIMSEQ__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),X: A,Y6: fun(nat,A),Y: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( filterlim(nat,A,Y6,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
           => ( ? [N8: nat] :
                ! [N2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N2)),aa(nat,A,Y6,N2))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ) ).

% LIMSEQ_le
tff(fact_6194_Lim__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F3: fun(nat,A),L: A,M6: nat,C5: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M6),N2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N2)),C5)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),C5)) ) ) ) ).

% Lim_bounded
tff(fact_6195_Lim__bounded2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F3: fun(nat,A),L: A,N3: nat,C5: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N3),N2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C5),aa(nat,A,F3,N2))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C5),L)) ) ) ) ).

% Lim_bounded2
tff(fact_6196_LIMSEQ__le__const,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),X: A,A2: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( ? [N8: nat] :
              ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(nat,A,X6,N2))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X)) ) ) ) ).

% LIMSEQ_le_const
tff(fact_6197_LIMSEQ__le__const2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),X: A,A2: A] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( ? [N8: nat] :
              ! [N2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N2)),A2)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A2)) ) ) ) ).

% LIMSEQ_le_const2
tff(fact_6198_Sup__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),S: set(A),A2: A] :
          ( ! [N2: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,B2,N2)),S))
         => ( filterlim(nat,A,B2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(set(A),A,complete_Sup_Sup(A),S))) ) ) ) ).

% Sup_lim
tff(fact_6199_Inf__lim,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: fun(nat,A),S: set(A),A2: A] :
          ( ! [N2: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,B2,N2)),S))
         => ( filterlim(nat,A,B2,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),S)),A2)) ) ) ) ).

% Inf_lim
tff(fact_6200_mult__nat__left__at__top,axiom,
    ! [C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_6201_mult__nat__right__at__top,axiom,
    ! [C2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),C2))
     => filterlim(nat,nat,aTP_Lamp_sa(nat,fun(nat,nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_right_at_top
tff(fact_6202_monoseq__convergent,axiom,
    ! [X6: fun(nat,real),B3: real] :
      ( topological_monoseq(real,X6)
     => ( ! [I4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(nat,real,X6,I4))),B3))
       => ~ ! [L6: real] : ~ filterlim(nat,real,X6,topolo7230453075368039082e_nhds(real,L6),at_top(nat)) ) ) ).

% monoseq_convergent
tff(fact_6203_LIMSEQ__root,axiom,
    filterlim(nat,real,aTP_Lamp_sb(nat,real),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ).

% LIMSEQ_root
tff(fact_6204_monoseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: fun(nat,A),X: A] :
          ( topological_monoseq(A,A2)
         => ( filterlim(nat,A,A2,topolo7230453075368039082e_nhds(A,X),at_top(nat))
           => ( ( ! [N7: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A2,N7)),X))
                & ! [M2: nat,N7: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N7))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A2,M2)),aa(nat,A,A2,N7))) ) )
              | ( ! [N7: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(nat,A,A2,N7)))
                & ! [M2: nat,N7: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N7))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A2,N7)),aa(nat,A,A2,M2))) ) ) ) ) ) ) ).

% monoseq_le
tff(fact_6205_lim__const__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A] : filterlim(nat,A,aTP_Lamp_sc(A,fun(nat,A),A2),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_const_over_n
tff(fact_6206_LIMSEQ__linear,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X6: fun(nat,A),X: A,L: nat] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,X),at_top(nat))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),L))
           => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_sd(fun(nat,A),fun(nat,fun(nat,A)),X6),L),topolo7230453075368039082e_nhds(A,X),at_top(nat)) ) ) ) ).

% LIMSEQ_linear
tff(fact_6207_telescope__summable,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => summable(A,aTP_Lamp_se(fun(nat,A),fun(nat,A),F3)) ) ) ).

% telescope_summable
tff(fact_6208_telescope__summable_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => summable(A,aTP_Lamp_sf(fun(nat,A),fun(nat,A),F3)) ) ) ).

% telescope_summable'
tff(fact_6209_nested__sequence__unique,axiom,
    ! [F3: fun(nat,real),G: fun(nat,real)] :
      ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F3,N2)),aa(nat,real,F3,aa(nat,nat,suc,N2))))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,G,aa(nat,nat,suc,N2))),aa(nat,real,G,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F3,N2)),aa(nat,real,G,N2)))
         => ( filterlim(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_sg(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F3),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => ? [L4: real] :
                ( ! [N7: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F3,N7)),L4))
                & filterlim(nat,real,F3,topolo7230453075368039082e_nhds(real,L4),at_top(nat))
                & ! [N7: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),L4),aa(nat,real,G,N7)))
                & filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,L4),at_top(nat)) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_6210_LIMSEQ__inverse__zero,axiom,
    ! [X6: fun(nat,real)] :
      ( ! [R2: real] :
        ? [N8: nat] :
        ! [N2: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N8),N2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),R2),aa(nat,real,X6,N2))) )
     => filterlim(nat,real,aTP_Lamp_sh(fun(nat,real),fun(nat,real),X6),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_zero
tff(fact_6211_lim__inverse__n_H,axiom,
    filterlim(nat,real,aTP_Lamp_si(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% lim_inverse_n'
tff(fact_6212_LIMSEQ__root__const,axiom,
    ! [C2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
     => filterlim(nat,real,aTP_Lamp_sj(real,fun(nat,real),C2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ) ).

% LIMSEQ_root_const
tff(fact_6213_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim(nat,real,aTP_Lamp_sk(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_6214_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R3: real] : filterlim(nat,real,aTP_Lamp_sl(real,fun(nat,real),R3),topolo7230453075368039082e_nhds(real,R3),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_6215_increasing__LIMSEQ,axiom,
    ! [F3: fun(nat,real),L: real] :
      ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F3,N2)),aa(nat,real,F3,aa(nat,nat,suc,N2))))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,F3,N2)),L))
       => ( ! [E: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
             => ? [N7: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),L),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F3,N7)),E))) )
         => filterlim(nat,real,F3,topolo7230453075368039082e_nhds(real,L),at_top(nat)) ) ) ) ).

% increasing_LIMSEQ
tff(fact_6216_lim__1__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_sm(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_1_over_n
tff(fact_6217_LIMSEQ__n__over__Suc__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_sn(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_n_over_Suc_n
tff(fact_6218_LIMSEQ__Suc__n__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_so(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_Suc_n_over_n
tff(fact_6219_LIMSEQ__realpow__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => filterlim(nat,real,power_power(real,X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_6220_telescope__sums_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => sums(A,aTP_Lamp_sf(fun(nat,A),fun(nat,A),F3),aa(A,A,minus_minus(A,aa(nat,A,F3,zero_zero(nat))),C2)) ) ) ).

% telescope_sums'
tff(fact_6221_telescope__sums,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C2: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => sums(A,aTP_Lamp_se(fun(nat,A),fun(nat,A),F3),aa(A,A,minus_minus(A,C2),aa(nat,A,F3,zero_zero(nat)))) ) ) ).

% telescope_sums
tff(fact_6222_LIMSEQ__divide__realpow__zero,axiom,
    ! [X: real,A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_sp(real,fun(real,fun(nat,real)),X),A2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_6223_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),C2)),one_one(real)))
     => filterlim(nat,real,power_power(real,C2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_6224_LIMSEQ__abs__realpow__zero,axiom,
    ! [C2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,abs_abs(real),C2)),one_one(real)))
     => filterlim(nat,real,power_power(real,aa(real,real,abs_abs(real),C2)),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_6225_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => filterlim(nat,real,aTP_Lamp_sq(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_6226_root__test__convergence,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),X: real] :
          ( filterlim(nat,real,aTP_Lamp_sr(fun(nat,A),fun(nat,real),F3),topolo7230453075368039082e_nhds(real,X),at_top(nat))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
           => summable(A,F3) ) ) ) ).

% root_test_convergence
tff(fact_6227_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R3: real] : filterlim(nat,real,aTP_Lamp_ss(real,fun(nat,real),R3),topolo7230453075368039082e_nhds(real,R3),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_6228_LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L5: A,R3: real] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
           => ? [No: nat] :
              ! [N7: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No),N7))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X6,N7)),L5))),R3)) ) ) ) ) ).

% LIMSEQ_D
tff(fact_6229_LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( ! [R2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
             => ? [No2: nat] :
                ! [N2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No2),N2))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X6,N2)),L5))),R2)) ) )
         => filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% LIMSEQ_I
tff(fact_6230_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] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [No3: nat] :
                ! [N5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No3),N5))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X6,N5)),L5))),R5)) ) ) ) ) ).

% LIMSEQ_iff
tff(fact_6231_LIMSEQ__power__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),one_one(real)))
         => filterlim(nat,A,power_power(A,X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_power_zero
tff(fact_6232_tendsto__exp__limit__sequentially,axiom,
    ! [X: real] : filterlim(nat,real,aTP_Lamp_st(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,exp(real,X)),at_top(nat)) ).

% tendsto_exp_limit_sequentially
tff(fact_6233_tendsto__power__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F3: fun(B,nat),F5: filter(B),X: A] :
          ( filterlim(B,nat,F3,at_top(nat),F5)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),one_one(real)))
           => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_su(fun(B,nat),fun(A,fun(B,A)),F3),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),F5) ) ) ) ).

% tendsto_power_zero
tff(fact_6234_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R3: real] : filterlim(nat,real,aTP_Lamp_sv(real,fun(nat,real),R3),topolo7230453075368039082e_nhds(real,R3),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_6235_LIMSEQ__norm__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,N2))),divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N2)))))
         => filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_norm_0
tff(fact_6236_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_rv(fun(nat,real),fun(nat,real),A2)) ) ) ).

% summable_Leibniz(1)
tff(fact_6237_field__derivative__lim__unique,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Df: A,Z: A,S: fun(nat,A),A2: A] :
          ( has_field_derivative(A,F3,Df,topolo174197925503356063within(A,Z,top_top(set(A))))
         => ( filterlim(nat,A,S,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
           => ( ! [N2: nat] : aa(nat,A,S,N2) != zero_zero(A)
             => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),aTP_Lamp_sw(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),F3),Z),S),topolo7230453075368039082e_nhds(A,A2),at_top(nat))
               => ( Df = A2 ) ) ) ) ) ) ).

% field_derivative_lim_unique
tff(fact_6238_powser__times__n__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),one_one(real)))
         => filterlim(nat,A,aTP_Lamp_sx(A,fun(nat,A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% powser_times_n_limit_0
tff(fact_6239_lim__n__over__pown,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,X)))
         => filterlim(nat,A,aTP_Lamp_sy(A,fun(nat,A),X),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% lim_n_over_pown
tff(fact_6240_summable,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,N2)))
         => summable(real,aTP_Lamp_rv(fun(nat,real),fun(nat,real),A2)) ) ) ) ).

% summable
tff(fact_6241_cos__diff__limit__1,axiom,
    ! [Theta: fun(nat,real),Theta2: real] :
      ( filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_sz(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_ta(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_6242_cos__limit__1,axiom,
    ! [Theta: fun(nat,real)] :
      ( filterlim(nat,real,aTP_Lamp_tb(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_ta(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_6243_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_tc(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_rv(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ).

% summable_Leibniz(4)
tff(fact_6244_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),X)),one_one(real)))
     => filterlim(nat,real,aTP_Lamp_ak(real,fun(nat,real),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% zeroseq_arctan_series
tff(fact_6245_summable__Leibniz_H_I2_J,axiom,
    ! [A2: fun(nat,real),N: nat] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,N2)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_rv(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N)))),suminf(real,aTP_Lamp_rv(fun(nat,real),fun(nat,real),A2)))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_6246_summable__Leibniz_H_I3_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,N2)))
         => filterlim(nat,real,aTP_Lamp_tc(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_rv(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_6247_sums__alternating__upper__lower,axiom,
    ! [A2: fun(nat,real)] :
      ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,N2)))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N2)))
       => ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
         => ? [L4: real] :
              ( ! [N7: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_rv(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),N7)))),L4))
              & filterlim(nat,real,aTP_Lamp_tc(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L4),at_top(nat))
              & ! [N7: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),L4),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_rv(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),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),aa(num,num,bit0,one2))),N7)),one_one(nat))))))
              & filterlim(nat,real,aTP_Lamp_td(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,L4),at_top(nat)) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_6248_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_td(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_rv(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ).

% summable_Leibniz(5)
tff(fact_6249_summable__Leibniz_H_I5_J,axiom,
    ! [A2: fun(nat,real)] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,N2)))
         => filterlim(nat,real,aTP_Lamp_td(fun(nat,real),fun(nat,real),A2),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_rv(fun(nat,real),fun(nat,real),A2))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_6250_summable__Leibniz_H_I4_J,axiom,
    ! [A2: fun(nat,real),N: nat] :
      ( filterlim(nat,real,A2,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,A2,N2)))
       => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,A2,aa(nat,nat,suc,N2))),aa(nat,real,A2,N2)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),suminf(real,aTP_Lamp_rv(fun(nat,real),fun(nat,real),A2))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_rv(fun(nat,real),fun(nat,real),A2)),aa(nat,set(nat),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),aa(num,num,bit0,one2))),N)),one_one(nat)))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_6251_summable__bounded__partials,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [F3: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_te(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),F3),G),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F3) ) ) ) ).

% summable_bounded_partials
tff(fact_6252_has__derivative__at2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F10: fun(A,B),X: A] :
          ( has_derivative(A,B,F3,F10,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F10)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_tf(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F3),F10),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,top_top(set(A)))) ) ) ) ).

% has_derivative_at2
tff(fact_6253_eventually__sequentially__Suc,axiom,
    ! [P: fun(nat,bool)] :
      ( eventually(nat,aTP_Lamp_tg(fun(nat,bool),fun(nat,bool),P),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_6254_eventually__sequentially__seg,axiom,
    ! [P: fun(nat,bool),K: nat] :
      ( eventually(nat,aa(nat,fun(nat,bool),aTP_Lamp_th(fun(nat,bool),fun(nat,fun(nat,bool)),P),K),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_seg
tff(fact_6255_le__sequentially,axiom,
    ! [F5: filter(nat)] :
      ( pp(aa(filter(nat),bool,aa(filter(nat),fun(filter(nat),bool),ord_less_eq(filter(nat)),F5),at_top(nat)))
    <=> ! [N6: nat] : eventually(nat,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),F5) ) ).

% le_sequentially
tff(fact_6256_eventually__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aa(A,fun(A,bool),ord_less_eq(A),C2),at_top(A)) ) ).

% eventually_ge_at_top
tff(fact_6257_eventually__sequentially,axiom,
    ! [P: fun(nat,bool)] :
      ( eventually(nat,P,at_top(nat))
    <=> ? [N6: nat] :
        ! [N5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),N5))
         => pp(aa(nat,bool,P,N5)) ) ) ).

% eventually_sequentially
tff(fact_6258_eventually__sequentiallyI,axiom,
    ! [C2: nat,P: fun(nat,bool)] :
      ( ! [X3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),C2),X3))
         => pp(aa(nat,bool,P,X3)) )
     => eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentiallyI
tff(fact_6259_eventually__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N6: A] :
            ! [N5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N6),N5))
             => pp(aa(A,bool,P,N5)) ) ) ) ).

% eventually_at_top_linorder
tff(fact_6260_eventually__at__top__linorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,P: fun(A,bool)] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),C2),X3))
             => pp(aa(A,bool,P,X3)) )
         => eventually(A,P,at_top(A)) ) ) ).

% eventually_at_top_linorderI
tff(fact_6261_sequentially__offset,axiom,
    ! [P: fun(nat,bool),K: nat] :
      ( eventually(nat,P,at_top(nat))
     => eventually(nat,aa(nat,fun(nat,bool),aTP_Lamp_th(fun(nat,bool),fun(nat,fun(nat,bool)),P),K),at_top(nat)) ) ).

% sequentially_offset
tff(fact_6262_eventually__gt__at__top,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,aa(A,fun(A,bool),ord_less(A),C2),at_top(A)) ) ).

% eventually_gt_at_top
tff(fact_6263_eventually__at__top__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N6: A] :
            ! [N5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N6),N5))
             => pp(aa(A,bool,P,N5)) ) ) ) ).

% eventually_at_top_dense
tff(fact_6264_eventually__ex,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),F5: filter(A)] :
      ( eventually(A,aTP_Lamp_nt(fun(A,fun(B,bool)),fun(A,bool),P),F5)
    <=> ? [Y7: fun(A,B)] : eventually(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_ti(fun(A,fun(B,bool)),fun(fun(A,B),fun(A,bool)),P),Y7),F5) ) ).

% eventually_ex
tff(fact_6265_bounded__linear__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),G: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ( real_V3181309239436604168linear(A,B,G)
           => real_V3181309239436604168linear(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pn(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G)) ) ) ) ).

% bounded_linear_add
tff(fact_6266_bounded__linear__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_tj(A,fun(A,A),Y)) ) ).

% bounded_linear_divide
tff(fact_6267_le__filter__def,axiom,
    ! [A: $tType,F5: filter(A),F11: filter(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F5),F11))
    <=> ! [P6: fun(A,bool)] :
          ( eventually(A,P6,F11)
         => eventually(A,P6,F5) ) ) ).

% le_filter_def
tff(fact_6268_filter__leI,axiom,
    ! [A: $tType,F11: filter(A),F5: filter(A)] :
      ( ! [P5: fun(A,bool)] :
          ( eventually(A,P5,F11)
         => eventually(A,P5,F5) )
     => pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F5),F11)) ) ).

% filter_leI
tff(fact_6269_filter__leD,axiom,
    ! [A: $tType,F5: filter(A),F11: filter(A),P: fun(A,bool)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F5),F11))
     => ( eventually(A,P,F11)
       => eventually(A,P,F5) ) ) ).

% filter_leD
tff(fact_6270_filterlim__mono__eventually,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),F5: filter(B),G4: filter(A),F11: filter(B),G5: filter(A),F10: fun(A,B)] :
      ( filterlim(A,B,F3,F5,G4)
     => ( pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),F5),F11))
       => ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),G5),G4))
         => ( eventually(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_tk(fun(A,B),fun(fun(A,B),fun(A,bool)),F3),F10),G5)
           => filterlim(A,B,F10,F11,G5) ) ) ) ) ).

% filterlim_mono_eventually
tff(fact_6271_eventually__nhds__top,axiom,
    ! [A: $tType] :
      ( ( order_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),top_top(A)))
         => ( eventually(A,P,topolo7230453075368039082e_nhds(A,top_top(A)))
          <=> ? [B6: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B6),top_top(A)))
                & ! [Z5: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B6),Z5))
                   => pp(aa(A,bool,P,Z5)) ) ) ) ) ) ).

% eventually_nhds_top
tff(fact_6272_filterlim__at__top__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [Q: fun(A,bool),F3: fun(A,B),P: fun(B,bool),G: fun(B,A)] :
          ( ! [X3: A,Y4: A] :
              ( pp(aa(A,bool,Q,X3))
             => ( pp(aa(A,bool,Q,Y4))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y4))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y4))) ) ) )
         => ( ! [X3: B] :
                ( pp(aa(B,bool,P,X3))
               => ( aa(A,B,F3,aa(B,A,G,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,P,X3))
                 => pp(aa(A,bool,Q,aa(B,A,G,X3))) )
             => ( eventually(A,Q,at_top(A))
               => ( eventually(B,P,at_top(B))
                 => filterlim(A,B,F3,at_top(B),at_top(A)) ) ) ) ) ) ) ).

% filterlim_at_top_at_top
tff(fact_6273_eventually__at__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Y: A,X: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( eventually(A,P,topolo174197925503356063within(A,X,aa(A,set(A),set_ord_lessThan(A),X)))
          <=> ? [B6: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B6),X))
                & ! [Y5: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B6),Y5))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X))
                     => pp(aa(A,bool,P,Y5)) ) ) ) ) ) ) ).

% eventually_at_left
tff(fact_6274_eventually__at__left__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,bool),X: A] :
          ( eventually(A,P,topolo174197925503356063within(A,X,aa(A,set(A),set_ord_lessThan(A),X)))
        <=> ? [B6: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B6),X))
              & ! [Y5: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B6),Y5))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X))
                   => pp(aa(A,bool,P,Y5)) ) ) ) ) ) ).

% eventually_at_left_field
tff(fact_6275_bounded__linear_Obounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ? [K9: real] :
            ! [X2: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X2))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X2)),K9))) ) ) ).

% bounded_linear.bounded
tff(fact_6276_tendsto__sandwich,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F3: fun(B,A),G: fun(B,A),Net: filter(B),H: fun(B,A),C2: A] :
          ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_tl(fun(B,A),fun(fun(B,A),fun(B,bool)),F3),G),Net)
         => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_tl(fun(B,A),fun(fun(B,A),fun(B,bool)),G),H),Net)
           => ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,C2),Net)
             => ( filterlim(B,A,H,topolo7230453075368039082e_nhds(A,C2),Net)
               => filterlim(B,A,G,topolo7230453075368039082e_nhds(A,C2),Net) ) ) ) ) ) ).

% tendsto_sandwich
tff(fact_6277_order__tendstoD_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F3: fun(B,A),Y: A,F5: filter(B),A2: A] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,Y),F5)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),A2))
           => eventually(B,aa(A,fun(B,bool),aTP_Lamp_tm(fun(B,A),fun(A,fun(B,bool)),F3),A2),F5) ) ) ) ).

% order_tendstoD(2)
tff(fact_6278_order__tendstoD_I1_J,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F3: fun(B,A),Y: A,F5: filter(B),A2: A] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,Y),F5)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),Y))
           => eventually(B,aa(A,fun(B,bool),aTP_Lamp_tn(fun(B,A),fun(A,fun(B,bool)),F3),A2),F5) ) ) ) ).

% order_tendstoD(1)
tff(fact_6279_order__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Y: A,F3: fun(B,A),F5: filter(B)] :
          ( ! [A5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A5),Y))
             => eventually(B,aa(A,fun(B,bool),aTP_Lamp_tn(fun(B,A),fun(A,fun(B,bool)),F3),A5),F5) )
         => ( ! [A5: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),A5))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_tm(fun(B,A),fun(A,fun(B,bool)),F3),A5),F5) )
           => filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,Y),F5) ) ) ) ).

% order_tendstoI
tff(fact_6280_order__tendsto__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F3: fun(B,A),X: A,F5: filter(B)] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,X),F5)
        <=> ( ! [L2: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L2),X))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_tn(fun(B,A),fun(A,fun(B,bool)),F3),L2),F5) )
            & ! [U4: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),U4))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_tm(fun(B,A),fun(A,fun(B,bool)),F3),U4),F5) ) ) ) ) ).

% order_tendsto_iff
tff(fact_6281_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_top(A))
         => eventually(A,aTP_Lamp_to(fun(A,bool),fun(A,bool),P),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_6282_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F5: filter(A)] :
          ( filterlim(A,B,F3,at_top(B),F5)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_tp(fun(A,B),fun(B,fun(A,bool)),F3),Z7),F5) ) ) ).

% filterlim_at_top
tff(fact_6283_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F5: filter(A),C2: B] :
          ( filterlim(A,B,F3,at_top(B),F5)
        <=> ! [Z7: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),C2),Z7))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_tp(fun(A,B),fun(B,fun(A,bool)),F3),Z7),F5) ) ) ) ).

% filterlim_at_top_ge
tff(fact_6284_filterlim__at__top__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),F5: filter(B),G: fun(B,A)] :
          ( filterlim(B,A,F3,at_top(A),F5)
         => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_tq(fun(B,A),fun(fun(B,A),fun(B,bool)),F3),G),F5)
           => filterlim(B,A,G,at_top(A),F5) ) ) ) ).

% filterlim_at_top_mono
tff(fact_6285_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F3: fun(A,B),F5: filter(A)] :
          ( filterlim(A,B,F3,at_top(B),F5)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_tr(fun(A,B),fun(B,fun(A,bool)),F3),Z7),F5) ) ) ).

% filterlim_at_top_dense
tff(fact_6286_real__tendsto__sandwich,axiom,
    ! [B: $tType,F3: fun(B,real),G: fun(B,real),Net: filter(B),H: fun(B,real),C2: real] :
      ( eventually(B,aa(fun(B,real),fun(B,bool),aTP_Lamp_ts(fun(B,real),fun(fun(B,real),fun(B,bool)),F3),G),Net)
     => ( eventually(B,aa(fun(B,real),fun(B,bool),aTP_Lamp_ts(fun(B,real),fun(fun(B,real),fun(B,bool)),G),H),Net)
       => ( filterlim(B,real,F3,topolo7230453075368039082e_nhds(real,C2),Net)
         => ( filterlim(B,real,H,topolo7230453075368039082e_nhds(real,C2),Net)
           => filterlim(B,real,G,topolo7230453075368039082e_nhds(real,C2),Net) ) ) ) ) ).

% real_tendsto_sandwich
tff(fact_6287_eventually__Inf__base,axiom,
    ! [A: $tType,B3: set(filter(A)),P: fun(A,bool)] :
      ( ( B3 != bot_bot(set(filter(A))) )
     => ( ! [F7: filter(A)] :
            ( pp(aa(set(filter(A)),bool,aa(filter(A),fun(set(filter(A)),bool),member(filter(A)),F7),B3))
           => ! [G2: filter(A)] :
                ( pp(aa(set(filter(A)),bool,aa(filter(A),fun(set(filter(A)),bool),member(filter(A)),G2),B3))
               => ? [X2: filter(A)] :
                    ( pp(aa(set(filter(A)),bool,aa(filter(A),fun(set(filter(A)),bool),member(filter(A)),X2),B3))
                    & pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),X2),aa(filter(A),filter(A),aa(filter(A),fun(filter(A),filter(A)),inf_inf(filter(A)),F7),G2))) ) ) )
       => ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B3))
        <=> ? [X4: filter(A)] :
              ( pp(aa(set(filter(A)),bool,aa(filter(A),fun(set(filter(A)),bool),member(filter(A)),X4),B3))
              & eventually(A,P,X4) ) ) ) ) ).

% eventually_Inf_base
tff(fact_6288_eventually__at__leftI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A,P: fun(A,bool)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),set_or5935395276787703475ssThan(A,A2,B2)))
             => pp(aa(A,bool,P,X3)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => eventually(A,P,topolo174197925503356063within(A,B2,aa(A,set(A),set_ord_lessThan(A),B2))) ) ) ) ).

% eventually_at_leftI
tff(fact_6289_bounded__linear_Ononneg__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ? [K9: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),K9))
              & ! [X2: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X2))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X2)),K9))) ) ) ) ).

% bounded_linear.nonneg_bounded
tff(fact_6290_eventually__at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,bool),A2: A] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> eventually(A,aa(A,fun(A,bool),aTP_Lamp_tt(fun(A,bool),fun(A,fun(A,bool)),P),A2),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% eventually_at_to_0
tff(fact_6291_increasing__tendsto,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [F3: fun(B,A),L: A,F5: filter(B)] :
          ( eventually(B,aa(A,fun(B,bool),aTP_Lamp_tu(fun(B,A),fun(A,fun(B,bool)),F3),L),F5)
         => ( ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),L))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_tn(fun(B,A),fun(A,fun(B,bool)),F3),X3),F5) )
           => filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F5) ) ) ) ).

% increasing_tendsto
tff(fact_6292_decreasing__tendsto,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [L: A,F3: fun(B,A),F5: filter(B)] :
          ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_tv(A,fun(fun(B,A),fun(B,bool)),L),F3),F5)
         => ( ! [X3: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),L),X3))
               => eventually(B,aa(A,fun(B,bool),aTP_Lamp_tm(fun(B,A),fun(A,fun(B,bool)),F3),X3),F5) )
           => filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F5) ) ) ) ).

% decreasing_tendsto
tff(fact_6293_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F3: fun(A,B),F5: filter(A),C2: B] :
          ( filterlim(A,B,F3,at_top(B),F5)
        <=> ! [Z7: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),C2),Z7))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_tw(fun(A,B),fun(B,fun(A,bool)),F3),Z7),F5) ) ) ) ).

% filterlim_at_top_gt
tff(fact_6294_tendsto__le,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F5: filter(B),F3: fun(B,A),X: A,G: fun(B,A),Y: A] :
          ( ( F5 != bot_bot(filter(B)) )
         => ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,X),F5)
           => ( filterlim(B,A,G,topolo7230453075368039082e_nhds(A,Y),F5)
             => ( eventually(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_tx(fun(B,A),fun(fun(B,A),fun(B,bool)),F3),G),F5)
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ) ) ) ).

% tendsto_le
tff(fact_6295_tendsto__lowerbound,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F3: fun(B,A),X: A,F5: filter(B),A2: A] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,X),F5)
         => ( eventually(B,aa(A,fun(B,bool),aTP_Lamp_ty(fun(B,A),fun(A,fun(B,bool)),F3),A2),F5)
           => ( ( F5 != bot_bot(filter(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X)) ) ) ) ) ).

% tendsto_lowerbound
tff(fact_6296_tendsto__upperbound,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [F3: fun(B,A),X: A,F5: filter(B),A2: A] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,X),F5)
         => ( eventually(B,aa(A,fun(B,bool),aTP_Lamp_tz(fun(B,A),fun(A,fun(B,bool)),F3),A2),F5)
           => ( ( F5 != bot_bot(filter(B)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A2)) ) ) ) ) ).

% tendsto_upperbound
tff(fact_6297_eventually__INF,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F5: fun(B,filter(A)),B3: set(B)] :
      ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image(B,filter(A),F5),B3)))
    <=> ? [X8: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),X8),B3))
          & pp(aa(set(B),bool,finite_finite(B),X8))
          & eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(B),set(filter(A)),image(B,filter(A),F5),X8))) ) ) ).

% eventually_INF
tff(fact_6298_bounded__linear_Opos__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ? [K9: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K9))
              & ! [X2: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X2))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X2)),K9))) ) ) ) ).

% bounded_linear.pos_bounded
tff(fact_6299_bounded__linear__intro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),K5: real] :
          ( ! [X3: A,Y4: A] : aa(A,B,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Y4)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F3,X3)),aa(A,B,F3,Y4))
         => ( ! [R2: real,X3: A] : aa(A,B,F3,aa(A,A,real_V8093663219630862766scaleR(A,R2),X3)) = aa(B,B,real_V8093663219630862766scaleR(B,R2),aa(A,B,F3,X3))
           => ( ! [X3: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F3,X3))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X3)),K5)))
             => real_V3181309239436604168linear(A,B,F3) ) ) ) ) ).

% bounded_linear_intro
tff(fact_6300_tendsto__imp__filterlim__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),L5: B,F5: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),F5)
         => ( eventually(A,aa(B,fun(A,bool),aTP_Lamp_ua(fun(A,B),fun(B,fun(A,bool)),F3),L5),F5)
           => filterlim(A,B,F3,topolo174197925503356063within(B,L5,aa(B,set(B),set_ord_lessThan(B),L5)),F5) ) ) ) ).

% tendsto_imp_filterlim_at_left
tff(fact_6301_eventually__Inf,axiom,
    ! [A: $tType,P: fun(A,bool),B3: set(filter(A))] :
      ( eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),B3))
    <=> ? [X8: set(filter(A))] :
          ( pp(aa(set(filter(A)),bool,aa(set(filter(A)),fun(set(filter(A)),bool),ord_less_eq(set(filter(A))),X8),B3))
          & pp(aa(set(filter(A)),bool,finite_finite(filter(A)),X8))
          & eventually(A,P,aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),X8)) ) ) ).

% eventually_Inf
tff(fact_6302_summable__comparison__test__ev,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_ub(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),F3),G),at_top(nat))
         => ( summable(real,G)
           => summable(A,F3) ) ) ) ).

% summable_comparison_test_ev
tff(fact_6303_tendsto__arcosh__strong,axiom,
    ! [B: $tType,F3: fun(B,real),A2: real,F5: filter(B)] :
      ( filterlim(B,real,F3,topolo7230453075368039082e_nhds(real,A2),F5)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),A2))
       => ( eventually(B,aTP_Lamp_uc(fun(B,real),fun(B,bool),F3),F5)
         => filterlim(B,real,aTP_Lamp_ra(fun(B,real),fun(B,real),F3),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A2)),F5) ) ) ) ).

% tendsto_arcosh_strong
tff(fact_6304_filterlim__at__top__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,bool),F3: fun(A,B),P: fun(B,bool),G: fun(B,A),A2: A] :
          ( ! [X3: A,Y4: A] :
              ( pp(aa(A,bool,Q,X3))
             => ( pp(aa(A,bool,Q,Y4))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y4))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y4))) ) ) )
         => ( ! [X3: B] :
                ( pp(aa(B,bool,P,X3))
               => ( aa(A,B,F3,aa(B,A,G,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,P,X3))
                 => pp(aa(A,bool,Q,aa(B,A,G,X3))) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_lessThan(A),A2)))
               => ( ! [B5: A] :
                      ( pp(aa(A,bool,Q,B5))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B5),A2)) )
                 => ( eventually(B,P,at_top(B))
                   => filterlim(A,B,F3,at_top(B),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_lessThan(A),A2))) ) ) ) ) ) ) ) ).

% filterlim_at_top_at_left
tff(fact_6305_eventually__INF__base,axiom,
    ! [B: $tType,A: $tType,B3: set(A),F5: fun(A,filter(B)),P: fun(B,bool)] :
      ( ( B3 != bot_bot(set(A)) )
     => ( ! [A5: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),B3))
           => ! [B5: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B5),B3))
               => ? [X2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),B3))
                    & pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),aa(A,filter(B),F5,X2)),aa(filter(B),filter(B),aa(filter(B),fun(filter(B),filter(B)),inf_inf(filter(B)),aa(A,filter(B),F5,A5)),aa(A,filter(B),F5,B5)))) ) ) )
       => ( eventually(B,P,aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),F5),B3)))
        <=> ? [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),B3))
              & eventually(B,P,aa(A,filter(B),F5,X4)) ) ) ) ) ).

% eventually_INF_base
tff(fact_6306_tendsto__0__le,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F5: filter(A),G: fun(A,C),K5: real] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,zero_zero(B)),F5)
         => ( eventually(A,aa(real,fun(A,bool),aa(fun(A,C),fun(real,fun(A,bool)),aTP_Lamp_ud(fun(A,B),fun(fun(A,C),fun(real,fun(A,bool))),F3),G),K5),F5)
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,zero_zero(C)),F5) ) ) ) ).

% tendsto_0_le
tff(fact_6307_tendsto__zero__powrI,axiom,
    ! [A: $tType,F3: fun(A,real),F5: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,zero_zero(real)),F5)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F5)
       => ( eventually(A,aTP_Lamp_ue(fun(A,real),fun(A,bool),F3),F5)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_uf(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F5) ) ) ) ) ).

% tendsto_zero_powrI
tff(fact_6308_tendsto__powr2,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F5: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F5)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F5)
       => ( eventually(A,aTP_Lamp_ue(fun(A,real),fun(A,bool),F3),F5)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_uf(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F5) ) ) ) ) ).

% tendsto_powr2
tff(fact_6309_tendsto__powr_H,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F5: filter(A),G: fun(A,real),B2: real] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F5)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B2),F5)
       => ( ( ( A2 != zero_zero(real) )
            | ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B2))
              & eventually(A,aTP_Lamp_ue(fun(A,real),fun(A,bool),F3),F5) ) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_uf(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G),topolo7230453075368039082e_nhds(real,powr(real,A2,B2)),F5) ) ) ) ).

% tendsto_powr'
tff(fact_6310_eventually__floor__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F3: fun(A,B),L: B,F5: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F5)
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),L),ring_1_Ints(B)))
           => eventually(A,aa(B,fun(A,bool),aTP_Lamp_ug(fun(A,B),fun(B,fun(A,bool)),F3),L),F5) ) ) ) ).

% eventually_floor_less
tff(fact_6311_eventually__less__ceiling,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [F3: fun(A,B),L: B,F5: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),F5)
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),L),ring_1_Ints(B)))
           => eventually(A,aa(B,fun(A,bool),aTP_Lamp_uh(fun(A,B),fun(B,fun(A,bool)),F3),L),F5) ) ) ) ).

% eventually_less_ceiling
tff(fact_6312_has__derivative__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F10: fun(A,B),X: A] :
          ( has_derivative(A,B,F3,F10,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F10)
            & ? [E3: fun(A,B)] :
                ( ! [H5: A] : aa(A,B,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F3,X)),aa(A,B,F10,H5))),aa(A,B,E3,H5))
                & filterlim(A,real,aTP_Lamp_ui(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_6313_has__derivative__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F10: fun(A,B),X: A,S: set(A)] :
          ( has_derivative(A,B,F3,F10,topolo174197925503356063within(A,X,S))
        <=> ( real_V3181309239436604168linear(A,B,F10)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_tf(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),F3),F10),X),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,X,S)) ) ) ) ).

% has_derivative_within
tff(fact_6314_has__derivative__at,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),D4: fun(A,B),X: A] :
          ( has_derivative(A,B,F3,D4,topolo174197925503356063within(A,X,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,D4)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_uj(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),F3),D4),X),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% has_derivative_at
tff(fact_6315_summable__Cauchy_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F3: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_uk(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),F3),G),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,F3) ) ) ) ).

% summable_Cauchy'
tff(fact_6316_polyfun__extremal,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [C2: fun(nat,A),K: nat,N: nat,B3: real] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),K))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N))
             => eventually(A,aa(real,fun(A,bool),aa(nat,fun(real,fun(A,bool)),aTP_Lamp_ul(fun(nat,A),fun(nat,fun(real,fun(A,bool))),C2),N),B3),at_infinity(A)) ) ) ) ) ).

% polyfun_extremal
tff(fact_6317_has__derivative__at__within__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [X: A,S2: set(A),F3: fun(A,B),F10: fun(A,B)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S2))
         => ( topolo1002775350975398744n_open(A,S2)
           => ( has_derivative(A,B,F3,F10,topolo174197925503356063within(A,X,S2))
            <=> ( real_V3181309239436604168linear(A,B,F10)
                & ? [E3: fun(A,B)] :
                    ( ! [H5: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H5)),S2))
                       => ( aa(A,B,F3,aa(A,A,aa(A,fun(A,A),plus_plus(A),X),H5)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,F3,X)),aa(A,B,F10,H5))),aa(A,B,E3,H5)) ) )
                    & filterlim(A,real,aTP_Lamp_ui(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_6318_filterlim__at__infinity__imp__filterlim__at__top,axiom,
    ! [A: $tType,F3: fun(A,real),F5: filter(A)] :
      ( filterlim(A,real,F3,at_infinity(real),F5)
     => ( eventually(A,aTP_Lamp_um(fun(A,real),fun(A,bool),F3),F5)
       => filterlim(A,real,F3,at_top(real),F5) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_top
tff(fact_6319_open__diagonal__complement,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => topolo1002775350975398744n_open(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_un(product_prod(A,A),bool))) ) ).

% open_diagonal_complement
tff(fact_6320_first__countable__basis,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [X: A] :
        ? [A7: fun(nat,set(A))] :
          ( ! [I: nat] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(nat,set(A),A7,I)))
              & topolo1002775350975398744n_open(A,aa(nat,set(A),A7,I)) )
          & ! [S8: set(A)] :
              ( ( topolo1002775350975398744n_open(A,S8)
                & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S8)) )
             => ? [I4: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),A7,I4)),S8)) ) ) ) ).

% first_countable_basis
tff(fact_6321_open__subopen,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A)] :
          ( topolo1002775350975398744n_open(A,S2)
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
             => ? [T9: set(A)] :
                  ( topolo1002775350975398744n_open(A,T9)
                  & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),T9))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T9),S2)) ) ) ) ) ).

% open_subopen
tff(fact_6322_openI,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
             => ? [T10: set(A)] :
                  ( topolo1002775350975398744n_open(A,T10)
                  & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),T10))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T10),S2)) ) )
         => topolo1002775350975398744n_open(A,S2) ) ) ).

% openI
tff(fact_6323_Inf__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A3: set(A),X: A] :
          ( topolo1002775350975398744n_open(A,A3)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X3)) )
           => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,complete_Inf_Inf(A),A3)),A3)) ) ) ) ).

% Inf_notin_open
tff(fact_6324_open__Collect__ex,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(B,fun(A,bool))] :
          ( ! [I4: B] : topolo1002775350975398744n_open(A,collect(A,aa(B,fun(A,bool),P,I4)))
         => topolo1002775350975398744n_open(A,collect(A,aTP_Lamp_uo(fun(B,fun(A,bool)),fun(A,bool),P))) ) ) ).

% open_Collect_ex
tff(fact_6325_Sup__notin__open,axiom,
    ! [A: $tType] :
      ( topolo8458572112393995274pology(A)
     => ! [A3: set(A),X: A] :
          ( topolo1002775350975398744n_open(A,A3)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),X)) )
           => ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,complete_Sup_Sup(A),A3)),A3)) ) ) ) ).

% Sup_notin_open
tff(fact_6326_at__within__open__subset,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A2: A,S2: set(A),T6: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),S2))
         => ( topolo1002775350975398744n_open(A,S2)
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),T6))
             => ( topolo174197925503356063within(A,A2,T6) = topolo174197925503356063within(A,A2,top_top(set(A))) ) ) ) ) ) ).

% at_within_open_subset
tff(fact_6327_open__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S2: set(A),X: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S2)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
             => ? [B5: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B5))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,X,B5)),S2)) ) ) ) ) ) ).

% open_right
tff(fact_6328_open__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S2: set(A),X: A,Y: A] :
          ( topolo1002775350975398744n_open(A,S2)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S2))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
             => ? [B5: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B5),X))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or3652927894154168847AtMost(A,B5,X)),S2)) ) ) ) ) ) ).

% open_left
tff(fact_6329_open__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_up(product_prod(A,A),bool))) ) ).

% open_subdiagonal
tff(fact_6330_open__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo1002775350975398744n_open(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_uq(product_prod(A,A),bool))) ) ).

% open_superdiagonal
tff(fact_6331_countable__basis__at__decseq,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [X: A] :
          ~ ! [A7: fun(nat,set(A))] :
              ( ! [I: nat] : topolo1002775350975398744n_open(A,aa(nat,set(A),A7,I))
             => ( ! [I: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(nat,set(A),A7,I)))
               => ~ ! [S8: set(A)] :
                      ( topolo1002775350975398744n_open(A,S8)
                     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S8))
                       => eventually(nat,aa(set(A),fun(nat,bool),aTP_Lamp_ur(fun(nat,set(A)),fun(set(A),fun(nat,bool)),A7),S8),at_top(nat)) ) ) ) ) ) ).

% countable_basis_at_decseq
tff(fact_6332_eventually__at__left__real,axiom,
    ! [B2: real,A2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),B2),A2))
     => eventually(real,aa(real,fun(real,bool),aTP_Lamp_us(real,fun(real,fun(real,bool)),B2),A2),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_lessThan(real),A2))) ) ).

% eventually_at_left_real
tff(fact_6333_lim__explicit,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [F3: fun(nat,A),F0: A] :
          ( filterlim(nat,A,F3,topolo7230453075368039082e_nhds(A,F0),at_top(nat))
        <=> ! [S9: set(A)] :
              ( topolo1002775350975398744n_open(A,S9)
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),F0),S9))
               => ? [N6: nat] :
                  ! [N5: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),N5))
                   => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,F3,N5)),S9)) ) ) ) ) ) ).

% lim_explicit
tff(fact_6334_eventually__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_infinity(A))
        <=> ? [B6: real] :
            ! [X4: A] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B6),real_V7770717601297561774m_norm(A,X4)))
             => pp(aa(A,bool,P,X4)) ) ) ) ).

% eventually_at_infinity
tff(fact_6335_filterlim__pow__at__top,axiom,
    ! [A: $tType,N: nat,F3: fun(A,real),F5: filter(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( filterlim(A,real,F3,at_top(real),F5)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_qu(nat,fun(fun(A,real),fun(A,real)),N),F3),at_top(real),F5) ) ) ).

% filterlim_pow_at_top
tff(fact_6336_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_6337_tendsto__add__filterlim__at__infinity_H,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),F5: filter(A),G: fun(A,B),C2: B] :
          ( filterlim(A,B,F3,at_infinity(B),F5)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),F5)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ut(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G),at_infinity(B),F5) ) ) ) ).

% tendsto_add_filterlim_at_infinity'
tff(fact_6338_tendsto__add__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),C2: B,F5: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,C2),F5)
         => ( filterlim(A,B,G,at_infinity(B),F5)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ut(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G),at_infinity(B),F5) ) ) ) ).

% tendsto_add_filterlim_at_infinity
tff(fact_6339_artanh__real__at__left__1,axiom,
    filterlim(real,real,artanh(real),at_top(real),topolo174197925503356063within(real,one_one(real),aa(real,set(real),set_ord_lessThan(real),one_one(real)))) ).

% artanh_real_at_left_1
tff(fact_6340_filterlim__tendsto__pos__mult__at__top,axiom,
    ! [A: $tType,F3: fun(A,real),C2: real,F5: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C2),F5)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
       => ( filterlim(A,real,G,at_top(real),F5)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_uu(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G),at_top(real),F5) ) ) ) ).

% filterlim_tendsto_pos_mult_at_top
tff(fact_6341_filterlim__at__top__mult__tendsto__pos,axiom,
    ! [A: $tType,F3: fun(A,real),C2: real,F5: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C2),F5)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
       => ( filterlim(A,real,G,at_top(real),F5)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_uv(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G),at_top(real),F5) ) ) ) ).

% filterlim_at_top_mult_tendsto_pos
tff(fact_6342_tendsto__neg__powr,axiom,
    ! [A: $tType,S: real,F3: fun(A,real),F5: filter(A)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),S),zero_zero(real)))
     => ( filterlim(A,real,F3,at_top(real),F5)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_uw(real,fun(fun(A,real),fun(A,real)),S),F3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F5) ) ) ).

% tendsto_neg_powr
tff(fact_6343_tendsto__divide__0,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F3: fun(C,A),C2: A,F5: filter(C),G: fun(C,A)] :
          ( filterlim(C,A,F3,topolo7230453075368039082e_nhds(A,C2),F5)
         => ( filterlim(C,A,G,at_infinity(A),F5)
           => filterlim(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_ux(fun(C,A),fun(fun(C,A),fun(C,A)),F3),G),topolo7230453075368039082e_nhds(A,zero_zero(A)),F5) ) ) ) ).

% tendsto_divide_0
tff(fact_6344_filterlim__power__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [F3: fun(A,B),F5: filter(A),N: nat] :
          ( filterlim(A,B,F3,at_infinity(B),F5)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_uy(fun(A,B),fun(nat,fun(A,B)),F3),N),at_infinity(B),F5) ) ) ) ).

% filterlim_power_at_infinity
tff(fact_6345_LIM__at__top__divide,axiom,
    ! [A: $tType,F3: fun(A,real),A2: real,F5: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,A2),F5)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),A2))
       => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),F5)
         => ( eventually(A,aTP_Lamp_um(fun(A,real),fun(A,bool),G),F5)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_uz(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G),at_top(real),F5) ) ) ) ) ).

% LIM_at_top_divide
tff(fact_6346_filterlim__inverse__at__top__iff,axiom,
    ! [A: $tType,F3: fun(A,real),F5: filter(A)] :
      ( eventually(A,aTP_Lamp_um(fun(A,real),fun(A,bool),F3),F5)
     => ( filterlim(A,real,aTP_Lamp_va(fun(A,real),fun(A,real),F3),at_top(real),F5)
      <=> filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,zero_zero(real)),F5) ) ) ).

% filterlim_inverse_at_top_iff
tff(fact_6347_filterlim__inverse__at__top,axiom,
    ! [A: $tType,F3: fun(A,real),F5: filter(A)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,zero_zero(real)),F5)
     => ( eventually(A,aTP_Lamp_um(fun(A,real),fun(A,bool),F3),F5)
       => filterlim(A,real,aTP_Lamp_va(fun(A,real),fun(A,real),F3),at_top(real),F5) ) ) ).

% filterlim_inverse_at_top
tff(fact_6348_eventually__at__infinity__pos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P2: fun(A,bool)] :
          ( eventually(A,P2,at_infinity(A))
        <=> ? [B6: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B6))
              & ! [X4: A] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B6),real_V7770717601297561774m_norm(A,X4)))
                 => pp(aa(A,bool,P2,X4)) ) ) ) ) ).

% eventually_at_infinity_pos
tff(fact_6349_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim(real,real,aTP_Lamp_vb(nat,fun(real,real),K),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% tendsto_power_div_exp_0
tff(fact_6350_tendsto__offset__zero__iff,axiom,
    ! [C: $tType,D: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(D)
        & zero(C) )
     => ! [A2: A,S2: set(A),F3: fun(A,D),L5: D] :
          ( nO_MATCH(C,A,zero_zero(C),A2)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),S2))
           => ( topolo1002775350975398744n_open(A,S2)
             => ( filterlim(A,D,F3,topolo7230453075368039082e_nhds(D,L5),topolo174197925503356063within(A,A2,S2))
              <=> filterlim(A,D,aa(fun(A,D),fun(A,D),aTP_Lamp_rm(A,fun(fun(A,D),fun(A,D)),A2),F3),topolo7230453075368039082e_nhds(D,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ) ).

% tendsto_offset_zero_iff
tff(fact_6351_tendsto__exp__limit__at__top,axiom,
    ! [X: real] : filterlim(real,real,aTP_Lamp_vc(real,fun(real,real),X),topolo7230453075368039082e_nhds(real,exp(real,X)),at_top(real)) ).

% tendsto_exp_limit_at_top
tff(fact_6352_filterlim__divide__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),C2: A,F5: filter(A),G: fun(A,A)] :
          ( filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,C2),F5)
         => ( filterlim(A,A,G,topolo174197925503356063within(A,zero_zero(A),top_top(set(A))),F5)
           => ( ( C2 != zero_zero(A) )
             => filterlim(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_og(fun(A,A),fun(fun(A,A),fun(A,A)),F3),G),at_infinity(A),F5) ) ) ) ) ).

% filterlim_divide_at_infinity
tff(fact_6353_filterlim__tan__at__left,axiom,
    filterlim(real,real,tan(real),at_top(real),topolo174197925503356063within(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))),aa(real,set(real),set_ord_lessThan(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))) ).

% filterlim_tan_at_left
tff(fact_6354_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B2: real,F3: fun(real,real),Flim: real] :
      ( ! [X3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B2),X3))
         => ? [Y3: real] :
              ( has_field_derivative(real,F3,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y3),zero_zero(real))) ) )
     => ( filterlim(real,real,F3,topolo7230453075368039082e_nhds(real,Flim),at_top(real))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Flim),aa(real,real,F3,B2))) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_6355_tendsto__arctan__at__top,axiom,
    filterlim(real,real,arctan,topolo7230453075368039082e_nhds(real,divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),at_top(real)) ).

% tendsto_arctan_at_top
tff(fact_6356_filterlim__at__infinity,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C2: real,F3: fun(C,A),F5: filter(C)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),C2))
         => ( filterlim(C,A,F3,at_infinity(A),F5)
          <=> ! [R5: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),R5))
               => eventually(C,aa(real,fun(C,bool),aTP_Lamp_vd(fun(C,A),fun(real,fun(C,bool)),F3),R5),F5) ) ) ) ) ).

% filterlim_at_infinity
tff(fact_6357_filterlim__realpow__sequentially__gt1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [X: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),real_V7770717601297561774m_norm(A,X)))
         => filterlim(nat,A,power_power(A,X),at_infinity(A),at_top(nat)) ) ) ).

% filterlim_realpow_sequentially_gt1
tff(fact_6358_lim__zero__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),L: A] :
          ( filterlim(A,A,aTP_Lamp_ve(fun(A,A),fun(A,A),F3),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,A,F3,topolo7230453075368039082e_nhds(A,L),at_infinity(A)) ) ) ).

% lim_zero_infinity
tff(fact_6359_has__derivativeI__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [E2: real,F10: fun(A,B),S: set(A),X: A,F3: fun(A,B),H6: fun(A,real)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
         => ( real_V3181309239436604168linear(A,B,F10)
           => ( ! [Y4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),S))
                 => ( ( Y4 != X )
                   => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Y4,X)),E2))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),divide_divide(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,F3,Y4)),aa(A,B,F3,X))),aa(A,B,F10,aa(A,A,minus_minus(A,Y4),X)))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Y4),X)))),aa(A,real,H6,Y4))) ) ) )
             => ( filterlim(A,real,H6,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,X,S))
               => has_derivative(A,B,F3,F10,topolo174197925503356063within(A,X,S)) ) ) ) ) ) ).

% has_derivativeI_sandwich
tff(fact_6360_BfunE,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),F5: filter(A)] :
          ( bfun(A,B,F3,F5)
         => ~ ! [B7: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),B7))
               => ~ eventually(A,aa(real,fun(A,bool),aTP_Lamp_vf(fun(A,B),fun(real,fun(A,bool)),F3),B7),F5) ) ) ) ).

% BfunE
tff(fact_6361_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_6362_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_6363_zero__less__dist__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X,Y)))
        <=> ( X != Y ) ) ) ).

% zero_less_dist_iff
tff(fact_6364_dist__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V557655796197034286t_dist(A,X,Y)),zero_zero(real)))
        <=> ( X = Y ) ) ) ).

% dist_le_zero_iff
tff(fact_6365_dist__triangle,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Z: A,Y: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V557655796197034286t_dist(A,X,Z)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,X,Y)),real_V557655796197034286t_dist(A,Y,Z)))) ) ).

% dist_triangle
tff(fact_6366_dist__triangle2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A,Z: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V557655796197034286t_dist(A,X,Y)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,X,Z)),real_V557655796197034286t_dist(A,Y,Z)))) ) ).

% dist_triangle2
tff(fact_6367_dist__triangle3,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A,A2: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V557655796197034286t_dist(A,X,Y)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,A2,X)),real_V557655796197034286t_dist(A,A2,Y)))) ) ).

% dist_triangle3
tff(fact_6368_dist__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Z: A,Y: A,E2: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,X,Z)),real_V557655796197034286t_dist(A,Y,Z))),E2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V557655796197034286t_dist(A,X,Y)),E2)) ) ) ).

% dist_triangle_le
tff(fact_6369_zero__le__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),real_V557655796197034286t_dist(A,X,Y))) ) ).

% zero_le_dist
tff(fact_6370_open__ball,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,D2: real] : topolo1002775350975398744n_open(A,collect(A,aa(real,fun(A,bool),aTP_Lamp_vg(A,fun(real,fun(A,bool)),X),D2))) ) ).

% open_ball
tff(fact_6371_at__top__le__at__infinity,axiom,
    pp(aa(filter(real),bool,aa(filter(real),fun(filter(real),bool),ord_less_eq(filter(real)),at_top(real)),at_infinity(real))) ).

% at_top_le_at_infinity
tff(fact_6372_dist__commute__lessI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,X: A,E2: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Y,X)),E2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X,Y)),E2)) ) ) ).

% dist_commute_lessI
tff(fact_6373_dist__pos__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] :
          ( ( X != Y )
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X,Y))) ) ) ).

% dist_pos_lt
tff(fact_6374_dist__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Y: A] : ~ pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X,Y)),zero_zero(real))) ) ).

% dist_not_less_zero
tff(fact_6375_dist__triangle__less__add,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X15: A,Y: A,E1: real,X22: A,E22: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X15,Y)),E1))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X22,Y)),E22))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X15,X22)),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22))) ) ) ) ).

% dist_triangle_less_add
tff(fact_6376_dist__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A,Z: A,Y: A,E2: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V557655796197034286t_dist(A,X,Z)),real_V557655796197034286t_dist(A,Y,Z))),E2))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X,Y)),E2)) ) ) ).

% dist_triangle_lt
tff(fact_6377_Bseq__add,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C2: A] :
          ( bfun(nat,A,F3,at_top(nat))
         => bfun(nat,A,aa(A,fun(nat,A),aTP_Lamp_vh(fun(nat,A),fun(A,fun(nat,A)),F3),C2),at_top(nat)) ) ) ).

% Bseq_add
tff(fact_6378_Bseq__add__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),C2: A] :
          ( bfun(nat,A,aa(A,fun(nat,A),aTP_Lamp_vh(fun(nat,A),fun(A,fun(nat,A)),F3),C2),at_top(nat))
        <=> bfun(nat,A,F3,at_top(nat)) ) ) ).

% Bseq_add_iff
tff(fact_6379_Bseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A)] :
          ( bfun(nat,A,aTP_Lamp_as(fun(nat,A),fun(nat,A),F3),at_top(nat))
        <=> bfun(nat,A,F3,at_top(nat)) ) ) ).

% Bseq_Suc_iff
tff(fact_6380_Bfun__metric__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [F3: fun(A,B),F5: filter(A)] :
          ( bfun(A,B,F3,F5)
        <=> ? [Y5: B,K6: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K6))
              & eventually(A,aa(real,fun(A,bool),aa(B,fun(real,fun(A,bool)),aTP_Lamp_vi(fun(A,B),fun(B,fun(real,fun(A,bool))),F3),Y5),K6),F5) ) ) ) ).

% Bfun_metric_def
tff(fact_6381_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_vj(fun(nat,A),fun(nat,fun(nat,A)),X6),K),at_top(nat))
         => bfun(nat,A,X6,at_top(nat)) ) ) ).

% Bseq_offset
tff(fact_6382_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_vj(fun(nat,A),fun(nat,fun(nat,A)),X6),K),at_top(nat)) ) ) ).

% Bseq_ignore_initial_segment
tff(fact_6383_open__dist,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S2: set(A)] :
          ( topolo1002775350975398744n_open(A,S2)
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
             => ? [E3: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
                  & ! [Y5: A] :
                      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Y5,X4)),E3))
                     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y5),S2)) ) ) ) ) ) ).

% open_dist
tff(fact_6384_abs__dist__diff__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [A2: A,B2: A,C2: A] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,real_V557655796197034286t_dist(A,A2,B2)),real_V557655796197034286t_dist(A,B2,C2)))),real_V557655796197034286t_dist(A,A2,C2))) ) ).

% abs_dist_diff_le
tff(fact_6385_BseqI_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A),K5: real] :
          ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N2))),K5))
         => bfun(nat,A,X6,at_top(nat)) ) ) ).

% BseqI'
tff(fact_6386_eventually__at,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,bool),A2: A,S2: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,S2))
        <=> ? [D5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D5))
              & ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
                 => ( ( ( X4 != A2 )
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X4,A2)),D5)) )
                   => pp(aa(A,bool,P,X4)) ) ) ) ) ) ).

% eventually_at
tff(fact_6387_eventually__nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,bool),A2: A] :
          ( eventually(A,P,topolo7230453075368039082e_nhds(A,A2))
        <=> ? [D5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D5))
              & ! [X4: A] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X4,A2)),D5))
                 => pp(aa(A,bool,P,X4)) ) ) ) ) ).

% eventually_nhds_metric
tff(fact_6388_has__field__derivative__transform__within,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),F10: A,A2: A,S2: set(A),D2: real,G: fun(A,A)] :
          ( has_field_derivative(A,F3,F10,topolo174197925503356063within(A,A2,S2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),S2))
             => ( ! [X3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                   => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),D2))
                     => ( aa(A,A,F3,X3) = aa(A,A,G,X3) ) ) )
               => has_field_derivative(A,G,F10,topolo174197925503356063within(A,A2,S2)) ) ) ) ) ) ).

% has_field_derivative_transform_within
tff(fact_6389_has__derivative__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F10: fun(A,B),X: A,S: set(A),D2: real,G: fun(A,B)] :
          ( has_derivative(A,B,F3,F10,topolo174197925503356063within(A,X,S))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),S))
             => ( ! [X9: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X9),S))
                   => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X9,X)),D2))
                     => ( aa(A,B,F3,X9) = aa(A,B,G,X9) ) ) )
               => has_derivative(A,B,G,F10,topolo174197925503356063within(A,X,S)) ) ) ) ) ) ).

% has_derivative_transform_within
tff(fact_6390_metric__CauchyI,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
             => ? [M10: nat] :
                ! [M5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M10),M5))
                 => ! [N2: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M10),N2))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M5),aa(nat,A,X6,N2))),E)) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% metric_CauchyI
tff(fact_6391_metric__CauchyD,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),E2: real] :
          ( topolo3814608138187158403Cauchy(A,X6)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
           => ? [M9: nat] :
              ! [M2: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),M2))
               => ! [N7: nat] :
                    ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M9),N7))
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M2),aa(nat,A,X6,N7))),E2)) ) ) ) ) ) ).

% metric_CauchyD
tff(fact_6392_Cauchy__altdef2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,S)
        <=> ! [E3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
             => ? [N6: nat] :
                ! [N5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),N5))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,S,N5),aa(nat,A,S,N6))),E3)) ) ) ) ) ).

% Cauchy_altdef2
tff(fact_6393_Cauchy__def,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [E3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
             => ? [M8: nat] :
                ! [M7: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),M7))
                 => ! [N5: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),N5))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M7),aa(nat,A,X6,N5))),E3)) ) ) ) ) ) ).

% Cauchy_def
tff(fact_6394_metric__tendsto__imp__tendsto,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(A) )
     => ! [F3: fun(C,A),A2: A,F5: filter(C),G: fun(C,B),B2: B] :
          ( filterlim(C,A,F3,topolo7230453075368039082e_nhds(A,A2),F5)
         => ( eventually(C,aa(B,fun(C,bool),aa(fun(C,B),fun(B,fun(C,bool)),aa(A,fun(fun(C,B),fun(B,fun(C,bool))),aTP_Lamp_vk(fun(C,A),fun(A,fun(fun(C,B),fun(B,fun(C,bool)))),F3),A2),G),B2),F5)
           => filterlim(C,B,G,topolo7230453075368039082e_nhds(B,B2),F5) ) ) ) ).

% metric_tendsto_imp_tendsto
tff(fact_6395_metric__LIM__imp__LIM,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & real_V7819770556892013058_space(B)
        & real_V7819770556892013058_space(A) )
     => ! [F3: fun(C,A),L: A,A2: C,G: fun(C,B),M: B] :
          ( filterlim(C,A,F3,topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(C,A2,top_top(set(C))))
         => ( ! [X3: C] :
                ( ( X3 != A2 )
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V557655796197034286t_dist(B,aa(C,B,G,X3),M)),real_V557655796197034286t_dist(A,aa(C,A,F3,X3),L))) )
           => filterlim(C,B,G,topolo7230453075368039082e_nhds(B,M),topolo174197925503356063within(C,A2,top_top(set(C)))) ) ) ) ).

% metric_LIM_imp_LIM
tff(fact_6396_Lim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),L: B,X: A,S2: set(A),D2: real,G: fun(A,B)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,X,S2))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
           => ( ! [X9: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X9),S2))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X9,X)))
                   => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X9,X)),D2))
                     => ( aa(A,B,F3,X9) = aa(A,B,G,X9) ) ) ) )
             => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,X,S2)) ) ) ) ) ).

% Lim_transform_within
tff(fact_6397_dist__triangle__half__r,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Y: A,X15: A,E2: real,X22: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Y,X15)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Y,X22)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X15,X22)),E2)) ) ) ) ).

% dist_triangle_half_r
tff(fact_6398_dist__triangle__half__l,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X15: A,Y: A,E2: real,X22: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X15,Y)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X22,Y)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X15,X22)),E2)) ) ) ) ).

% dist_triangle_half_l
tff(fact_6399_eventually__at__le,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [P: fun(A,bool),A2: A,S2: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A2,S2))
        <=> ? [D5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D5))
              & ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
                 => ( ( ( X4 != A2 )
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V557655796197034286t_dist(A,X4,A2)),D5)) )
                   => pp(aa(A,bool,P,X4)) ) ) ) ) ) ).

% eventually_at_le
tff(fact_6400_dist__triangle__third,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X15: A,X22: A,E2: real,X32: A,X42: A] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X15,X22)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X22,X32)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X32,X42)),divide_divide(real,E2,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X15,X42)),E2)) ) ) ) ) ).

% dist_triangle_third
tff(fact_6401_filterlim__transform__within,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [G: fun(A,B),G4: filter(B),X: A,S2: set(A),F5: filter(B),D2: real,F3: fun(A,B)] :
          ( filterlim(A,B,G,G4,topolo174197925503356063within(A,X,S2))
         => ( pp(aa(filter(B),bool,aa(filter(B),fun(filter(B),bool),ord_less_eq(filter(B)),G4),F5))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
             => ( ! [X9: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X9),S2))
                   => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),real_V557655796197034286t_dist(A,X9,X)))
                     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X9,X)),D2))
                       => ( aa(A,B,F3,X9) = aa(A,B,G,X9) ) ) ) )
               => filterlim(A,B,F3,F5,topolo174197925503356063within(A,X,S2)) ) ) ) ) ) ).

% filterlim_transform_within
tff(fact_6402_Cauchy__altdef,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F3: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,F3)
        <=> ! [E3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
             => ? [M8: nat] :
                ! [M7: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),M7))
                 => ! [N5: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M7),N5))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,F3,M7),aa(nat,A,F3,N5))),E3)) ) ) ) ) ) ).

% Cauchy_altdef
tff(fact_6403_CauchyI_H,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( ! [E: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
             => ? [M10: nat] :
                ! [M5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M10),M5))
                 => ! [N2: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N2))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M5),aa(nat,A,X6,N2))),E)) ) ) )
         => topolo3814608138187158403Cauchy(A,X6) ) ) ).

% CauchyI'
tff(fact_6404_Bseq__eventually__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(nat,A),G: fun(nat,B)] :
          ( eventually(nat,aa(fun(nat,B),fun(nat,bool),aTP_Lamp_vl(fun(nat,A),fun(fun(nat,B),fun(nat,bool)),F3),G),at_top(nat))
         => ( bfun(nat,B,G,at_top(nat))
           => bfun(nat,A,F3,at_top(nat)) ) ) ) ).

% Bseq_eventually_mono
tff(fact_6405_Bseq__eq__bounded,axiom,
    ! [F3: fun(nat,real),A2: real,B2: real] :
      ( pp(aa(set(real),bool,aa(set(real),fun(set(real),bool),ord_less_eq(set(real)),aa(set(nat),set(real),image(nat,real,F3),top_top(set(nat)))),set_or1337092689740270186AtMost(real,A2,B2)))
     => bfun(nat,real,F3,at_top(nat)) ) ).

% Bseq_eq_bounded
tff(fact_6406_tendsto__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F3: fun(B,A),L: A,F5: filter(B)] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F5)
        <=> ! [E3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
             => eventually(B,aa(real,fun(B,bool),aa(A,fun(real,fun(B,bool)),aTP_Lamp_vm(fun(B,A),fun(A,fun(real,fun(B,bool))),F3),L),E3),F5) ) ) ) ).

% tendsto_iff
tff(fact_6407_tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F3: fun(B,A),L: A,F5: filter(B)] :
          ( ! [E: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E))
             => eventually(B,aa(real,fun(B,bool),aa(A,fun(real,fun(B,bool)),aTP_Lamp_vm(fun(B,A),fun(A,fun(real,fun(B,bool))),F3),L),E),F5) )
         => filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F5) ) ) ).

% tendstoI
tff(fact_6408_tendstoD,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [F3: fun(B,A),L: A,F5: filter(B),E2: real] :
          ( filterlim(B,A,F3,topolo7230453075368039082e_nhds(A,L),F5)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E2))
           => eventually(B,aa(real,fun(B,bool),aa(A,fun(real,fun(B,bool)),aTP_Lamp_vm(fun(B,A),fun(A,fun(real,fun(B,bool))),F3),L),E2),F5) ) ) ) ).

% tendstoD
tff(fact_6409_Bseq__def,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [K6: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K6))
              & ! [N5: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N5))),K6)) ) ) ) ).

% Bseq_def
tff(fact_6410_BseqI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [K5: real,X6: fun(nat,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K5))
         => ( ! [N2: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N2))),K5))
           => bfun(nat,A,X6,at_top(nat)) ) ) ) ).

% BseqI
tff(fact_6411_BseqE,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
         => ~ ! [K9: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K9))
               => ~ ! [N7: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N7))),K9)) ) ) ) ).

% BseqE
tff(fact_6412_BseqD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
         => ? [K9: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K9))
              & ! [N7: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N7))),K9)) ) ) ) ).

% BseqD
tff(fact_6413_Bseq__iff1a,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [N6: nat] :
            ! [N5: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N5))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6)))) ) ) ).

% Bseq_iff1a
tff(fact_6414_Bseq__realpow,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),one_one(real)))
       => bfun(nat,real,power_power(real,X),at_top(nat)) ) ) ).

% Bseq_realpow
tff(fact_6415_Bseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [N6: nat] :
            ! [N5: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,X6,N5))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N6)))) ) ) ).

% Bseq_iff
tff(fact_6416_metric__LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [G: fun(A,B),L: B,A2: A,R: real,F3: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R))
           => ( ! [X3: A] :
                  ( ( X3 != A2 )
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),R))
                   => ( aa(A,B,F3,X3) = aa(A,B,G,X3) ) ) )
             => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_equal2
tff(fact_6417_metric__LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [A2: A,F3: fun(A,B),L5: B] :
          ( ! [R2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
             => ? [S7: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S7))
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),S7)) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F3,X3),L5)),R2)) ) ) )
         => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% metric_LIM_I
tff(fact_6418_metric__LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F3: fun(A,B),L5: B,A2: A,R3: real] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
           => ? [S3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S3))
                & ! [X2: A] :
                    ( ( ( X2 != A2 )
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X2,A2)),S3)) )
                   => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F3,X2),L5)),R3)) ) ) ) ) ) ).

% metric_LIM_D
tff(fact_6419_LIM__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [F3: fun(A,B),L5: B,A2: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A2,top_top(set(A))))
        <=> ! [R5: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [S6: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),S6))
                  & ! [X4: A] :
                      ( ( ( X4 != A2 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X4,A2)),S6)) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(B,aa(A,B,F3,X4),L5)),R5)) ) ) ) ) ) ).

% LIM_def
tff(fact_6420_lim__sequentially,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] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [No3: nat] :
                ! [N5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No3),N5))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,N5),L5)),R5)) ) ) ) ) ).

% lim_sequentially
tff(fact_6421_metric__LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L5: A] :
          ( ! [R2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R2))
             => ? [No2: nat] :
                ! [N2: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No2),N2))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,N2),L5)),R2)) ) )
         => filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% metric_LIMSEQ_I
tff(fact_6422_metric__LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A),L5: A,R3: real] :
          ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R3))
           => ? [No: nat] :
              ! [N7: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No),N7))
               => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,N7),L5)),R3)) ) ) ) ) ).

% metric_LIMSEQ_D
tff(fact_6423_metric__Cauchy__iff2,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [J3: nat] :
            ? [M8: nat] :
            ! [M7: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),M7))
             => ! [N5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M8),N5))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,M7),aa(nat,A,X6,N5))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3))))) ) ) ) ) ).

% metric_Cauchy_iff2
tff(fact_6424_metric__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F3: fun(A,B),B2: B,A2: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,F3,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] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D6))
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),D6)) )
                     => ( aa(A,B,F3,X3) != B2 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_vn(fun(A,B),fun(fun(B,C),fun(A,C)),F3),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_LIM_compose2
tff(fact_6425_BfunI,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),K5: real,F5: filter(A)] :
          ( eventually(A,aa(real,fun(A,bool),aTP_Lamp_vf(fun(A,B),fun(real,fun(A,bool)),F3),K5),F5)
         => bfun(A,B,F3,F5) ) ) ).

% BfunI
tff(fact_6426_Bseq__iff3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [K3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K3))
              & ? [N6: nat] :
                ! [N5: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X6,N5)),aa(A,A,uminus_uminus(A),aa(nat,A,X6,N6))))),K3)) ) ) ) ).

% Bseq_iff3
tff(fact_6427_Bseq__iff2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X6: fun(nat,A)] :
          ( bfun(nat,A,X6,at_top(nat))
        <=> ? [K3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K3))
              & ? [X4: A] :
                ! [N5: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,X6,N5)),aa(A,A,uminus_uminus(A),X4)))),K3)) ) ) ) ).

% Bseq_iff2
tff(fact_6428_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] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),R5))
             => ? [No3: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),No3))
                  & ! [N5: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),No3),N5))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(nat,A,X6,N5),L5)),R5)) ) ) ) ) ) ).

% LIMSEQ_iff_nz
tff(fact_6429_Bfun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [F3: fun(A,B),F5: filter(A)] :
          ( bfun(A,B,F3,F5)
        <=> ? [K6: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),K6))
              & eventually(A,aa(real,fun(A,bool),aTP_Lamp_vf(fun(A,B),fun(real,fun(A,bool)),F3),K6),F5) ) ) ) ).

% Bfun_def
tff(fact_6430_totally__bounded__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [S2: set(A)] :
          ( topolo6688025880775521714ounded(A,S2)
        <=> ! [E3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
             => ? [K3: set(A)] :
                  ( pp(aa(set(A),bool,finite_finite(A),K3))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(A),set(set(A)),image(A,set(A),aTP_Lamp_vp(real,fun(A,set(A)),E3)),K3)))) ) ) ) ) ).

% totally_bounded_metric
tff(fact_6431_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F3: fun(real,real),F5: filter(real)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( filterlim(real,real,F3,at_bot(real),F5)
       => ( pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vq(nat,fun(fun(real,real),fun(real,real)),N),F3),at_top(real),F5) ) ) ) ).

% filterlim_pow_at_bot_even
tff(fact_6432_at__bot__le__at__infinity,axiom,
    pp(aa(filter(real),bool,aa(filter(real),fun(filter(real),bool),ord_less_eq(filter(real)),at_bot(real)),at_infinity(real))) ).

% at_bot_le_at_infinity
tff(fact_6433_totally__bounded__subset,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [S2: set(A),T6: set(A)] :
          ( topolo6688025880775521714ounded(A,S2)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T6),S2))
           => topolo6688025880775521714ounded(A,T6) ) ) ) ).

% totally_bounded_subset
tff(fact_6434_eventually__at__bot__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N6: A] :
            ! [N5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),N5),N6))
             => pp(aa(A,bool,P,N5)) ) ) ) ).

% eventually_at_bot_linorder
tff(fact_6435_eventually__at__bot__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_bot(A) )
     => ! [P: fun(A,bool)] :
          ( eventually(A,P,at_bot(A))
        <=> ? [N6: A] :
            ! [N5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N5),N6))
             => pp(aa(A,bool,P,N5)) ) ) ) ).

% eventually_at_bot_dense
tff(fact_6436_eventually__le__at__bot,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,aa(A,fun(A,bool),aTP_Lamp_vr(A,fun(A,bool)),C2),at_bot(A)) ) ).

% eventually_le_at_bot
tff(fact_6437_eventually__gt__at__bot,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [C2: A] : eventually(A,aTP_Lamp_vs(A,fun(A,bool),C2),at_bot(A)) ) ).

% eventually_gt_at_bot
tff(fact_6438_filterlim__at__bot,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F5: filter(A)] :
          ( filterlim(A,B,F3,at_bot(B),F5)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_vt(fun(A,B),fun(B,fun(A,bool)),F3),Z7),F5) ) ) ).

% filterlim_at_bot
tff(fact_6439_filterlim__at__bot__le,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),F5: filter(A),C2: B] :
          ( filterlim(A,B,F3,at_bot(B),F5)
        <=> ! [Z7: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Z7),C2))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_vt(fun(A,B),fun(B,fun(A,bool)),F3),Z7),F5) ) ) ) ).

% filterlim_at_bot_le
tff(fact_6440_filterlim__at__bot__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [F3: fun(A,B),F5: filter(A)] :
          ( filterlim(A,B,F3,at_bot(B),F5)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,bool),aTP_Lamp_vu(fun(A,B),fun(B,fun(A,bool)),F3),Z7),F5) ) ) ).

% filterlim_at_bot_dense
tff(fact_6441_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_6442_filterlim__tendsto__pos__mult__at__bot,axiom,
    ! [A: $tType,F3: fun(A,real),C2: real,F5: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C2),F5)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),C2))
       => ( filterlim(A,real,G,at_bot(real),F5)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_uu(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G),at_bot(real),F5) ) ) ) ).

% filterlim_tendsto_pos_mult_at_bot
tff(fact_6443_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
    ! [A: $tType,F3: fun(A,real),F5: filter(A)] :
      ( filterlim(A,real,F3,at_infinity(real),F5)
     => ( eventually(A,aTP_Lamp_vv(fun(A,real),fun(A,bool),F3),F5)
       => filterlim(A,real,F3,at_bot(real),F5) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_bot
tff(fact_6444_filterlim__at__bot__lt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [F3: fun(A,B),F5: filter(A),C2: B] :
          ( filterlim(A,B,F3,at_bot(B),F5)
        <=> ! [Z7: B] :
              ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Z7),C2))
             => eventually(A,aa(B,fun(A,bool),aTP_Lamp_vw(fun(A,B),fun(B,fun(A,bool)),F3),Z7),F5) ) ) ) ).

% filterlim_at_bot_lt
tff(fact_6445_filterlim__inverse__at__bot,axiom,
    ! [A: $tType,F3: fun(A,real),F5: filter(A)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,zero_zero(real)),F5)
     => ( eventually(A,aTP_Lamp_vv(fun(A,real),fun(A,bool),F3),F5)
       => filterlim(A,real,aTP_Lamp_va(fun(A,real),fun(A,real),F3),at_bot(real),F5) ) ) ).

% filterlim_inverse_at_bot
tff(fact_6446_filterlim__tendsto__neg__mult__at__bot,axiom,
    ! [A: $tType,F3: fun(A,real),C2: real,F5: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,F3,topolo7230453075368039082e_nhds(real,C2),F5)
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C2),zero_zero(real)))
       => ( filterlim(A,real,G,at_top(real),F5)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_uu(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G),at_bot(real),F5) ) ) ) ).

% filterlim_tendsto_neg_mult_at_bot
tff(fact_6447_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B2: real,F3: fun(real,real),Flim: real] :
      ( ! [X3: real] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
         => ? [Y3: real] :
              ( has_field_derivative(real,F3,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y3)) ) )
     => ( filterlim(real,real,F3,topolo7230453075368039082e_nhds(real,Flim),at_bot(real))
       => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Flim),aa(real,real,F3,B2))) ) ) ).

% DERIV_pos_imp_increasing_at_bot
tff(fact_6448_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F3: fun(real,real),F5: filter(real)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( filterlim(real,real,F3,at_bot(real),F5)
       => ( ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),N))
         => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vq(nat,fun(fun(real,real),fun(real,real)),N),F3),at_bot(real),F5) ) ) ) ).

% filterlim_pow_at_bot_odd
tff(fact_6449_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),aa(num,num,bit0,one2))))),at_bot(real)) ).

% tendsto_arctan_at_bot
tff(fact_6450_tendsto__exp__limit__at__right,axiom,
    ! [X: real] : filterlim(real,real,aTP_Lamp_vx(real,fun(real,real),X),topolo7230453075368039082e_nhds(real,exp(real,X)),topolo174197925503356063within(real,zero_zero(real),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ).

% tendsto_exp_limit_at_right
tff(fact_6451_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),aa(num,num,bit0,one2)))),aa(real,set(real),set_ord_greaterThan(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))))) ).

% filterlim_tan_at_right
tff(fact_6452_greaterThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,set(A),set_ord_greaterThan(A),X) = aa(A,set(A),set_ord_greaterThan(A),Y) )
        <=> ( X = Y ) ) ) ).

% greaterThan_eq_iff
tff(fact_6453_greaterThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),aa(A,set(A),set_ord_greaterThan(A),K)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),K),I2)) ) ) ).

% greaterThan_iff
tff(fact_6454_Inf__greaterThan,axiom,
    ! [A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & dense_linorder(A) )
     => ! [X: A] : aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_greaterThan(A),X)) = X ) ).

% Inf_greaterThan
tff(fact_6455_greaterThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_greaterThan(A),X)),aa(A,set(A),set_ord_greaterThan(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% greaterThan_subset_iff
tff(fact_6456_Compl__atMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_atMost(A),K)) = aa(A,set(A),set_ord_greaterThan(A),K) ) ).

% Compl_atMost
tff(fact_6457_Compl__greaterThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_greaterThan(A),K)) = aa(A,set(A),set_ord_atMost(A),K) ) ).

% Compl_greaterThan
tff(fact_6458_Sup__greaterThanAtLeast,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),top_top(A)))
         => ( aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_greaterThan(A),X)) = top_top(A) ) ) ) ).

% Sup_greaterThanAtLeast
tff(fact_6459_image__uminus__greaterThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A] : aa(set(A),set(A),image(A,A,uminus_uminus(A)),aa(A,set(A),set_ord_greaterThan(A),X)) = aa(A,set(A),set_ord_lessThan(A),aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_greaterThan
tff(fact_6460_image__uminus__lessThan,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A] : aa(set(A),set(A),image(A,A,uminus_uminus(A)),aa(A,set(A),set_ord_lessThan(A),X)) = aa(A,set(A),set_ord_greaterThan(A),aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_lessThan
tff(fact_6461_greaterThan__non__empty,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [X: A] : aa(A,set(A),set_ord_greaterThan(A),X) != bot_bot(set(A)) ) ).

% greaterThan_non_empty
tff(fact_6462_lessThan__Int__lessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_greaterThan(A),A2)),aa(A,set(A),set_ord_greaterThan(A),B2)) = aa(A,set(A),set_ord_greaterThan(A),aa(A,A,aa(A,fun(A,A),ord_max(A),A2),B2)) ) ).

% lessThan_Int_lessThan
tff(fact_6463_infinite__Ioi,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [A2: A] : ~ pp(aa(set(A),bool,finite_finite(A),aa(A,set(A),set_ord_greaterThan(A),A2))) ) ).

% infinite_Ioi
tff(fact_6464_greaterThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : aa(A,set(A),set_ord_greaterThan(A),L) = collect(A,aa(A,fun(A,bool),ord_less(A),L)) ) ).

% greaterThan_def
tff(fact_6465_eventually__at__right__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,bool),X: A] :
          ( eventually(A,P,topolo174197925503356063within(A,X,aa(A,set(A),set_ord_greaterThan(A),X)))
        <=> ? [B6: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B6))
              & ! [Y5: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y5))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),B6))
                   => pp(aa(A,bool,P,Y5)) ) ) ) ) ) ).

% eventually_at_right_field
tff(fact_6466_eventually__at__right,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X: A,Y: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( eventually(A,P,topolo174197925503356063within(A,X,aa(A,set(A),set_ord_greaterThan(A),X)))
          <=> ? [B6: A] :
                ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B6))
                & ! [Y5: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y5))
                   => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),B6))
                     => pp(aa(A,bool,P,Y5)) ) ) ) ) ) ) ).

% eventually_at_right
tff(fact_6467_at__within__Icc__at__right,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( topolo174197925503356063within(A,A2,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2)) ) ) ) ).

% at_within_Icc_at_right
tff(fact_6468_ivl__disj__int__one_I7_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(7)
tff(fact_6469_ivl__disj__un__one_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or3652927894154168847AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = aa(A,set(A),set_ord_greaterThan(A),L) ) ) ) ).

% ivl_disj_un_one(5)
tff(fact_6470_ivl__disj__int__one_I5_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(5)
tff(fact_6471_greaterThanLessThan__eq,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] : set_or5935395276787703475ssThan(A,A2,B2) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_greaterThan(A),A2)),aa(A,set(A),set_ord_lessThan(A),B2)) ) ).

% greaterThanLessThan_eq
tff(fact_6472_greaterThanLessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A,U: A] : set_or5935395276787703475ssThan(A,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_greaterThan(A),L)),aa(A,set(A),set_ord_lessThan(A),U)) ) ).

% greaterThanLessThan_def
tff(fact_6473_greaterThanAtMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A,U: A] : set_or3652927894154168847AtMost(A,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_greaterThan(A),L)),aa(A,set(A),set_ord_atMost(A),U)) ) ).

% greaterThanAtMost_def
tff(fact_6474_eventually__at__right__less,axiom,
    ! [A: $tType] :
      ( ( no_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [X: A] : eventually(A,aa(A,fun(A,bool),ord_less(A),X),topolo174197925503356063within(A,X,aa(A,set(A),set_ord_greaterThan(A),X))) ) ).

% eventually_at_right_less
tff(fact_6475_less__separate,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ? [A5: A,B5: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(A,set(A),set_ord_lessThan(A),A5)))
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),aa(A,set(A),set_ord_greaterThan(A),B5)))
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),A5)),aa(A,set(A),set_ord_greaterThan(A),B5)) = bot_bot(set(A)) ) ) ) ) ).

% less_separate
tff(fact_6476_eventually__at__rightI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A,P: fun(A,bool)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),set_or5935395276787703475ssThan(A,A2,B2)))
             => pp(aa(A,bool,P,X3)) )
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => eventually(A,P,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ).

% eventually_at_rightI
tff(fact_6477_eventually__at__right__real,axiom,
    ! [A2: real,B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => eventually(real,aa(real,fun(real,bool),aTP_Lamp_us(real,fun(real,fun(real,bool)),A2),B2),topolo174197925503356063within(real,A2,aa(real,set(real),set_ord_greaterThan(real),A2))) ) ).

% eventually_at_right_real
tff(fact_6478_filterlim__times__pos,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [F3: fun(B,A),P2: A,F12: filter(B),C2: A,L: A] :
          ( filterlim(B,A,F3,topolo174197925503356063within(A,P2,aa(A,set(A),set_ord_greaterThan(A),P2)),F12)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),C2))
           => ( ( L = aa(A,A,aa(A,fun(A,A),times_times(A),C2),P2) )
             => filterlim(B,A,aa(A,fun(B,A),aTP_Lamp_vy(fun(B,A),fun(A,fun(B,A)),F3),C2),topolo174197925503356063within(A,L,aa(A,set(A),set_ord_greaterThan(A),L)),F12) ) ) ) ) ).

% filterlim_times_pos
tff(fact_6479_tendsto__imp__filterlim__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [F3: fun(A,B),L5: B,F5: filter(A)] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,L5),F5)
         => ( eventually(A,aa(B,fun(A,bool),aTP_Lamp_vz(fun(A,B),fun(B,fun(A,bool)),F3),L5),F5)
           => filterlim(A,B,F3,topolo174197925503356063within(B,L5,aa(B,set(B),set_ord_greaterThan(B),L5)),F5) ) ) ) ).

% tendsto_imp_filterlim_at_right
tff(fact_6480_filterlim__at__bot__at__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,bool),F3: fun(A,B),P: fun(B,bool),G: fun(B,A),A2: A] :
          ( ! [X3: A,Y4: A] :
              ( pp(aa(A,bool,Q,X3))
             => ( pp(aa(A,bool,Q,Y4))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y4))
                 => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y4))) ) ) )
         => ( ! [X3: B] :
                ( pp(aa(B,bool,P,X3))
               => ( aa(A,B,F3,aa(B,A,G,X3)) = X3 ) )
           => ( ! [X3: B] :
                  ( pp(aa(B,bool,P,X3))
                 => pp(aa(A,bool,Q,aa(B,A,G,X3))) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2)))
               => ( ! [B5: A] :
                      ( pp(aa(A,bool,Q,B5))
                     => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B5)) )
                 => ( eventually(B,P,at_bot(B))
                   => filterlim(A,B,F3,at_bot(B),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ) ) ) ) ).

% filterlim_at_bot_at_right
tff(fact_6481_tendsto__arcosh__at__left__1,axiom,
    filterlim(real,real,arcosh(real),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,one_one(real),aa(real,set(real),set_ord_greaterThan(real),one_one(real)))) ).

% tendsto_arcosh_at_left_1
tff(fact_6482_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)),aa(real,set(real),set_ord_greaterThan(real),aa(real,real,uminus_uminus(real),one_one(real))))) ).

% artanh_real_at_right_1
tff(fact_6483_interval__cases,axiom,
    ! [A: $tType] :
      ( condit6923001295902523014norder(A)
     => ! [S2: set(A)] :
          ( ! [A5: A,B5: A,X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),S2))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B5),S2))
               => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),X3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B5))
                   => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2)) ) ) ) )
         => ? [A5: A,B5: A] :
              ( ( S2 = bot_bot(set(A)) )
              | ( S2 = top_top(set(A)) )
              | ( S2 = aa(A,set(A),set_ord_lessThan(A),B5) )
              | ( S2 = aa(A,set(A),set_ord_atMost(A),B5) )
              | ( S2 = aa(A,set(A),set_ord_greaterThan(A),A5) )
              | ( S2 = aa(A,set(A),set_ord_atLeast(A),A5) )
              | ( S2 = set_or5935395276787703475ssThan(A,A5,B5) )
              | ( S2 = set_or3652927894154168847AtMost(A,A5,B5) )
              | ( S2 = set_or7035219750837199246ssThan(A,A5,B5) )
              | ( S2 = set_or1337092689740270186AtMost(A,A5,B5) ) ) ) ) ).

% interval_cases
tff(fact_6484_sequentially__imp__eventually__at__right,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: A,B2: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( ! [F4: fun(nat,A)] :
                ( ! [N7: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(nat,A,F4,N7)))
               => ( ! [N7: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F4,N7)),B2))
                 => ( order_antimono(nat,A,F4)
                   => ( filterlim(nat,A,F4,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,bool),aTP_Lamp_wa(fun(A,bool),fun(fun(nat,A),fun(nat,bool)),P),F4),at_top(nat)) ) ) ) )
           => eventually(A,P,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ).

% sequentially_imp_eventually_at_right
tff(fact_6485_atLeast__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X: A,Y: A] :
          ( ( aa(A,set(A),set_ord_atLeast(A),X) = aa(A,set(A),set_ord_atLeast(A),Y) )
        <=> ( X = Y ) ) ) ).

% atLeast_eq_iff
tff(fact_6486_atLeast__0,axiom,
    aa(nat,set(nat),set_ord_atLeast(nat),zero_zero(nat)) = top_top(set(nat)) ).

% atLeast_0
tff(fact_6487_atLeast__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I2: A,K: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I2),aa(A,set(A),set_ord_atLeast(A),K)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),K),I2)) ) ) ).

% atLeast_iff
tff(fact_6488_Inf__atLeast,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] : aa(set(A),A,complete_Inf_Inf(A),aa(A,set(A),set_ord_atLeast(A),X)) = X ) ).

% Inf_atLeast
tff(fact_6489_atLeast__empty__triv,axiom,
    ! [A: $tType] : aa(set(A),set(set(A)),set_ord_atLeast(set(A)),bot_bot(set(A))) = top_top(set(set(A))) ).

% atLeast_empty_triv
tff(fact_6490_atLeast__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [X: A,Y: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),X)),aa(A,set(A),set_ord_atLeast(A),Y)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X)) ) ) ).

% atLeast_subset_iff
tff(fact_6491_image__add__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),K)),aa(A,set(A),set_ord_atLeast(A),I2)) = aa(A,set(A),set_ord_atLeast(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),K),I2)) ) ).

% image_add_atLeast
tff(fact_6492_Sup__atLeast,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [X: A] : aa(set(A),A,complete_Sup_Sup(A),aa(A,set(A),set_ord_atLeast(A),X)) = top_top(A) ) ).

% Sup_atLeast
tff(fact_6493_Compl__lessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_lessThan(A),K)) = aa(A,set(A),set_ord_atLeast(A),K) ) ).

% Compl_lessThan
tff(fact_6494_Compl__atLeast,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [K: A] : aa(set(A),set(A),uminus_uminus(set(A)),aa(A,set(A),set_ord_atLeast(A),K)) = aa(A,set(A),set_ord_lessThan(A),K) ) ).

% Compl_atLeast
tff(fact_6495_Icc__subset__Ici__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,L3: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),aa(A,set(A),set_ord_atLeast(A),L3)))
        <=> ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),H))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L3),L)) ) ) ) ).

% Icc_subset_Ici_iff
tff(fact_6496_image__minus__const__AtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] : aa(set(A),set(A),image(A,A,minus_minus(A,C2)),aa(A,set(A),set_ord_atMost(A),B2)) = aa(A,set(A),set_ord_atLeast(A),aa(A,A,minus_minus(A,C2),B2)) ) ).

% image_minus_const_AtMost
tff(fact_6497_image__minus__const__atLeast,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] : aa(set(A),set(A),image(A,A,minus_minus(A,C2)),aa(A,set(A),set_ord_atLeast(A),A2)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,minus_minus(A,C2),A2)) ) ).

% image_minus_const_atLeast
tff(fact_6498_image__uminus__atMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A] : aa(set(A),set(A),image(A,A,uminus_uminus(A)),aa(A,set(A),set_ord_atMost(A),X)) = aa(A,set(A),set_ord_atLeast(A),aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_atMost
tff(fact_6499_image__uminus__atLeast,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A] : aa(set(A),set(A),image(A,A,uminus_uminus(A)),aa(A,set(A),set_ord_atLeast(A),X)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,uminus_uminus(A),X)) ) ).

% image_uminus_atLeast
tff(fact_6500_Int__atLeastAtMostL2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(A,set(A),set_ord_atLeast(A),C2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C2),B2) ) ).

% Int_atLeastAtMostL2
tff(fact_6501_Int__atLeastAtMostR2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,D2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atLeast(A),A2)),set_or1337092689740270186AtMost(A,C2,D2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C2),D2) ) ).

% Int_atLeastAtMostR2
tff(fact_6502_Ioi__le__Ico,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_greaterThan(A),A2)),aa(A,set(A),set_ord_atLeast(A),A2))) ) ).

% Ioi_le_Ico
tff(fact_6503_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_atLeast(nat),aa(nat,nat,suc,K)) = aa(nat,set(nat),set_ord_greaterThan(nat),K) ).

% atLeast_Suc_greaterThan
tff(fact_6504_not__UNIV__eq__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [L3: A] : top_top(set(A)) != aa(A,set(A),set_ord_atLeast(A),L3) ) ).

% not_UNIV_eq_Ici
tff(fact_6505_not__Iic__eq__Ici,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [H: A,L3: A] : aa(A,set(A),set_ord_atMost(A),H) != aa(A,set(A),set_ord_atLeast(A),L3) ) ).

% not_Iic_eq_Ici
tff(fact_6506_infinite__Ici,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [A2: A] : ~ pp(aa(set(A),bool,finite_finite(A),aa(A,set(A),set_ord_atLeast(A),A2))) ) ).

% infinite_Ici
tff(fact_6507_not__Ici__eq__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L3: A,L: A,H: A] : aa(A,set(A),set_ord_atLeast(A),L3) != set_or1337092689740270186AtMost(A,L,H) ) ).

% not_Ici_eq_Icc
tff(fact_6508_not__empty__eq__Ici__eq__empty,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A] : bot_bot(set(A)) != aa(A,set(A),set_ord_atLeast(A),L) ) ).

% not_empty_eq_Ici_eq_empty
tff(fact_6509_atLeast__eq__UNIV__iff,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [X: A] :
          ( ( aa(A,set(A),set_ord_atLeast(A),X) = top_top(set(A)) )
        <=> ( X = bot_bot(A) ) ) ) ).

% atLeast_eq_UNIV_iff
tff(fact_6510_not__Ici__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,L3: A,H2: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),L)),set_or1337092689740270186AtMost(A,L3,H2))) ) ).

% not_Ici_le_Icc
tff(fact_6511_antimono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( order_antimono(A,B,F3)
        <=> ! [X4: A,Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y5))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Y5)),aa(A,B,F3,X4))) ) ) ) ).

% antimono_def
tff(fact_6512_antimonoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( ! [X3: A,Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y4))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Y4)),aa(A,B,F3,X3))) )
         => order_antimono(A,B,F3) ) ) ).

% antimonoI
tff(fact_6513_antimonoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Y)),aa(A,B,F3,X))) ) ) ) ).

% antimonoE
tff(fact_6514_antimonoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_antimono(A,B,F3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Y)),aa(A,B,F3,X))) ) ) ) ).

% antimonoD
tff(fact_6515_atLeast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A] : aa(A,set(A),set_ord_atLeast(A),L) = collect(A,aa(A,fun(A,bool),ord_less_eq(A),L)) ) ).

% atLeast_def
tff(fact_6516_not__UNIV__le__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [L: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),top_top(set(A))),aa(A,set(A),set_ord_atLeast(A),L))) ) ).

% not_UNIV_le_Ici
tff(fact_6517_not__Ici__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H2: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),L)),aa(A,set(A),set_ord_atMost(A),H2))) ) ).

% not_Ici_le_Iic
tff(fact_6518_not__Iic__le__Ici,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L3: A] : ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atMost(A),H)),aa(A,set(A),set_ord_atLeast(A),L3))) ) ).

% not_Iic_le_Ici
tff(fact_6519_decseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( order_antimono(nat,A,F3)
        <=> ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,aa(nat,nat,suc,N5))),aa(nat,A,F3,N5))) ) ) ).

% decseq_Suc_iff
tff(fact_6520_decseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,aa(nat,nat,suc,N2))),aa(nat,A,X6,N2)))
         => order_antimono(nat,A,X6) ) ) ).

% decseq_SucI
tff(fact_6521_decseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: fun(nat,A),I2: nat] :
          ( order_antimono(nat,A,A3)
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A3,aa(nat,nat,suc,I2))),aa(nat,A,A3,I2))) ) ) ).

% decseq_SucD
tff(fact_6522_decseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( order_antimono(nat,A,X6)
        <=> ! [M7: nat,N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),N5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N5)),aa(nat,A,X6,M7))) ) ) ) ).

% decseq_def
tff(fact_6523_decseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),I2: nat,J: nat] :
          ( order_antimono(nat,A,F3)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,J)),aa(nat,A,F3,I2))) ) ) ) ).

% decseqD
tff(fact_6524_ivl__disj__un__one_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = aa(A,set(A),set_ord_atLeast(A),L) ) ) ) ).

% ivl_disj_un_one(8)
tff(fact_6525_Ici__subset__Ioi__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(A,set(A),set_ord_atLeast(A),A2)),aa(A,set(A),set_ord_greaterThan(A),B2)))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2)) ) ) ).

% Ici_subset_Ioi_iff
tff(fact_6526_ivl__disj__int__one_I8_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(8)
tff(fact_6527_atLeastLessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A,U: A] : set_or7035219750837199246ssThan(A,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atLeast(A),L)),aa(A,set(A),set_ord_lessThan(A),U)) ) ).

% atLeastLessThan_def
tff(fact_6528_atLeastAtMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [L: A,U: A] : set_or1337092689740270186AtMost(A,L,U) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atLeast(A),L)),aa(A,set(A),set_ord_atMost(A),U)) ) ).

% atLeastAtMost_def
tff(fact_6529_ivl__disj__int__one_I6_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,U: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,L,U)),aa(A,set(A),set_ord_atLeast(A),U)) = bot_bot(set(A)) ) ).

% ivl_disj_int_one(6)
tff(fact_6530_decseq__bounded,axiom,
    ! [X6: fun(nat,real),B3: real] :
      ( order_antimono(nat,real,X6)
     => ( ! [I4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B3),aa(nat,real,X6,I4)))
       => bfun(nat,real,X6,at_top(nat)) ) ) ).

% decseq_bounded
tff(fact_6531_INT__greaterThan__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Inf_Inf(set(nat)),aa(set(nat),set(set(nat)),image(nat,set(nat),set_ord_greaterThan(nat)),top_top(set(nat)))) = bot_bot(set(nat)) ).

% INT_greaterThan_UNIV
tff(fact_6532_greaterThan__0,axiom,
    aa(nat,set(nat),set_ord_greaterThan(nat),zero_zero(nat)) = aa(set(nat),set(nat),image(nat,nat,suc),top_top(set(nat))) ).

% greaterThan_0
tff(fact_6533_atMost__Int__atLeast,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [N: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),N)),aa(A,set(A),set_ord_atLeast(A),N)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),N),bot_bot(set(A))) ) ).

% atMost_Int_atLeast
tff(fact_6534_ivl__disj__un__one_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L),U))
         => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),set_or1337092689740270186AtMost(A,L,U)),aa(A,set(A),set_ord_greaterThan(A),U)) = aa(A,set(A),set_ord_atLeast(A),L) ) ) ) ).

% ivl_disj_un_one(7)
tff(fact_6535_ivl__disj__un__singleton_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),L),bot_bot(set(A)))),aa(A,set(A),set_ord_greaterThan(A),L)) = aa(A,set(A),set_ord_atLeast(A),L) ) ).

% ivl_disj_un_singleton(1)
tff(fact_6536_ivl__disj__un__one_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [L: A,U: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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(A,set(A),set_ord_atLeast(A),U)) = aa(A,set(A),set_ord_greaterThan(A),L) ) ) ) ).

% ivl_disj_un_one(6)
tff(fact_6537_decseq__ge,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),L5: A,N: nat] :
          ( order_antimono(nat,A,X6)
         => ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),L5),aa(nat,A,X6,N))) ) ) ) ).

% decseq_ge
tff(fact_6538_greaterThan__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_greaterThan(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),minus_minus(set(nat),aa(nat,set(nat),set_ord_greaterThan(nat),K)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),aa(nat,nat,suc,K)),bot_bot(set(nat)))) ).

% greaterThan_Suc
tff(fact_6539_decseq__convergent,axiom,
    ! [X6: fun(nat,real),B3: real] :
      ( order_antimono(nat,real,X6)
     => ( ! [I4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),B3),aa(nat,real,X6,I4)))
       => ~ ! [L6: real] :
              ( filterlim(nat,real,X6,topolo7230453075368039082e_nhds(real,L6),at_top(nat))
             => ~ ! [I: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),L6),aa(nat,real,X6,I))) ) ) ) ).

% decseq_convergent
tff(fact_6540_UN__atLeast__UNIV,axiom,
    aa(set(set(nat)),set(nat),complete_Sup_Sup(set(nat)),aa(set(nat),set(set(nat)),image(nat,set(nat),set_ord_atLeast(nat)),top_top(set(nat)))) = top_top(set(nat)) ).

% UN_atLeast_UNIV
tff(fact_6541_atLeast__Suc,axiom,
    ! [K: nat] : aa(nat,set(nat),set_ord_atLeast(nat),aa(nat,nat,suc,K)) = aa(set(nat),set(nat),minus_minus(set(nat),aa(nat,set(nat),set_ord_atLeast(nat),K)),aa(set(nat),set(nat),aa(nat,fun(set(nat),set(nat)),insert(nat),K),bot_bot(set(nat)))) ).

% atLeast_Suc
tff(fact_6542_tendsto__at__right__sequentially,axiom,
    ! [C: $tType,B: $tType] :
      ( ( topolo3112930676232923870pology(B)
        & topolo1944317154257567458pology(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: B,B2: B,X6: fun(B,C),L5: C] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),A2),B2))
         => ( ! [S5: fun(nat,B)] :
                ( ! [N7: nat] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),A2),aa(nat,B,S5,N7)))
               => ( ! [N7: nat] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(nat,B,S5,N7)),B2))
                 => ( order_antimono(nat,B,S5)
                   => ( filterlim(nat,B,S5,topolo7230453075368039082e_nhds(B,A2),at_top(nat))
                     => filterlim(nat,C,aa(fun(nat,B),fun(nat,C),aTP_Lamp_wb(fun(B,C),fun(fun(nat,B),fun(nat,C)),X6),S5),topolo7230453075368039082e_nhds(C,L5),at_top(nat)) ) ) ) )
           => filterlim(B,C,X6,topolo7230453075368039082e_nhds(C,L5),topolo174197925503356063within(B,A2,aa(B,set(B),set_ord_greaterThan(B),A2))) ) ) ) ).

% tendsto_at_right_sequentially
tff(fact_6543_cauchy__filter__metric,axiom,
    ! [A: $tType] :
      ( ( real_V768167426530841204y_dist(A)
        & topolo7287701948861334536_space(A) )
     => ! [F5: filter(A)] :
          ( topolo6773858410816713723filter(A,F5)
        <=> ! [E3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
             => ? [P6: fun(A,bool)] :
                  ( eventually(A,P6,F5)
                  & ! [X4: A,Y5: A] :
                      ( ( pp(aa(A,bool,P6,X4))
                        & pp(aa(A,bool,P6,Y5)) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X4,Y5)),E3)) ) ) ) ) ) ).

% cauchy_filter_metric
tff(fact_6544_extract__Some__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Ys: list(A),Y: A,Zs: 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)),Ys),aa(list(A),product_prod(A,list(A)),product_Pair(A,list(A),Y),Zs))) )
    <=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) )
        & pp(aa(A,bool,P,Y))
        & ~ ? [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Ys)))
              & pp(aa(A,bool,P,X4)) ) ) ) ).

% extract_Some_iff
tff(fact_6545_extract__None__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( extract(A,P,Xs) = none(product_prod(list(A),product_prod(A,list(A)))) )
    <=> ~ ? [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
            & pp(aa(A,bool,P,X4)) ) ) ).

% extract_None_iff
tff(fact_6546_extract__Nil__code,axiom,
    ! [A: $tType,P: fun(A,bool)] : extract(A,P,nil(A)) = none(product_prod(list(A),product_prod(A,list(A)))) ).

% extract_Nil_code
tff(fact_6547_nhds__imp__cauchy__filter,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [F5: filter(A),X: A] :
          ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F5),topolo7230453075368039082e_nhds(A,X)))
         => topolo6773858410816713723filter(A,F5) ) ) ).

% nhds_imp_cauchy_filter
tff(fact_6548_extract__SomeE,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Ys: list(A),Y: A,Zs: 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)),Ys),aa(list(A),product_prod(A,list(A)),product_Pair(A,list(A),Y),Zs))) )
     => ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) )
        & pp(aa(A,bool,P,Y))
        & ~ ? [X2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Ys)))
              & pp(aa(A,bool,P,X2)) ) ) ) ).

% extract_SomeE
tff(fact_6549_lexord__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : lexord(A,R3) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aTP_Lamp_wc(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R3))) ).

% lexord_def
tff(fact_6550_relpow__finite__bounded1,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),K: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite(product_prod(A,A)),R))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),K))
       => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),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))),aa(set(nat),set(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_wd(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),collect(nat,aTP_Lamp_we(set(product_prod(A,A)),fun(nat,bool),R)))))) ) ) ).

% relpow_finite_bounded1
tff(fact_6551_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_6552_lexord__cons__cons,axiom,
    ! [A: $tType,A2: A,X: list(A),B2: A,Y: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),X)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y))),lexord(A,R3)))
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,A2),B2)),R3))
        | ( ( A2 = B2 )
          & pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),X),Y)),lexord(A,R3))) ) ) ) ).

% lexord_cons_cons
tff(fact_6553_lexord__Nil__left,axiom,
    ! [A: $tType,Y: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),nil(A)),Y)),lexord(A,R3)))
    <=> ? [A6: A,X4: list(A)] : Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),X4) ) ).

% lexord_Nil_left
tff(fact_6554_finite__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),N: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite(product_prod(A,A)),R))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
       => pp(aa(set(product_prod(A,A)),bool,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))),N),R))) ) ) ).

% finite_relpow
tff(fact_6555_lexord__irreflexive,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X3: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X3),X3)),R3))
     => ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Xs)),lexord(A,R3))) ) ).

% lexord_irreflexive
tff(fact_6556_lexord__linear,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X: list(A),Y: list(A)] :
      ( ! [A5: A,B5: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,A5),B5)),R3))
          | ( A5 = B5 )
          | pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,B5),A5)),R3)) )
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),X),Y)),lexord(A,R3)))
        | ( X = Y )
        | pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Y),X)),lexord(A,R3))) ) ) ).

% lexord_linear
tff(fact_6557_relpow__Suc__I2,axiom,
    ! [A: $tType,X: A,Y: A,R: set(product_prod(A,A)),Z: A,N: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y)),R))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),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))),N),R)))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),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,N)),R))) ) ) ).

% relpow_Suc_I2
tff(fact_6558_relpow__Suc__E2,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),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,N)),R)))
     => ~ ! [Y4: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y4)),R))
           => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Y4),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))),N),R))) ) ) ).

% relpow_Suc_E2
tff(fact_6559_relpow__Suc__D2,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),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,N)),R)))
     => ? [Y4: A] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y4)),R))
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Y4),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))),N),R))) ) ) ).

% relpow_Suc_D2
tff(fact_6560_relpow__Suc__I,axiom,
    ! [A: $tType,X: A,Y: A,N: nat,R: set(product_prod(A,A)),Z: A] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),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))),N),R)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Y),Z)),R))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),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,N)),R))) ) ) ).

% relpow_Suc_I
tff(fact_6561_relpow__Suc__E,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),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,N)),R)))
     => ~ ! [Y4: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y4)),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))),N),R)))
           => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Y4),Z)),R)) ) ) ).

% relpow_Suc_E
tff(fact_6562_lexord__Nil__right,axiom,
    ! [A: $tType,X: list(A),R3: set(product_prod(A,A))] : ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),X),nil(A))),lexord(A,R3))) ).

% lexord_Nil_right
tff(fact_6563_lexord__append__leftI,axiom,
    ! [A: $tType,U: list(A),V: list(A),R3: set(product_prod(A,A)),X: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),U),V)),lexord(A,R3)))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),V))),lexord(A,R3))) ) ).

% lexord_append_leftI
tff(fact_6564_relpow__E,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),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))),N),R)))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y4: A,M5: nat] :
              ( ( N = aa(nat,nat,suc,M5) )
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y4)),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))),M5),R)))
               => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Y4),Z)),R)) ) ) ) ) ).

% relpow_E
tff(fact_6565_relpow__E2,axiom,
    ! [A: $tType,X: A,Z: A,N: nat,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),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))),N),R)))
     => ( ( ( N = zero_zero(nat) )
         => ( X != Z ) )
       => ~ ! [Y4: A,M5: nat] :
              ( ( N = aa(nat,nat,suc,M5) )
             => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y4)),R))
               => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Y4),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))),M5),R))) ) ) ) ) ).

% relpow_E2
tff(fact_6566_relpow__empty,axiom,
    ! [A: $tType,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( 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))),N),bot_bot(set(product_prod(A,A)))) = bot_bot(set(product_prod(A,A))) ) ) ).

% relpow_empty
tff(fact_6567_lexord__partial__trans,axiom,
    ! [A: $tType,Xs: list(A),R3: set(product_prod(A,A)),Ys: list(A),Zs: list(A)] :
      ( ! [X3: A,Y4: A,Z2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X3),Y4)),R3))
           => ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Y4),Z2)),R3))
             => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X3),Z2)),R3)) ) ) )
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),lexord(A,R3)))
       => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Ys),Zs)),lexord(A,R3)))
         => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Zs)),lexord(A,R3))) ) ) ) ).

% lexord_partial_trans
tff(fact_6568_lexord__append__leftD,axiom,
    ! [A: $tType,X: list(A),U: list(A),V: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),U)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),V))),lexord(A,R3)))
     => ( ! [A5: A] : ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,A5),A5)),R3))
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),U),V)),lexord(A,R3))) ) ) ).

% lexord_append_leftD
tff(fact_6569_lexord__append__rightI,axiom,
    ! [A: $tType,Y: list(A),X: list(A),R3: set(product_prod(A,A))] :
      ( ? [B10: A,Z3: list(A)] : Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B10),Z3)
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),X),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),X),Y))),lexord(A,R3))) ) ).

% lexord_append_rightI
tff(fact_6570_lexord__sufE,axiom,
    ! [A: $tType,Xs: list(A),Zs: list(A),Ys: list(A),Qs: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Qs))),lexord(A,R3)))
     => ( ( Xs != Ys )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
         => ( ( aa(list(A),nat,size_size(list(A)),Zs) = aa(list(A),nat,size_size(list(A)),Qs) )
           => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),lexord(A,R3))) ) ) ) ) ).

% lexord_sufE
tff(fact_6571_lexord__lex,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),X),Y)),lex(A,R3)))
    <=> ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),X),Y)),lexord(A,R3)))
        & ( aa(list(A),nat,size_size(list(A)),X) = aa(list(A),nat,size_size(list(A)),Y) ) ) ) ).

% lexord_lex
tff(fact_6572_relpow__fun__conv,axiom,
    ! [A: $tType,A2: A,B2: A,N: nat,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),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))),N),R)))
    <=> ? [F6: fun(nat,A)] :
          ( ( aa(nat,A,F6,zero_zero(nat)) = A2 )
          & ( aa(nat,A,F6,N) = B2 )
          & ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),N))
             => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,F6,I3)),aa(nat,A,F6,aa(nat,nat,suc,I3)))),R)) ) ) ) ).

% relpow_fun_conv
tff(fact_6573_lexord__append__left__rightI,axiom,
    ! [A: $tType,A2: A,B2: A,R3: set(product_prod(A,A)),U: list(A),X: list(A),Y: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,A2),B2)),R3))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),X))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B2),Y)))),lexord(A,R3))) ) ).

% lexord_append_left_rightI
tff(fact_6574_lexord__same__pref__iff,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lexord(A,R3)))
    <=> ( ? [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X4),X4)),R3)) )
        | pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Ys),Zs)),lexord(A,R3))) ) ) ).

% lexord_same_pref_iff
tff(fact_6575_lexord__sufI,axiom,
    ! [A: $tType,U: list(A),W: list(A),R3: set(product_prod(A,A)),V: list(A),Z: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),U),W)),lexord(A,R3)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),W)),aa(list(A),nat,size_size(list(A)),U)))
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U),V)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),W),Z))),lexord(A,R3))) ) ) ).

% lexord_sufI
tff(fact_6576_relpow__finite__bounded,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),K: nat] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite(product_prod(A,A)),R))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),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))),aa(set(nat),set(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_wd(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),collect(nat,aTP_Lamp_wf(set(product_prod(A,A)),fun(nat,bool),R)))))) ) ).

% relpow_finite_bounded
tff(fact_6577_List_Olexordp__def,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
      ( lexordp(A,R3,Xs,Ys)
    <=> pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),lexord(A,collect(product_prod(A,A),product_case_prod(A,A,bool,R3))))) ) ).

% List.lexordp_def
tff(fact_6578_ntrancl__def,axiom,
    ! [A: $tType,N: nat,R: set(product_prod(A,A))] : transitive_ntrancl(A,N,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_wd(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),collect(nat,aTP_Lamp_wg(nat,fun(nat,bool),N)))) ).

% ntrancl_def
tff(fact_6579_trancl__finite__eq__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,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))),aa(set(nat),set(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_wd(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),collect(nat,aTP_Lamp_we(set(product_prod(A,A)),fun(nat,bool),R)))) ) ) ).

% trancl_finite_eq_relpow
tff(fact_6580_isCont__powser_H,axiom,
    ! [A: $tType,Aa: $tType] :
      ( ( real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa)
        & topological_t2_space(A) )
     => ! [A2: A,F3: fun(A,Aa),C2: fun(nat,Aa),K5: Aa] :
          ( topolo3448309680560233919inuous(A,Aa,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( summable(Aa,aa(Aa,fun(nat,Aa),aTP_Lamp_wh(fun(nat,Aa),fun(Aa,fun(nat,Aa)),C2),K5))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(Aa,aa(A,Aa,F3,A2))),real_V7770717601297561774m_norm(Aa,K5)))
             => topolo3448309680560233919inuous(A,Aa,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(nat,Aa),fun(A,Aa),aTP_Lamp_wj(fun(A,Aa),fun(fun(nat,Aa),fun(A,Aa)),F3),C2)) ) ) ) ) ).

% isCont_powser'
tff(fact_6581_continuous__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F5: filter(A),F3: fun(A,B),N: nat] :
          ( topolo3448309680560233919inuous(A,B,F5,F3)
         => topolo3448309680560233919inuous(A,B,F5,aa(nat,fun(A,B),aTP_Lamp_wk(fun(A,B),fun(nat,fun(A,B)),F3),N)) ) ) ).

% continuous_power
tff(fact_6582_continuous__power_H,axiom,
    ! [B: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & topolo1898628316856586783d_mult(B) )
     => ! [F5: filter(C),F3: fun(C,B),G: fun(C,nat)] :
          ( topolo3448309680560233919inuous(C,B,F5,F3)
         => ( topolo3448309680560233919inuous(C,nat,F5,G)
           => topolo3448309680560233919inuous(C,B,F5,aa(fun(C,nat),fun(C,B),aTP_Lamp_wl(fun(C,B),fun(fun(C,nat),fun(C,B)),F3),G)) ) ) ) ).

% continuous_power'
tff(fact_6583_continuous__add,axiom,
    ! [B: $tType,D: $tType] :
      ( ( topological_t2_space(D)
        & topolo6943815403480290642id_add(B) )
     => ! [F5: filter(D),F3: fun(D,B),G: fun(D,B)] :
          ( topolo3448309680560233919inuous(D,B,F5,F3)
         => ( topolo3448309680560233919inuous(D,B,F5,G)
           => topolo3448309680560233919inuous(D,B,F5,aa(fun(D,B),fun(D,B),aTP_Lamp_wm(fun(D,B),fun(fun(D,B),fun(D,B)),F3),G)) ) ) ) ).

% continuous_add
tff(fact_6584_trancl__mono,axiom,
    ! [A: $tType,P2: product_prod(A,A),R3: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),P2),transitive_trancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S))
       => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),P2),transitive_trancl(A,S))) ) ) ).

% trancl_mono
tff(fact_6585_IVT2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & topolo8458572112393995274pology(A) )
     => ! [F3: fun(A,B),B2: A,Y: B,A2: A] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,B2)),Y))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y),aa(A,B,F3,A2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
             => ( ! [X3: A] :
                    ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2)) )
                   => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X3,top_top(set(A))),F3) )
               => ? [X3: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2))
                    & ( aa(A,B,F3,X3) = Y ) ) ) ) ) ) ) ).

% IVT2
tff(fact_6586_IVT,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & topolo8458572112393995274pology(A) )
     => ! [F3: fun(A,B),A2: A,Y: B,B2: A] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,A2)),Y))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y),aa(A,B,F3,B2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
             => ( ! [X3: A] :
                    ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2)) )
                   => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X3,top_top(set(A))),F3) )
               => ? [X3: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2))
                    & ( aa(A,B,F3,X3) = Y ) ) ) ) ) ) ) ).

% IVT
tff(fact_6587_isCont__Lb__Ub,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
     => ( ! [X3: real] :
            ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X3,top_top(set(real))),F3) )
       => ? [L6: real,M9: real] :
            ( ! [X2: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X2))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),B2)) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),L6),aa(real,real,F3,X2)))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,F3,X2)),M9)) ) )
            & ! [Y3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),L6),Y3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Y3),M9)) )
               => ? [X3: real] :
                    ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
                    & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
                    & ( aa(real,real,F3,X3) = Y3 ) ) ) ) ) ) ).

% isCont_Lb_Ub
tff(fact_6588_trancl__power,axiom,
    ! [A: $tType,P2: product_prod(A,A),R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),P2),transitive_trancl(A,R)))
    <=> ? [N5: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N5))
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),P2),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))),N5),R))) ) ) ).

% trancl_power
tff(fact_6589_continuous__at__within__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,S: set(A),F3: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,S),F3)
         => ( 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_wn(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G)) ) ) ) ) ).

% continuous_at_within_divide
tff(fact_6590_isCont__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [A2: A,F3: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( 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_wo(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G)) ) ) ) ).

% isCont_add
tff(fact_6591_isCont__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [A2: A,F3: fun(A,B),N: nat] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(nat,fun(A,B),aTP_Lamp_wk(fun(A,B),fun(nat,fun(A,B)),F3),N)) ) ) ).

% isCont_power
tff(fact_6592_less__eq,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(set(product_prod(nat,nat)),bool,aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),bool),member(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,M),N)),transitive_trancl(nat,pred_nat)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),N)) ) ).

% less_eq
tff(fact_6593_isCont__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F3: fun(real,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => ( ! [X3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F3) )
           => ? [M9: A] :
              ! [X2: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X2))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),B2)) )
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(real,A,F3,X2)),M9)) ) ) ) ) ).

% isCont_bounded
tff(fact_6594_isCont__eq__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F3: fun(real,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => ( ! [X3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F3) )
           => ? [M9: A] :
                ( ! [X2: real] :
                    ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X2))
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),B2)) )
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(real,A,F3,X2)),M9)) )
                & ? [X3: real] :
                    ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
                    & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
                    & ( aa(real,A,F3,X3) = M9 ) ) ) ) ) ) ).

% isCont_eq_Ub
tff(fact_6595_isCont__eq__Lb,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F3: fun(real,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => ( ! [X3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F3) )
           => ? [M9: A] :
                ( ! [X2: real] :
                    ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X2))
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),B2)) )
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M9),aa(real,A,F3,X2))) )
                & ? [X3: real] :
                    ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
                    & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
                    & ( aa(real,A,F3,X3) = M9 ) ) ) ) ) ) ).

% isCont_eq_Lb
tff(fact_6596_isCont__inverse__function2,axiom,
    ! [A2: real,X: real,B2: real,G: fun(real,real),F3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),B2))
       => ( ! [Z2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),Z2))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Z2),B2))
               => ( aa(real,real,G,aa(real,real,F3,Z2)) = Z2 ) ) )
         => ( ! [Z2: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),Z2))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Z2),B2))
                 => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F3) ) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F3,X),top_top(set(real))),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_6597_isCont__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A2: A,F3: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( 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_wn(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G)) ) ) ) ) ).

% isCont_divide
tff(fact_6598_continuous__arcosh__strong,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F5: filter(A),F3: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F5,F3)
         => ( eventually(A,aTP_Lamp_wp(fun(A,real),fun(A,bool),F3),F5)
           => topolo3448309680560233919inuous(A,real,F5,aTP_Lamp_wq(fun(A,real),fun(A,real),F3)) ) ) ) ).

% continuous_arcosh_strong
tff(fact_6599_isCont__has__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A2: real,B2: real,F3: fun(real,A)] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
         => ( ! [X3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X3,top_top(set(real))),F3) )
           => ? [M9: A] :
                ( ! [X2: real] :
                    ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X2))
                      & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X2),B2)) )
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(real,A,F3,X2)),M9)) )
                & ! [N8: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N8),M9))
                   => ? [X3: real] :
                        ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2))
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),N8),aa(real,A,F3,X3))) ) ) ) ) ) ) ).

% isCont_has_Ub
tff(fact_6600_isCont__arcosh,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),X))
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),arcosh(real)) ) ).

% isCont_arcosh
tff(fact_6601_continuous__at__within__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,S: set(A),F3: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S),F3)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,S),G)
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,F3,A2)))
             => ( ( aa(A,real,F3,A2) != one_one(real) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),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_wr(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G)) ) ) ) ) ) ) ).

% continuous_at_within_log
tff(fact_6602_isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [X: A,F3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,X,top_top(set(A))),F3)
        <=> filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ws(A,fun(fun(A,B),fun(A,B)),X),F3),topolo7230453075368039082e_nhds(B,aa(A,B,F3,X)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% isCont_iff
tff(fact_6603_DERIV__inverse__function,axiom,
    ! [F3: fun(real,real),D4: real,G: fun(real,real),X: real,A2: real,B2: real] :
      ( has_field_derivative(real,F3,D4,topolo174197925503356063within(real,aa(real,real,G,X),top_top(set(real))))
     => ( ( D4 != zero_zero(real) )
       => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),B2))
           => ( ! [Y4: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Y4))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y4),B2))
                   => ( aa(real,real,F3,aa(real,real,G,Y4)) = Y4 ) ) )
             => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),G)
               => has_field_derivative(real,G,aa(real,real,inverse_inverse(real),D4),topolo174197925503356063within(real,X,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_6604_isCont__polynom,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: A,C2: fun(nat,A),N: nat] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A2,top_top(set(A))),aa(nat,fun(A,A),aTP_Lamp_wt(fun(nat,A),fun(nat,fun(A,A)),C2),N)) ) ).

% isCont_polynom
tff(fact_6605_finite__trancl__ntranl,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite(product_prod(A,A)),R))
     => ( transitive_trancl(A,R) = transitive_ntrancl(A,aa(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_6606_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,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_6607_isCont__arccos,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),arccos) ) ) ).

% isCont_arccos
tff(fact_6608_isCont__arcsin,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),arcsin) ) ) ).

% isCont_arcsin
tff(fact_6609_isCont__powser__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),X: A] :
          ( ! [Y4: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(A,fun(nat,A)),C2),Y4))
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_om(fun(nat,A),fun(A,A),C2)) ) ) ).

% isCont_powser_converges_everywhere
tff(fact_6610_LIM__less__bound,axiom,
    ! [B2: real,X: real,F3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),B2),X))
     => ( ! [X3: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),set_or5935395276787703475ssThan(real,B2,X)))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,F3,X3))) )
       => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),F3)
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(real,real,F3,X))) ) ) ) ).

% LIM_less_bound
tff(fact_6611_isCont__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A2: A,F3: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A2,top_top(set(A))),G)
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,F3,A2)))
             => ( ( aa(A,real,F3,A2) != one_one(real) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),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_wr(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G)) ) ) ) ) ) ) ).

% isCont_log
tff(fact_6612_isCont__artanh,axiom,
    ! [X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),one_one(real))),X))
     => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),one_one(real)))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X,top_top(set(real))),artanh(real)) ) ) ).

% isCont_artanh
tff(fact_6613_isCont__inverse__function,axiom,
    ! [D2: real,X: real,G: fun(real,real),F3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D2))
     => ( ! [Z2: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Z2),X))),D2))
           => ( aa(real,real,G,aa(real,real,F3,Z2)) = Z2 ) )
       => ( ! [Z2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Z2),X))),D2))
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F3) )
         => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,F3,X),top_top(set(real))),G) ) ) ) ).

% isCont_inverse_function
tff(fact_6614_GMVT_H,axiom,
    ! [A2: real,B2: real,F3: fun(real,real),G: fun(real,real),G3: fun(real,real),F10: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [Z2: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),Z2))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Z2),B2))
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),F3) ) )
       => ( ! [Z2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),Z2))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Z2),B2))
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),G) ) )
         => ( ! [Z2: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Z2))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z2),B2))
                 => has_field_derivative(real,G,aa(real,real,G3,Z2),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) )
           => ( ! [Z2: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Z2))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z2),B2))
                   => has_field_derivative(real,F3,aa(real,real,F10,Z2),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) )
             => ? [C4: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),C4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C4),B2))
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,F3,B2)),aa(real,real,F3,A2))),aa(real,real,G3,C4)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,G,B2)),aa(real,real,G,A2))),aa(real,real,F10,C4)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_6615_isCont__If__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,G: fun(A,B),F3: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_lessThan(A),A2)),G)
         => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,G,A2)),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2)))
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_wu(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),G),F3)) ) ) ) ).

% isCont_If_ge
tff(fact_6616_metric__isCont__LIM__compose2,axiom,
    ! [D: $tType,C: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(D) )
     => ! [A2: A,F3: fun(A,C),G: fun(C,D),L: D] :
          ( topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( filterlim(C,D,G,topolo7230453075368039082e_nhds(D,L),topolo174197925503356063within(C,aa(A,C,F3,A2),top_top(set(C))))
           => ( ? [D6: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D6))
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X3,A2)),D6)) )
                     => ( aa(A,C,F3,X3) != aa(A,C,F3,A2) ) ) )
             => filterlim(A,D,aa(fun(C,D),fun(A,D),aTP_Lamp_wv(fun(A,C),fun(fun(C,D),fun(A,D)),F3),G),topolo7230453075368039082e_nhds(D,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% metric_isCont_LIM_compose2
tff(fact_6617_isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A2: A,F3: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A2,top_top(set(A))),F3)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,F3,A2),top_top(set(B))))
           => ( ? [D6: real] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),D6))
                  & ! [X3: A] :
                      ( ( ( X3 != A2 )
                        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X3),A2))),D6)) )
                     => ( aa(A,B,F3,X3) != aa(A,B,F3,A2) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_rp(fun(A,B),fun(fun(B,C),fun(A,C)),F3),G),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ) ) ).

% isCont_LIM_compose2
tff(fact_6618_isCont__powser,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,X: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V7770717601297561774m_norm(A,X)),real_V7770717601297561774m_norm(A,K5)))
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,X,top_top(set(A))),aTP_Lamp_om(fun(nat,A),fun(A,A),C2)) ) ) ) ).

% isCont_powser
tff(fact_6619_compactE__image,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),C5: set(B),F3: fun(B,set(A))] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( ! [T7: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),T7),C5))
               => topolo1002775350975398744n_open(A,aa(B,set(A),F3,T7)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),F3),C5))))
             => ~ ! [C7: set(B)] :
                    ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C7),C5))
                   => ( pp(aa(set(B),bool,finite_finite(B),C7))
                     => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),F3),C7)))) ) ) ) ) ) ) ).

% compactE_image
tff(fact_6620_GMVT,axiom,
    ! [A2: real,B2: real,F3: fun(real,real),G: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [X3: real] :
            ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X3,top_top(set(real))),F3) )
       => ( ! [X3: real] :
              ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X3))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2)) )
             => differentiable(real,real,F3,topolo174197925503356063within(real,X3,top_top(set(real)))) )
         => ( ! [X3: real] :
                ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X3))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X3),B2)) )
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,X3,top_top(set(real))),G) )
           => ( ! [X3: real] :
                  ( ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X3))
                    & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2)) )
                 => differentiable(real,real,G,topolo174197925503356063within(real,X3,top_top(set(real)))) )
             => ? [G_c: real,F_c: real,C4: real] :
                  ( has_field_derivative(real,G,G_c,topolo174197925503356063within(real,C4,top_top(set(real))))
                  & has_field_derivative(real,F3,F_c,topolo174197925503356063within(real,C4,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),C4))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),C4),B2))
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,F3,B2)),aa(real,real,F3,A2))),G_c) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,G,B2)),aa(real,real,G,A2))),F_c) ) ) ) ) ) ) ) ).

% GMVT
tff(fact_6621_differentiable__within__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),X: A,S: set(A),T2: set(A)] :
          ( differentiable(A,B,F3,topolo174197925503356063within(A,X,S))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),S))
           => differentiable(A,B,F3,topolo174197925503356063within(A,X,T2)) ) ) ) ).

% differentiable_within_subset
tff(fact_6622_differentiable__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),F5: filter(A),G: fun(A,B)] :
          ( differentiable(A,B,F3,F5)
         => ( differentiable(A,B,G,F5)
           => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_pn(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G),F5) ) ) ) ).

% differentiable_add
tff(fact_6623_differentiable__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: fun(A,B),X: A,S: set(A),N: nat] :
          ( differentiable(A,B,F3,topolo174197925503356063within(A,X,S))
         => differentiable(A,B,aa(nat,fun(A,B),aTP_Lamp_pu(fun(A,B),fun(nat,fun(A,B)),F3),N),topolo174197925503356063within(A,X,S)) ) ) ).

% differentiable_power
tff(fact_6624_compact__attains__inf,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S2: set(A)] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                & ! [Xa2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),S2))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa2)) ) ) ) ) ) ).

% compact_attains_inf
tff(fact_6625_compact__attains__sup,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [S2: set(A)] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ? [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S2))
                & ! [Xa2: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),S2))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Xa2),X3)) ) ) ) ) ) ).

% compact_attains_sup
tff(fact_6626_differentiable__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [F3: fun(A,B),X: A,S: set(A),G: fun(A,B)] :
          ( differentiable(A,B,F3,topolo174197925503356063within(A,X,S))
         => ( differentiable(A,B,G,topolo174197925503356063within(A,X,S))
           => ( ( aa(A,B,G,X) != zero_zero(B) )
             => differentiable(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ww(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G),topolo174197925503356063within(A,X,S)) ) ) ) ) ).

% differentiable_divide
tff(fact_6627_compact__eq__Heine__Borel,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A)] :
          ( topolo2193935891317330818ompact(A,S2)
        <=> ! [C8: set(set(A))] :
              ( ( ! [X4: set(A)] :
                    ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),C8))
                   => topolo1002775350975398744n_open(A,X4) )
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C8))) )
             => ? [D7: set(set(A))] :
                  ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),D7),C8))
                  & pp(aa(set(set(A)),bool,finite_finite(set(A)),D7))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),D7))) ) ) ) ) ).

% compact_eq_Heine_Borel
tff(fact_6628_compactI,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A)] :
          ( ! [C6: set(set(A))] :
              ( ! [X2: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X2),C6))
                 => topolo1002775350975398744n_open(A,X2) )
             => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C6)))
               => ? [C9: set(set(A))] :
                    ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),C9),C6))
                    & pp(aa(set(set(A)),bool,finite_finite(set(A)),C9))
                    & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),C9))) ) ) )
         => topolo2193935891317330818ompact(A,S) ) ) ).

% compactI
tff(fact_6629_compactE,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),T11: set(set(A))] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),T11)))
           => ( ! [B7: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),B7),T11))
                 => topolo1002775350975398744n_open(A,B7) )
             => ~ ! [T12: set(set(A))] :
                    ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),T12),T11))
                   => ( pp(aa(set(set(A)),bool,finite_finite(set(A)),T12))
                     => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),T12))) ) ) ) ) ) ) ).

% compactE
tff(fact_6630_MVT,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F3)
       => ( ! [X3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X3))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2))
               => differentiable(real,real,F3,topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ? [L4: real,Z2: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Z2))
              & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z2),B2))
              & has_field_derivative(real,F3,L4,topolo174197925503356063within(real,Z2,top_top(set(real))))
              & ( aa(real,real,minus_minus(real,aa(real,real,F3,B2)),aa(real,real,F3,A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,B2),A2)),L4) ) ) ) ) ) ).

% MVT
tff(fact_6631_continuous__artanh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F5: filter(A),F3: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F5,F3)
         => ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(A,real,F3,topolo3827282254853284352ce_Lim(A,A,F5,aTP_Lamp_wx(A,A)))),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real))))
           => topolo3448309680560233919inuous(A,real,F5,aTP_Lamp_wy(fun(A,real),fun(A,real),F3)) ) ) ) ).

% continuous_artanh
tff(fact_6632_continuous__on__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [S: set(A),F3: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,S,F3)
         => ( topolo81223032696312382ous_on(A,B,S,G)
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
                 => ( aa(A,B,G,X3) != zero_zero(B) ) )
             => topolo81223032696312382ous_on(A,B,S,aa(fun(A,B),fun(A,B),aTP_Lamp_wz(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G)) ) ) ) ) ).

% continuous_on_divide
tff(fact_6633_continuous__on__power_H,axiom,
    ! [B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo1898628316856586783d_mult(B) )
     => ! [A3: set(C),F3: fun(C,B),G: fun(C,nat)] :
          ( topolo81223032696312382ous_on(C,B,A3,F3)
         => ( topolo81223032696312382ous_on(C,nat,A3,G)
           => topolo81223032696312382ous_on(C,B,A3,aa(fun(C,nat),fun(C,B),aTP_Lamp_xa(fun(C,B),fun(fun(C,nat),fun(C,B)),F3),G)) ) ) ) ).

% continuous_on_power'
tff(fact_6634_continuous__on__power,axiom,
    ! [C: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(C) )
     => ! [S: set(C),F3: fun(C,B),N: nat] :
          ( topolo81223032696312382ous_on(C,B,S,F3)
         => topolo81223032696312382ous_on(C,B,S,aa(nat,fun(C,B),aTP_Lamp_xb(fun(C,B),fun(nat,fun(C,B)),F3),N)) ) ) ).

% continuous_on_power
tff(fact_6635_continuous__on__add,axiom,
    ! [B: $tType,D: $tType] :
      ( ( topolo4958980785337419405_space(D)
        & topolo6943815403480290642id_add(B) )
     => ! [S: set(D),F3: fun(D,B),G: fun(D,B)] :
          ( topolo81223032696312382ous_on(D,B,S,F3)
         => ( topolo81223032696312382ous_on(D,B,S,G)
           => topolo81223032696312382ous_on(D,B,S,aa(fun(D,B),fun(D,B),aTP_Lamp_xc(fun(D,B),fun(fun(D,B),fun(D,B)),F3),G)) ) ) ) ).

% continuous_on_add
tff(fact_6636_IVT2_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & topolo8458572112393995274pology(A) )
     => ! [F3: fun(A,B),B2: A,Y: B,A2: A] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,B2)),Y))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y),aa(A,B,F3,A2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
             => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F3)
               => ? [X3: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2))
                    & ( aa(A,B,F3,X3) = Y ) ) ) ) ) ) ) ).

% IVT2'
tff(fact_6637_IVT_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo1944317154257567458pology(B)
        & topolo8458572112393995274pology(A) )
     => ! [F3: fun(A,B),A2: A,Y: B,B2: A] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,A2)),Y))
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Y),aa(A,B,F3,B2)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
             => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F3)
               => ? [X3: A] :
                    ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3))
                    & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2))
                    & ( aa(A,B,F3,X3) = Y ) ) ) ) ) ) ) ).

% IVT'
tff(fact_6638_continuous__on__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S: set(A),F3: fun(A,B),T2: set(A)] :
          ( topolo81223032696312382ous_on(A,B,S,F3)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T2),S))
           => topolo81223032696312382ous_on(A,B,T2,F3) ) ) ) ).

% continuous_on_subset
tff(fact_6639_continuous__on__compose2,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [T2: set(A),G: fun(A,B),S: set(C),F3: fun(C,A)] :
          ( topolo81223032696312382ous_on(A,B,T2,G)
         => ( topolo81223032696312382ous_on(C,A,S,F3)
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,F3),S)),T2))
             => topolo81223032696312382ous_on(C,B,S,aa(fun(C,A),fun(C,B),aTP_Lamp_xd(fun(A,B),fun(fun(C,A),fun(C,B)),G),F3)) ) ) ) ) ).

% continuous_on_compose2
tff(fact_6640_continuous__onI__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & dense_order(B)
        & topolo1944317154257567458pology(B) )
     => ! [F3: fun(A,B),A3: set(A)] :
          ( topolo1002775350975398744n_open(B,aa(set(A),set(B),image(A,B,F3),A3))
         => ( ! [X3: A,Y4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),A3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y4))
                   => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y4))) ) ) )
           => topolo81223032696312382ous_on(A,B,A3,F3) ) ) ) ).

% continuous_onI_mono
tff(fact_6641_continuous__attains__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [S: set(A),F3: fun(A,B)] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ( topolo81223032696312382ous_on(A,B,S,F3)
             => ? [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
                  & ! [Xa2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),S))
                     => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,Xa2)),aa(A,B,F3,X3))) ) ) ) ) ) ) ).

% continuous_attains_sup
tff(fact_6642_continuous__attains__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [S: set(A),F3: fun(A,B)] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ( S != bot_bot(set(A)) )
           => ( topolo81223032696312382ous_on(A,B,S,F3)
             => ? [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
                  & ! [Xa2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),S))
                     => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Xa2))) ) ) ) ) ) ) ).

% continuous_attains_inf
tff(fact_6643_open__Collect__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F3: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F3)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G)
           => topolo1002775350975398744n_open(A,collect(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_xe(fun(A,B),fun(fun(A,B),fun(A,bool)),F3),G))) ) ) ) ).

% open_Collect_less
tff(fact_6644_open__Collect__less__Int,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F3: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F3)
         => ( topolo81223032696312382ous_on(A,real,S,G)
           => ? [A7: set(A)] :
                ( topolo1002775350975398744n_open(A,A7)
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A7),S) = collect(A,aa(fun(A,real),fun(A,bool),aa(fun(A,real),fun(fun(A,real),fun(A,bool)),aTP_Lamp_xf(set(A),fun(fun(A,real),fun(fun(A,real),fun(A,bool))),S),F3),G)) ) ) ) ) ) ).

% open_Collect_less_Int
tff(fact_6645_continuous__on__arcosh_H,axiom,
    ! [A3: set(real),F3: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A3,F3)
     => ( ! [X3: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),A3))
           => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(real,real,F3,X3))) )
       => topolo81223032696312382ous_on(real,real,A3,aTP_Lamp_xg(fun(real,real),fun(real,real),F3)) ) ) ).

% continuous_on_arcosh'
tff(fact_6646_continuous__image__closed__interval,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F3)
       => ? [C4: real,D3: real] :
            ( ( aa(set(real),set(real),image(real,real,F3),set_or1337092689740270186AtMost(real,A2,B2)) = set_or1337092689740270186AtMost(real,C4,D3) )
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),C4),D3)) ) ) ) ).

% continuous_image_closed_interval
tff(fact_6647_continuous__on__arcosh,axiom,
    ! [A3: set(real)] :
      ( pp(aa(set(real),bool,aa(set(real),fun(set(real),bool),ord_less_eq(set(real)),A3),aa(real,set(real),set_ord_atLeast(real),one_one(real))))
     => topolo81223032696312382ous_on(real,real,A3,arcosh(real)) ) ).

% continuous_on_arcosh
tff(fact_6648_open__Collect__positive,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F3: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F3)
         => ? [A7: set(A)] :
              ( topolo1002775350975398744n_open(A,A7)
              & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A7),S) = collect(A,aa(fun(A,real),fun(A,bool),aTP_Lamp_xh(set(A),fun(fun(A,real),fun(A,bool)),S),F3)) ) ) ) ) ).

% open_Collect_positive
tff(fact_6649_continuous__on__powr_H,axiom,
    ! [C: $tType] :
      ( topolo4958980785337419405_space(C)
     => ! [S: set(C),F3: fun(C,real),G: fun(C,real)] :
          ( topolo81223032696312382ous_on(C,real,S,F3)
         => ( topolo81223032696312382ous_on(C,real,S,G)
           => ( ! [X3: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X3),S))
                 => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(C,real,F3,X3)))
                    & ( ( aa(C,real,F3,X3) = zero_zero(real) )
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(C,real,G,X3))) ) ) )
             => topolo81223032696312382ous_on(C,real,S,aa(fun(C,real),fun(C,real),aTP_Lamp_xi(fun(C,real),fun(fun(C,real),fun(C,real)),F3),G)) ) ) ) ) ).

% continuous_on_powr'
tff(fact_6650_continuous__on__log,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F3: fun(A,real),G: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F3)
         => ( topolo81223032696312382ous_on(A,real,S,G)
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,F3,X3))) )
             => ( ! [X3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
                   => ( aa(A,real,F3,X3) != one_one(real) ) )
               => ( ! [X3: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
                     => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G,X3))) )
                 => topolo81223032696312382ous_on(A,real,S,aa(fun(A,real),fun(A,real),aTP_Lamp_xj(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G)) ) ) ) ) ) ) ).

% continuous_on_log
tff(fact_6651_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_6652_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_6653_continuous__on__arccos,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F3: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F3)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,F3,X3)))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(A,real,F3,X3)),one_one(real))) ) )
           => topolo81223032696312382ous_on(A,real,S,aTP_Lamp_xk(fun(A,real),fun(A,real),F3)) ) ) ) ).

% continuous_on_arccos
tff(fact_6654_continuous__on__arcsin,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),F3: fun(A,real)] :
          ( topolo81223032696312382ous_on(A,real,S,F3)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),S))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,F3,X3)))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(A,real,F3,X3)),one_one(real))) ) )
           => topolo81223032696312382ous_on(A,real,S,aTP_Lamp_xl(fun(A,real),fun(A,real),F3)) ) ) ) ).

% continuous_on_arcsin
tff(fact_6655_continuous__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [F5: filter(A),F3: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F5,F3)
         => ( topolo3448309680560233919inuous(A,B,F5,G)
           => ( ( aa(A,B,G,topolo3827282254853284352ce_Lim(A,A,F5,aTP_Lamp_wx(A,A))) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,F5,aa(fun(A,B),fun(A,B),aTP_Lamp_wn(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G)) ) ) ) ) ).

% continuous_divide
tff(fact_6656_DERIV__atLeastAtMost__imp__continuous__on,axiom,
    ! [A: $tType] :
      ( ( ord(A)
        & real_V3459762299906320749_field(A) )
     => ! [A2: A,B2: A,F3: fun(A,A)] :
          ( ! [X3: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),X3))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),B2))
               => ? [Y3: A] : has_field_derivative(A,F3,Y3,topolo174197925503356063within(A,X3,top_top(set(A)))) ) )
         => topolo81223032696312382ous_on(A,A,set_or1337092689740270186AtMost(A,A2,B2),F3) ) ) ).

% DERIV_atLeastAtMost_imp_continuous_on
tff(fact_6657_continuous__on__artanh_H,axiom,
    ! [A3: set(real),F3: fun(real,real)] :
      ( topolo81223032696312382ous_on(real,real,A3,F3)
     => ( ! [X3: real] :
            ( pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),X3),A3))
           => pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),aa(real,real,F3,X3)),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)))) )
       => topolo81223032696312382ous_on(real,real,A3,aTP_Lamp_xm(fun(real,real),fun(real,real),F3)) ) ) ).

% continuous_on_artanh'
tff(fact_6658_Rolle__deriv,axiom,
    ! [A2: real,B2: real,F3: fun(real,real),F10: fun(real,fun(real,real))] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ( aa(real,real,F3,A2) = aa(real,real,F3,B2) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F3)
         => ( ! [X3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X3))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2))
                 => has_derivative(real,real,F3,aa(real,fun(real,real),F10,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
           => ? [Z2: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Z2))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z2),B2))
                & ! [X2: real] : aa(real,real,aa(real,fun(real,real),F10,Z2),X2) = zero_zero(real) ) ) ) ) ) ).

% Rolle_deriv
tff(fact_6659_mvt,axiom,
    ! [A2: real,B2: real,F3: fun(real,real),F10: fun(real,fun(real,real))] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F3)
       => ( ! [X3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X3))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2))
               => has_derivative(real,real,F3,aa(real,fun(real,real),F10,X3),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ~ ! [Xi: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Xi))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Xi),B2))
                 => ( aa(real,real,minus_minus(real,aa(real,real,F3,B2)),aa(real,real,F3,A2)) != aa(real,real,aa(real,fun(real,real),F10,Xi),aa(real,real,minus_minus(real,B2),A2)) ) ) ) ) ) ) ).

% mvt
tff(fact_6660_continuous__on__Icc__at__leftD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,F3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,B2)),topolo174197925503356063within(A,B2,aa(A,set(A),set_ord_lessThan(A),B2))) ) ) ) ).

% continuous_on_Icc_at_leftD
tff(fact_6661_continuous__on__Icc__at__rightD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A2: A,B2: A,F3: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
           => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,A2)),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2))) ) ) ) ).

% continuous_on_Icc_at_rightD
tff(fact_6662_DERIV__pos__imp__increasing__open,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2))
             => ? [Y3: real] :
                  ( has_field_derivative(real,F3,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),Y3)) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F3)
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F3,A2)),aa(real,real,F3,B2))) ) ) ) ).

% DERIV_pos_imp_increasing_open
tff(fact_6663_DERIV__neg__imp__decreasing__open,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ! [X3: real] :
            ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X3))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2))
             => ? [Y3: real] :
                  ( has_field_derivative(real,F3,Y3,topolo174197925503356063within(real,X3,top_top(set(real))))
                  & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Y3),zero_zero(real))) ) ) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F3)
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,F3,B2)),aa(real,real,F3,A2))) ) ) ) ).

% DERIV_neg_imp_decreasing_open
tff(fact_6664_DERIV__isconst__end,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F3)
       => ( ! [X3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X3))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2))
               => has_field_derivative(real,F3,zero_zero(real),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ( aa(real,real,F3,B2) = aa(real,real,F3,A2) ) ) ) ) ).

% DERIV_isconst_end
tff(fact_6665_continuous__on__artanh,axiom,
    ! [A3: set(real)] :
      ( pp(aa(set(real),bool,aa(set(real),fun(set(real),bool),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_6666_continuous__arcosh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F5: filter(A),F3: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F5,F3)
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),aa(A,real,F3,topolo3827282254853284352ce_Lim(A,A,F5,aTP_Lamp_wx(A,A)))))
           => topolo3448309680560233919inuous(A,real,F5,aTP_Lamp_wq(fun(A,real),fun(A,real),F3)) ) ) ) ).

% continuous_arcosh
tff(fact_6667_DERIV__isconst2,axiom,
    ! [A2: real,B2: real,F3: fun(real,real),X: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F3)
       => ( ! [X3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X3))
             => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2))
               => has_field_derivative(real,F3,zero_zero(real),topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
         => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),A2),X))
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),X),B2))
             => ( aa(real,real,F3,X) = aa(real,real,F3,A2) ) ) ) ) ) ) ).

% DERIV_isconst2
tff(fact_6668_continuous__on__IccI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [F3: fun(A,B),A2: A,B2: A] :
          ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,A2)),topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_greaterThan(A),A2)))
         => ( filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,B2)),topolo174197925503356063within(A,B2,aa(A,set(A),set_ord_lessThan(A),B2)))
           => ( ! [X3: A] :
                  ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),X3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),B2))
                   => filterlim(A,B,F3,topolo7230453075368039082e_nhds(B,aa(A,B,F3,X3)),topolo174197925503356063within(A,X3,top_top(set(A)))) ) )
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
               => topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F3) ) ) ) ) ) ).

% continuous_on_IccI
tff(fact_6669_Rolle,axiom,
    ! [A2: real,B2: real,F3: fun(real,real)] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),B2))
     => ( ( aa(real,real,F3,A2) = aa(real,real,F3,B2) )
       => ( topolo81223032696312382ous_on(real,real,set_or1337092689740270186AtMost(real,A2,B2),F3)
         => ( ! [X3: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),X3))
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X3),B2))
                 => differentiable(real,real,F3,topolo174197925503356063within(real,X3,top_top(set(real)))) ) )
           => ? [Z2: real] :
                ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),A2),Z2))
                & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Z2),B2))
                & has_field_derivative(real,F3,zero_zero(real),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) ) ) ) ) ).

% Rolle
tff(fact_6670_continuous__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F5: filter(A),F3: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F5,F3)
         => ( topolo3448309680560233919inuous(A,real,F5,G)
           => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,F3,topolo3827282254853284352ce_Lim(A,A,F5,aTP_Lamp_wx(A,A)))))
             => ( ( aa(A,real,F3,topolo3827282254853284352ce_Lim(A,A,F5,aTP_Lamp_wx(A,A))) != one_one(real) )
               => ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,G,topolo3827282254853284352ce_Lim(A,A,F5,aTP_Lamp_wx(A,A)))))
                 => topolo3448309680560233919inuous(A,real,F5,aa(fun(A,real),fun(A,real),aTP_Lamp_wr(fun(A,real),fun(fun(A,real),fun(A,real)),F3),G)) ) ) ) ) ) ) ).

% continuous_log
tff(fact_6671_listrel1__def,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : listrel1(A,R3) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),bool,aTP_Lamp_xn(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),R3))) ).

% listrel1_def
tff(fact_6672_the__elem__set,axiom,
    ! [A: $tType,X: A] : the_elem(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) = X ).

% the_elem_set
tff(fact_6673_Cons__listrel1__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A),Y: A,Ys: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))),listrel1(A,R3)))
    <=> ( ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y)),R3))
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),listrel1(A,R3))) ) ) ) ).

% Cons_listrel1_Cons
tff(fact_6674_listrel1__mono,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,R3)),listrel1(A,S))) ) ).

% listrel1_mono
tff(fact_6675_append__listrel1I,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A)),Us: list(A),Vs: list(A)] :
      ( ( ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),listrel1(A,R3)))
          & ( Us = Vs ) )
        | ( ( Xs = Ys )
          & pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Us),Vs)),listrel1(A,R3))) ) )
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Vs))),listrel1(A,R3))) ) ).

% append_listrel1I
tff(fact_6676_listrel1I2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A)),X: A] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),listrel1(A,R3)))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))),listrel1(A,R3))) ) ).

% listrel1I2
tff(fact_6677_listrel1__eq__len,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),listrel1(A,R3)))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) ) ).

% listrel1_eq_len
tff(fact_6678_not__listrel1__Nil,axiom,
    ! [A: $tType,Xs: list(A),R3: set(product_prod(A,A))] : ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),nil(A))),listrel1(A,R3))) ).

% not_listrel1_Nil
tff(fact_6679_not__Nil__listrel1,axiom,
    ! [A: $tType,Xs: list(A),R3: set(product_prod(A,A))] : ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),nil(A)),Xs)),listrel1(A,R3))) ).

% not_Nil_listrel1
tff(fact_6680_listrel1I1,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),Xs: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y)),R3))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs))),listrel1(A,R3))) ) ).

% listrel1I1
tff(fact_6681_Cons__listrel1E1,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys)),listrel1(A,R3)))
     => ( ! [Y4: A] :
            ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs) )
           => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y4)),R3)) )
       => ~ ! [Zs2: list(A)] :
              ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs2) )
             => ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Zs2)),listrel1(A,R3))) ) ) ) ).

% Cons_listrel1E1
tff(fact_6682_Cons__listrel1E2,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))),listrel1(A,R3)))
     => ( ! [X3: A] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys) )
           => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X3),Y)),R3)) )
       => ~ ! [Zs2: list(A)] :
              ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs2) )
             => ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Zs2),Ys)),listrel1(A,R3))) ) ) ) ).

% Cons_listrel1E2
tff(fact_6683_listrel1I,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),Xs: list(A),Us: list(A),Vs: list(A),Ys: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y)),R3))
     => ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs)) )
       => ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Vs)) )
         => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),listrel1(A,R3))) ) ) ) ).

% listrel1I
tff(fact_6684_listrel1E,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),listrel1(A,R3)))
     => ~ ! [X3: A,Y4: A] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X3),Y4)),R3))
           => ! [Us3: list(A),Vs2: list(A)] :
                ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Vs2)) )
               => ( Ys != aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Vs2)) ) ) ) ) ).

% listrel1E
tff(fact_6685_snoc__listrel1__snoc__iff,axiom,
    ! [A: $tType,Xs: list(A),X: A,Ys: list(A),Y: A,R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))),listrel1(A,R3)))
    <=> ( ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),listrel1(A,R3)))
          & ( X = Y ) )
        | ( ( Xs = Ys )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y)),R3)) ) ) ) ).

% snoc_listrel1_snoc_iff
tff(fact_6686_listrel1__iff__update,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),listrel1(A,R3)))
    <=> ? [Y5: A,N5: nat] :
          ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,nth(A,Xs),N5)),Y5)),R3))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
          & ( Ys = list_update(A,Xs,N5,Y5) ) ) ) ).

% listrel1_iff_update
tff(fact_6687_listrel1p__def,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
      ( listrel1p(A,R3,Xs,Ys)
    <=> pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),listrel1(A,collect(product_prod(A,A),product_case_prod(A,A,bool,R3))))) ) ).

% listrel1p_def
tff(fact_6688_sequentially__imp__eventually__at__left,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [B2: A,A2: A,P: fun(A,bool)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( ! [F4: fun(nat,A)] :
                ( ! [N7: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),aa(nat,A,F4,N7)))
               => ( ! [N7: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F4,N7)),A2))
                 => ( order_mono(nat,A,F4)
                   => ( filterlim(nat,A,F4,topolo7230453075368039082e_nhds(A,A2),at_top(nat))
                     => eventually(nat,aa(fun(nat,A),fun(nat,bool),aTP_Lamp_wa(fun(A,bool),fun(fun(nat,A),fun(nat,bool)),P),F4),at_top(nat)) ) ) ) )
           => eventually(A,P,topolo174197925503356063within(A,A2,aa(A,set(A),set_ord_lessThan(A),A2))) ) ) ) ).

% sequentially_imp_eventually_at_left
tff(fact_6689_mono__Suc,axiom,
    order_mono(nat,nat,suc) ).

% mono_Suc
tff(fact_6690_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_6691_mono__strict__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_mono(A,B,F3)
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X)),aa(A,B,F3,Y)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).

% mono_strict_invE
tff(fact_6692_mono__invE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_mono(A,B,F3)
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X)),aa(A,B,F3,Y)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% mono_invE
tff(fact_6693_incseqD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A),I2: nat,J: nat] :
          ( order_mono(nat,A,F3)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,I2)),aa(nat,A,F3,J))) ) ) ) ).

% incseqD
tff(fact_6694_incseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( order_mono(nat,A,X6)
        <=> ! [M7: nat,N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M7),N5))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,M7)),aa(nat,A,X6,N5))) ) ) ) ).

% incseq_def
tff(fact_6695_incseq__SucD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: fun(nat,A),I2: nat] :
          ( order_mono(nat,A,A3)
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,A3,I2)),aa(nat,A,A3,aa(nat,nat,suc,I2)))) ) ) ).

% incseq_SucD
tff(fact_6696_incseq__SucI,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X6: fun(nat,A)] :
          ( ! [N2: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N2)),aa(nat,A,X6,aa(nat,nat,suc,N2))))
         => order_mono(nat,A,X6) ) ) ).

% incseq_SucI
tff(fact_6697_incseq__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( order_mono(nat,A,F3)
        <=> ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,F3,N5)),aa(nat,A,F3,aa(nat,nat,suc,N5)))) ) ) ).

% incseq_Suc_iff
tff(fact_6698_monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_mono(A,B,F3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,Y))) ) ) ) ).

% monoD
tff(fact_6699_monoE,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_mono(A,B,F3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,Y))) ) ) ) ).

% monoE
tff(fact_6700_monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( ! [X3: A,Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y4))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,F3,Y4))) )
         => order_mono(A,B,F3) ) ) ).

% monoI
tff(fact_6701_mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( order_mono(A,B,F3)
        <=> ! [X4: A,Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y5))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X4)),aa(A,B,F3,Y5))) ) ) ) ).

% mono_def
tff(fact_6702_mono__inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_inf(A)
        & semilattice_inf(B) )
     => ! [F3: fun(A,B),A3: A,B3: A] :
          ( order_mono(A,B,F3)
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,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,F3,A3)),aa(A,B,F3,B3)))) ) ) ).

% mono_inf
tff(fact_6703_mono__sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semilattice_sup(A)
        & semilattice_sup(B) )
     => ! [F3: fun(A,B),A3: A,B3: A] :
          ( order_mono(A,B,F3)
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(B,B,aa(B,fun(B,B),sup_sup(B),aa(A,B,F3,A3)),aa(A,B,F3,B3))),aa(A,B,F3,aa(A,A,aa(A,fun(A,A),sup_sup(A),A3),B3)))) ) ) ).

% mono_sup
tff(fact_6704_funpow__mono,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(A,A),A3: A,B3: A,N: nat] :
          ( order_mono(A,A,F3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A3),B3))
           => pp(aa(A,bool,aa(A,fun(A,bool),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)),N),F3),A3)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),B3))) ) ) ) ).

% funpow_mono
tff(fact_6705_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_xo(fun(A,A),fun(nat,A),Q)) ) ) ).

% mono_funpow
tff(fact_6706_mono__pow,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(A,A),N: nat] :
          ( order_mono(A,A,F3)
         => order_mono(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3)) ) ) ).

% mono_pow
tff(fact_6707_mono__times__nat,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => order_mono(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N)) ) ).

% mono_times_nat
tff(fact_6708_mono__mult,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_6709_mono__image__least,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [F3: fun(A,B),M: A,N: A,M4: B,N4: B] :
          ( order_mono(A,B,F3)
         => ( ( aa(set(A),set(B),image(A,B,F3),set_or7035219750837199246ssThan(A,M,N)) = set_or7035219750837199246ssThan(B,M4,N4) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),M),N))
             => ( aa(A,B,F3,M) = M4 ) ) ) ) ) ).

% mono_image_least
tff(fact_6710_Kleene__iter__gpfp,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [F3: fun(A,A),P2: A,K: nat] :
          ( order_mono(A,A,F3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P2),aa(A,A,F3,P2)))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),P2),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),K),F3),top_top(A)))) ) ) ) ).

% Kleene_iter_gpfp
tff(fact_6711_Kleene__iter__lpfp,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [F3: fun(A,A),P2: A,K: nat] :
          ( order_mono(A,A,F3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,F3,P2)),P2))
           => pp(aa(A,bool,aa(A,fun(A,bool),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),F3),bot_bot(A))),P2)) ) ) ) ).

% Kleene_iter_lpfp
tff(fact_6712_funpow__mono2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(A,A),I2: nat,J: nat,X: A,Y: A] :
          ( order_mono(A,A,F3)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
             => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(A,A,F3,X)))
               => pp(aa(A,bool,aa(A,fun(A,bool),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)),I2),F3),X)),aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),J),F3),Y))) ) ) ) ) ) ).

% funpow_mono2
tff(fact_6713_incseq__bounded,axiom,
    ! [X6: fun(nat,real),B3: real] :
      ( order_mono(nat,real,X6)
     => ( ! [I4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,X6,I4)),B3))
       => bfun(nat,real,X6,at_top(nat)) ) ) ).

% incseq_bounded
tff(fact_6714_mono__Sup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F3: fun(A,B),A3: set(A)] :
          ( order_mono(A,B,F3)
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F3),A3))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),A3)))) ) ) ).

% mono_Sup
tff(fact_6715_mono__SUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F3: fun(A,B),A3: fun(C,A),I5: set(C)] :
          ( order_mono(A,B,F3)
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_xp(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A3)),I5))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,A3),I5))))) ) ) ).

% mono_SUP
tff(fact_6716_mono__INF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(B)
        & comple6319245703460814977attice(A) )
     => ! [F3: fun(A,B),A3: fun(C,A),I5: set(C)] :
          ( order_mono(A,B,F3)
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image(C,A,A3),I5)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_xp(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A3)),I5)))) ) ) ).

% mono_INF
tff(fact_6717_mono__Inf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [F3: fun(A,B),A3: set(A)] :
          ( order_mono(A,B,F3)
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F3),A3)))) ) ) ).

% mono_Inf
tff(fact_6718_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_xq(fun(A,A),fun(nat,A),Q)) ) ) ).

% antimono_funpow
tff(fact_6719_incseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X6: fun(nat,A),L5: A,N: nat] :
          ( order_mono(nat,A,X6)
         => ( filterlim(nat,A,X6,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,X6,N)),L5)) ) ) ) ).

% incseq_le
tff(fact_6720_funpow__increasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [M: nat,N: nat,F3: fun(A,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( order_mono(A,A,F3)
           => pp(aa(A,bool,aa(A,fun(A,bool),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)),N),F3),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)),M),F3),top_top(A)))) ) ) ) ).

% funpow_increasing
tff(fact_6721_funpow__decreasing,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [M: nat,N: nat,F3: fun(A,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( order_mono(A,A,F3)
           => pp(aa(A,bool,aa(A,fun(A,bool),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)),M),F3),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)),N),F3),bot_bot(A)))) ) ) ) ).

% funpow_decreasing
tff(fact_6722_incseq__convergent,axiom,
    ! [X6: fun(nat,real),B3: real] :
      ( order_mono(nat,real,X6)
     => ( ! [I4: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,X6,I4)),B3))
       => ~ ! [L6: real] :
              ( filterlim(nat,real,X6,topolo7230453075368039082e_nhds(real,L6),at_top(nat))
             => ~ ! [I: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(nat,real,X6,I)),L6)) ) ) ) ).

% incseq_convergent
tff(fact_6723_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),K))
     => order_mono(nat,nat,aTP_Lamp_xr(nat,fun(nat,nat),K)) ) ).

% mono_ge2_power_minus_self
tff(fact_6724_finite__mono__remains__stable__implies__strict__prefix,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( pp(aa(set(A),bool,finite_finite(A),aa(set(nat),set(A),image(nat,A,F3),top_top(set(nat)))))
         => ( order_mono(nat,A,F3)
           => ( ! [N2: nat] :
                  ( ( aa(nat,A,F3,N2) = aa(nat,A,F3,aa(nat,nat,suc,N2)) )
                 => ( aa(nat,A,F3,aa(nat,nat,suc,N2)) = aa(nat,A,F3,aa(nat,nat,suc,aa(nat,nat,suc,N2))) ) )
             => ? [N9: nat] :
                  ( ! [N7: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N7),N9))
                     => ! [M2: nat] :
                          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M2),N9))
                         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M2),N7))
                           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,M2)),aa(nat,A,F3,N7))) ) ) )
                  & ! [N7: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N9),N7))
                     => ( aa(nat,A,F3,N9) = aa(nat,A,F3,N7) ) ) ) ) ) ) ) ).

% finite_mono_remains_stable_implies_strict_prefix
tff(fact_6725_tendsto__at__left__sequentially,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo3112930676232923870pology(B)
        & topolo1944317154257567458pology(B)
        & topolo4958980785337419405_space(A) )
     => ! [B2: B,A2: B,X6: fun(B,A),L5: A] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),A2))
         => ( ! [S5: fun(nat,B)] :
                ( ! [N7: nat] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(nat,B,S5,N7)),A2))
               => ( ! [N7: nat] : pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),B2),aa(nat,B,S5,N7)))
                 => ( order_mono(nat,B,S5)
                   => ( filterlim(nat,B,S5,topolo7230453075368039082e_nhds(B,A2),at_top(nat))
                     => filterlim(nat,A,aa(fun(nat,B),fun(nat,A),aTP_Lamp_xs(fun(B,A),fun(fun(nat,B),fun(nat,A)),X6),S5),topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ) )
           => filterlim(B,A,X6,topolo7230453075368039082e_nhds(A,L5),topolo174197925503356063within(B,A2,aa(B,set(B),set_ord_lessThan(B),A2))) ) ) ) ).

% tendsto_at_left_sequentially
tff(fact_6726_remdups__adj__altdef,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( remdups_adj(A,Xs) = Ys )
    <=> ? [F6: fun(nat,nat)] :
          ( order_mono(nat,nat,F6)
          & ( aa(set(nat),set(nat),image(nat,nat,F6),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)),Ys)) )
          & ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
             => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys),aa(nat,nat,F6,I3)) ) )
          & ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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,F6,I3) = aa(nat,nat,F6,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat))) ) ) ) ) ) ).

% remdups_adj_altdef
tff(fact_6727_compact__imp__fip__image,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S: set(A),I5: set(B),F3: fun(B,set(A))] :
          ( topolo2193935891317330818ompact(A,S)
         => ( ! [I4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),I5))
               => topolo7761053866217962861closed(A,aa(B,set(A),F3,I4)) )
           => ( ! [I8: set(B)] :
                  ( pp(aa(set(B),bool,finite_finite(B),I8))
                 => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),I8),I5))
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),F3),I8))) != bot_bot(set(A)) ) ) )
             => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(B),set(set(A)),image(B,set(A),F3),I5))) != bot_bot(set(A)) ) ) ) ) ) ).

% compact_imp_fip_image
tff(fact_6728_remdups__adj__Nil__iff,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( remdups_adj(A,Xs) = nil(A) )
    <=> ( Xs = nil(A) ) ) ).

% remdups_adj_Nil_iff
tff(fact_6729_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_6730_rtranclp_Omono,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool))] : order_mono(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aTP_Lamp_xt(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),R3)) ).

% rtranclp.mono
tff(fact_6731_tranclp_Omono,axiom,
    ! [A: $tType,R3: fun(A,fun(A,bool))] : order_mono(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aTP_Lamp_xu(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),R3)) ).

% tranclp.mono
tff(fact_6732_mono__Int,axiom,
    ! [B: $tType,A: $tType,F3: fun(set(A),set(B)),A3: set(A),B3: set(A)] :
      ( order_mono(set(A),set(B),F3)
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),F3,A3)),aa(set(A),set(B),F3,B3)))) ) ).

% mono_Int
tff(fact_6733_remdups__adj__length,axiom,
    ! [A: $tType,Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% remdups_adj_length
tff(fact_6734_mono__Un,axiom,
    ! [B: $tType,A: $tType,F3: fun(set(A),set(B)),A3: set(A),B3: set(A)] :
      ( order_mono(set(A),set(B),F3)
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),sup_sup(set(B)),aa(set(A),set(B),F3,A3)),aa(set(A),set(B),F3,B3))),aa(set(A),set(B),F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)))) ) ).

% mono_Un
tff(fact_6735_remdups__adj__distinct,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => ( remdups_adj(A,Xs) = Xs ) ) ).

% remdups_adj_distinct
tff(fact_6736_remdups__adj_Osimps_I3_J,axiom,
    ! [A: $tType,X: A,Y: A,Xs: list(A)] :
      ( ( ( X = Y )
       => ( remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs))) = remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) ) )
      & ( ( X != Y )
       => ( remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Xs))) ) ) ) ).

% remdups_adj.simps(3)
tff(fact_6737_remdups__adj_Osimps_I1_J,axiom,
    ! [A: $tType] : remdups_adj(A,nil(A)) = nil(A) ).

% remdups_adj.simps(1)
tff(fact_6738_remdups__adj_Osimps_I2_J,axiom,
    ! [A: $tType,X: A] : remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ).

% remdups_adj.simps(2)
tff(fact_6739_remdups__adj_Oelims,axiom,
    ! [A: $tType,X: list(A),Y: list(A)] :
      ( ( remdups_adj(A,X) = Y )
     => ( ( ( X = nil(A) )
         => ( Y != nil(A) ) )
       => ( ! [X3: A] :
              ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
             => ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) ) )
         => ~ ! [X3: A,Y4: A,Xs2: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs2)) )
               => ~ ( ( ( X3 = Y4 )
                     => ( Y = remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)) ) )
                    & ( ( X3 != Y4 )
                     => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs2))) ) ) ) ) ) ) ) ).

% remdups_adj.elims
tff(fact_6740_closed__diagonal,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => topolo7761053866217962861closed(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_xv(product_prod(A,A),bool))) ) ).

% closed_diagonal
tff(fact_6741_closed__superdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo7761053866217962861closed(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_xw(product_prod(A,A),bool))) ) ).

% closed_superdiagonal
tff(fact_6742_closed__subdiagonal,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => topolo7761053866217962861closed(product_prod(A,A),collect(product_prod(A,A),aTP_Lamp_xx(product_prod(A,A),bool))) ) ).

% closed_subdiagonal
tff(fact_6743_remdups__adj__append__two,axiom,
    ! [A: $tType,Xs: list(A),X: A,Y: A] : remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))))),if(list(A),aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y),nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A)))) ).

% remdups_adj_append_two
tff(fact_6744_finite_Omono,axiom,
    ! [A: $tType] : order_mono(fun(set(A),bool),fun(set(A),bool),aTP_Lamp_xy(fun(set(A),bool),fun(set(A),bool))) ).

% finite.mono
tff(fact_6745_closed__Collect__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F3: fun(A,B),G: fun(A,B)] :
          ( topolo81223032696312382ous_on(A,B,top_top(set(A)),F3)
         => ( topolo81223032696312382ous_on(A,B,top_top(set(A)),G)
           => topolo7761053866217962861closed(A,collect(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_xz(fun(A,B),fun(fun(A,B),fun(A,bool)),F3),G))) ) ) ) ).

% closed_Collect_le
tff(fact_6746_t3__space,axiom,
    ! [A: $tType] :
      ( topological_t3_space(A)
     => ! [S2: set(A),Y: A] :
          ( topolo7761053866217962861closed(A,S2)
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),S2))
           => ? [U5: set(A),V5: set(A)] :
                ( topolo1002775350975398744n_open(A,U5)
                & topolo1002775350975398744n_open(A,V5)
                & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),U5))
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),V5))
                & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U5),V5) = bot_bot(set(A)) ) ) ) ) ) ).

% t3_space
tff(fact_6747_t4__space,axiom,
    ! [A: $tType] :
      ( topological_t4_space(A)
     => ! [S2: set(A),T6: set(A)] :
          ( topolo7761053866217962861closed(A,S2)
         => ( topolo7761053866217962861closed(A,T6)
           => ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),T6) = bot_bot(set(A)) )
             => ? [U5: set(A),V5: set(A)] :
                  ( topolo1002775350975398744n_open(A,U5)
                  & topolo1002775350975398744n_open(A,V5)
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),U5))
                  & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T6),V5))
                  & ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U5),V5) = bot_bot(set(A)) ) ) ) ) ) ) ).

% t4_space
tff(fact_6748_ord_Olexordp_Omono,axiom,
    ! [A: $tType,Less: fun(A,fun(A,bool))] : order_mono(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)),aTP_Lamp_ya(fun(A,fun(A,bool)),fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Less)) ).

% ord.lexordp.mono
tff(fact_6749_remdups__adj__adjacent,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I2)),aa(list(A),nat,size_size(list(A)),remdups_adj(A,Xs))))
     => ( aa(nat,A,nth(A,remdups_adj(A,Xs)),I2) != aa(nat,A,nth(A,remdups_adj(A,Xs)),aa(nat,nat,suc,I2)) ) ) ).

% remdups_adj_adjacent
tff(fact_6750_nhds__closed,axiom,
    ! [A: $tType] :
      ( topological_t3_space(A)
     => ! [X: A,A3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => ( topolo1002775350975398744n_open(A,A3)
           => ? [A10: set(A)] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A10))
                & topolo7761053866217962861closed(A,A10)
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A10),A3))
                & eventually(A,aTP_Lamp_yb(set(A),fun(A,bool),A10),topolo7230453075368039082e_nhds(A,X)) ) ) ) ) ).

% nhds_closed
tff(fact_6751_remdups__adj__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( ( N = zero_zero(nat) )
       => ( remdups_adj(A,replicate(A,N,X)) = nil(A) ) )
      & ( ( N != zero_zero(nat) )
       => ( remdups_adj(A,replicate(A,N,X)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) ) ) ).

% remdups_adj_replicate
tff(fact_6752_remdups__adj__singleton,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( ( remdups_adj(A,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
     => ( Xs = replicate(A,aa(list(A),nat,size_size(list(A)),Xs),X) ) ) ).

% remdups_adj_singleton
tff(fact_6753_lexordp_Omono,axiom,
    ! [A: $tType] :
      ( ord(A)
     => order_mono(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)),aTP_Lamp_yc(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)))) ) ).

% lexordp.mono
tff(fact_6754_remdups__adj__length__ge1,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),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_6755_compact__imp__fip,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S2: set(A),F5: set(set(A))] :
          ( topolo2193935891317330818ompact(A,S2)
         => ( ! [T7: set(A)] :
                ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),T7),F5))
               => topolo7761053866217962861closed(A,T7) )
           => ( ! [F14: set(set(A))] :
                  ( pp(aa(set(set(A)),bool,finite_finite(set(A)),F14))
                 => ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),F14),F5))
                   => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),F14)) != bot_bot(set(A)) ) ) )
             => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),S2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),F5)) != bot_bot(set(A)) ) ) ) ) ) ).

% compact_imp_fip
tff(fact_6756_compact__fip,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [U2: set(A)] :
          ( topolo2193935891317330818ompact(A,U2)
        <=> ! [A11: set(set(A))] :
              ( ! [X4: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),A11))
                 => topolo7761053866217962861closed(A,X4) )
             => ( ! [B9: set(set(A))] :
                    ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),B9),A11))
                   => ( pp(aa(set(set(A)),bool,finite_finite(set(A)),B9))
                     => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),B9)) != bot_bot(set(A)) ) ) )
               => ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),U2),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),A11)) != bot_bot(set(A)) ) ) ) ) ) ).

% compact_fip
tff(fact_6757_remdups__adj_Opelims,axiom,
    ! [A: $tType,X: list(A),Y: list(A)] :
      ( ( remdups_adj(A,X) = Y )
     => ( pp(aa(list(A),bool,accp(list(A),remdups_adj_rel(A)),X))
       => ( ( ( X = nil(A) )
           => ( ( Y = nil(A) )
             => ~ pp(aa(list(A),bool,accp(list(A),remdups_adj_rel(A)),nil(A))) ) )
         => ( ! [X3: A] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
               => ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
                 => ~ pp(aa(list(A),bool,accp(list(A),remdups_adj_rel(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)))) ) )
           => ~ ! [X3: A,Y4: A,Xs2: list(A)] :
                  ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs2)) )
                 => ( ( ( ( X3 = Y4 )
                       => ( Y = remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)) ) )
                      & ( ( X3 != Y4 )
                       => ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs2))) ) ) )
                   => ~ pp(aa(list(A),bool,accp(list(A),remdups_adj_rel(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Xs2)))) ) ) ) ) ) ) ).

% remdups_adj.pelims
tff(fact_6758_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F3: fun(nat,real),G: fun(nat,nat)] :
      ( ! [X3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,F3,X3)))
     => ( order_mono(nat,real,F3)
       => ( order_strict_mono(nat,nat,G)
         => ( bfun(nat,real,aa(fun(nat,nat),fun(nat,real),aTP_Lamp_yd(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),F3),G),at_top(nat))
          <=> bfun(nat,real,F3,at_top(nat)) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
tff(fact_6759_strict__mono__add,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A] : order_strict_mono(A,A,aTP_Lamp_mh(A,fun(A,A),K)) ) ).

% strict_mono_add
tff(fact_6760_strict__mono__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_strict_mono(A,B,F3)
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X)),aa(A,B,F3,Y)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y)) ) ) ) ).

% strict_mono_less
tff(fact_6761_strict__mono__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( order_strict_mono(A,B,F3)
        <=> ! [X4: A,Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y5))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X4)),aa(A,B,F3,Y5))) ) ) ) ).

% strict_mono_def
tff(fact_6762_strict__monoI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B)] :
          ( ! [X3: A,Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y4))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X3)),aa(A,B,F3,Y4))) )
         => order_strict_mono(A,B,F3) ) ) ).

% strict_monoI
tff(fact_6763_strict__monoD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_strict_mono(A,B,F3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X)),aa(A,B,F3,Y))) ) ) ) ).

% strict_monoD
tff(fact_6764_strict__mono__less__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & order(B) )
     => ! [F3: fun(A,B),X: A,Y: A] :
          ( order_strict_mono(A,B,F3)
         => ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,Y)))
          <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y)) ) ) ) ).

% strict_mono_less_eq
tff(fact_6765_strict__mono__leD,axiom,
    ! [B: $tType,A: $tType] :
      ( ( order(A)
        & order(B) )
     => ! [R3: fun(A,B),M: A,N: A] :
          ( order_strict_mono(A,B,R3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),N))
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,R3,M)),aa(A,B,R3,N))) ) ) ) ).

% strict_mono_leD
tff(fact_6766_strict__mono__imp__increasing,axiom,
    ! [F3: fun(nat,nat),N: nat] :
      ( order_strict_mono(nat,nat,F3)
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(nat,nat,F3,N))) ) ).

% strict_mono_imp_increasing
tff(fact_6767_strict__mono__Suc__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F3: fun(nat,A)] :
          ( order_strict_mono(nat,A,F3)
        <=> ! [N5: nat] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(nat,A,F3,N5)),aa(nat,A,F3,aa(nat,nat,suc,N5)))) ) ) ).

% strict_mono_Suc_iff
tff(fact_6768_increasing__Bseq__subseq__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(nat,A),G: fun(nat,nat)] :
          ( ! [X3: nat,Y4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),Y4))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F3,X3))),real_V7770717601297561774m_norm(A,aa(nat,A,F3,Y4)))) )
         => ( order_strict_mono(nat,nat,G)
           => ( bfun(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_ye(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),F3),G),at_top(nat))
            <=> bfun(nat,A,F3,at_top(nat)) ) ) ) ) ).

% increasing_Bseq_subseq_iff
tff(fact_6769_lenlex__append2,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),Us: list(A),Xs: list(A),Ys: list(A)] :
      ( irrefl(A,R)
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us),Ys))),lenlex(A,R)))
      <=> pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),lenlex(A,R))) ) ) ).

% lenlex_append2
tff(fact_6770_inj__sgn__power,axiom,
    ! [N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => inj_on(real,real,aTP_Lamp_nz(nat,fun(real,real),N),top_top(set(real))) ) ).

% inj_sgn_power
tff(fact_6771_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_6772_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] :
          ( inj_on(A,A,aTP_Lamp_yf(A,fun(A,A),A2),top_top(set(A)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_6773_lexord__same__pref__if__irrefl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( irrefl(A,R3)
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs))),lexord(A,R3)))
      <=> pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Ys),Zs)),lexord(A,R3))) ) ) ).

% lexord_same_pref_if_irrefl
tff(fact_6774_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A3: set(A),F3: fun(A,B)] :
          ( ! [X3: A,Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y4))
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),A3))
                 => ( aa(A,B,F3,X3) != aa(A,B,F3,Y4) ) ) ) )
         => ( ! [X3: A,Y4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),A3))
                 => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Y4))
                    | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X3)) ) ) )
           => inj_on(A,B,F3,A3) ) ) ) ).

% linorder_inj_onI
tff(fact_6775_inj__on__subset,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A3: set(A),B3: set(A)] :
      ( inj_on(A,B,F3,A3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
       => inj_on(A,B,F3,B3) ) ) ).

% inj_on_subset
tff(fact_6776_subset__inj__on,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),B3: set(A),A3: set(A)] :
      ( inj_on(A,B,F3,B3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
       => inj_on(A,B,F3,A3) ) ) ).

% subset_inj_on
tff(fact_6777_inj__on__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),C5: set(A),A3: set(A),B3: set(A)] :
      ( inj_on(A,B,F3,C5)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C5))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C5))
         => ( ( aa(set(A),set(B),image(A,B,F3),A3) = aa(set(A),set(B),image(A,B,F3),B3) )
          <=> ( A3 = B3 ) ) ) ) ) ).

% inj_on_image_eq_iff
tff(fact_6778_inj__on__image__mem__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),B3: set(A),A2: A,A3: set(A)] :
      ( inj_on(A,B,F3,B3)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),B3))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),aa(A,B,F3,A2)),aa(set(A),set(B),image(A,B,F3),A3)))
          <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3)) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_6779_subset__image__inj,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(B,A),T6: set(B)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),S2),aa(set(B),set(A),image(B,A,F3),T6)))
    <=> ? [U6: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),U6),T6))
          & inj_on(B,A,F3,U6)
          & ( S2 = aa(set(B),set(A),image(B,A,F3),U6) ) ) ) ).

% subset_image_inj
tff(fact_6780_inj__fn,axiom,
    ! [A: $tType,F3: fun(A,A),N: nat] :
      ( inj_on(A,A,F3,top_top(set(A)))
     => inj_on(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3),top_top(set(A))) ) ).

% inj_fn
tff(fact_6781_lexord__irrefl,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( irrefl(A,R)
     => irrefl(list(A),lexord(A,R)) ) ).

% lexord_irrefl
tff(fact_6782_inj__on__strict__subset,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),B3: set(A),A3: set(A)] :
      ( inj_on(A,B,F3,B3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),A3),B3))
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less(set(B)),aa(set(A),set(B),image(A,B,F3),A3)),aa(set(A),set(B),image(A,B,F3),B3))) ) ) ).

% inj_on_strict_subset
tff(fact_6783_inj__on__add_H,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A3: set(A)] : inj_on(A,A,aTP_Lamp_yg(A,fun(A,A),A2),A3) ) ).

% inj_on_add'
tff(fact_6784_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_6785_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_6786_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_6787_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [F3: fun(A,B)] :
          ( ! [X3: A,Y4: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X3),Y4))
             => ( aa(A,B,F3,X3) != aa(A,B,F3,Y4) ) )
         => inj_on(A,B,F3,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_6788_sorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => inj_on(A,A,aTP_Lamp_ma(A,A),top_top(set(A))) ) ).

% sorted_list_of_set.inj_on
tff(fact_6789_irrefl__lex,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] :
      ( irrefl(A,R3)
     => irrefl(list(A),lex(A,R3)) ) ).

% irrefl_lex
tff(fact_6790_inj__image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A3: set(A),B3: set(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),A3)),aa(set(A),set(B),image(A,B,F3),B3)))
      <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ) ).

% inj_image_subset_iff
tff(fact_6791_inj__on__iff__surj,axiom,
    ! [B: $tType,A: $tType,A3: set(A),A8: set(B)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ? [F6: fun(A,B)] :
            ( inj_on(A,B,F6,A3)
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F6),A3)),A8)) )
      <=> ? [G6: fun(B,A)] : aa(set(B),set(A),image(B,A,G6),A8) = A3 ) ) ).

% inj_on_iff_surj
tff(fact_6792_finite__surj__inj,axiom,
    ! [A: $tType,A3: set(A),F3: fun(A,A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),image(A,A,F3),A3)))
       => inj_on(A,A,F3,A3) ) ) ).

% finite_surj_inj
tff(fact_6793_inj__on__finite,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A3: set(A),B3: set(B)] :
      ( inj_on(A,B,F3,A3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),A3)),B3))
       => ( pp(aa(set(B),bool,finite_finite(B),B3))
         => pp(aa(set(A),bool,finite_finite(A),A3)) ) ) ) ).

% inj_on_finite
tff(fact_6794_endo__inj__surj,axiom,
    ! [A: $tType,A3: set(A),F3: fun(A,A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,F3),A3)),A3))
       => ( inj_on(A,A,F3,A3)
         => ( aa(set(A),set(A),image(A,A,F3),A3) = A3 ) ) ) ) ).

% endo_inj_surj
tff(fact_6795_inj__on__image__Int,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),C5: set(A),A3: set(A),B3: set(A)] :
      ( inj_on(A,B,F3,C5)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C5))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C5))
         => ( aa(set(A),set(B),image(A,B,F3),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = aa(set(B),set(B),aa(set(B),fun(set(B),set(B)),inf_inf(set(B)),aa(set(A),set(B),image(A,B,F3),A3)),aa(set(A),set(B),image(A,B,F3),B3)) ) ) ) ) ).

% inj_on_image_Int
tff(fact_6796_inj__on__image__set__diff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),C5: set(A),A3: set(A),B3: set(A)] :
      ( inj_on(A,B,F3,C5)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),B3)),C5))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),C5))
         => ( aa(set(A),set(B),image(A,B,F3),aa(set(A),set(A),minus_minus(set(A),A3),B3)) = aa(set(B),set(B),minus_minus(set(B),aa(set(A),set(B),image(A,B,F3),A3)),aa(set(A),set(B),image(A,B,F3),B3)) ) ) ) ) ).

% inj_on_image_set_diff
tff(fact_6797_pigeonhole,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A3: set(B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(set(A),nat,finite_card(A),aa(set(B),set(A),image(B,A,F3),A3))),aa(set(B),nat,finite_card(B),A3)))
     => ~ inj_on(B,A,F3,A3) ) ).

% pigeonhole
tff(fact_6798_continuous__inj__imp__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo8458572112393995274pology(A)
        & topolo1944317154257567458pology(B) )
     => ! [A2: A,X: A,B2: A,F3: fun(A,B)] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),X))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),B2))
           => ( topolo81223032696312382ous_on(A,B,set_or1337092689740270186AtMost(A,A2,B2),F3)
             => ( inj_on(A,B,F3,set_or1337092689740270186AtMost(A,A2,B2))
               => ( ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,A2)),aa(A,B,F3,X)))
                    & pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X)),aa(A,B,F3,B2))) )
                  | ( pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,B2)),aa(A,B,F3,X)))
                    & pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X)),aa(A,B,F3,A2))) ) ) ) ) ) ) ) ).

% continuous_inj_imp_mono
tff(fact_6799_the__inv__into__into,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A3: set(A),X: B,B3: set(A)] :
      ( inj_on(A,B,F3,A3)
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),aa(set(A),set(B),image(A,B,F3),A3)))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),the_inv_into(A,B,A3,F3,X)),B3)) ) ) ) ).

% the_inv_into_into
tff(fact_6800_inj__on__UNION__chain,axiom,
    ! [C: $tType,B: $tType,A: $tType,I5: set(A),A3: fun(A,set(B)),F3: fun(B,C)] :
      ( ! [I4: A,J2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I4),I5))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),J2),I5))
           => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A3,I4)),aa(A,set(B),A3,J2)))
              | pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),A3,J2)),aa(A,set(B),A3,I4))) ) ) )
     => ( ! [I4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I4),I5))
           => inj_on(B,C,F3,aa(A,set(B),A3,I4)) )
       => inj_on(B,C,F3,aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),aa(set(A),set(set(B)),image(A,set(B),A3),I5))) ) ) ).

% inj_on_UNION_chain
tff(fact_6801_surjective__iff__injective__gen,axiom,
    ! [B: $tType,A: $tType,S2: set(A),T6: set(B),F3: fun(A,B)] :
      ( pp(aa(set(A),bool,finite_finite(A),S2))
     => ( pp(aa(set(B),bool,finite_finite(B),T6))
       => ( ( aa(set(A),nat,finite_card(A),S2) = aa(set(B),nat,finite_card(B),T6) )
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),S2)),T6))
           => ( ! [X4: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),T6))
                 => ? [Xa4: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),S2))
                      & ( aa(A,B,F3,Xa4) = X4 ) ) )
            <=> inj_on(A,B,F3,S2) ) ) ) ) ) ).

% surjective_iff_injective_gen
tff(fact_6802_card__bij__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A3: set(A),B3: set(B),G: fun(B,A)] :
      ( inj_on(A,B,F3,A3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),A3)),B3))
       => ( inj_on(B,A,G,B3)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,G),B3)),A3))
           => ( pp(aa(set(A),bool,finite_finite(A),A3))
             => ( pp(aa(set(B),bool,finite_finite(B),B3))
               => ( aa(set(A),nat,finite_card(A),A3) = aa(set(B),nat,finite_card(B),B3) ) ) ) ) ) ) ) ).

% card_bij_eq
tff(fact_6803_inj__image__Compl__subset,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A3: set(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),aa(set(A),set(A),uminus_uminus(set(A)),A3))),aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),image(A,B,F3),A3)))) ) ).

% inj_image_Compl_subset
tff(fact_6804_lexl__not__refl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),X: list(A)] :
      ( irrefl(A,R3)
     => ~ pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),X),X)),lex(A,R3))) ) ).

% lexl_not_refl
tff(fact_6805_image__INT,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(A,B),C5: set(A),A3: set(C),B3: fun(C,set(A)),J: C] :
      ( inj_on(A,B,F3,C5)
     => ( ! [X3: C] :
            ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X3),A3))
           => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(C,set(A),B3,X3)),C5)) )
       => ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),J),A3))
         => ( aa(set(A),set(B),image(A,B,F3),aa(set(set(A)),set(A),complete_Inf_Inf(set(A)),aa(set(C),set(set(A)),image(C,set(A),B3),A3))) = aa(set(set(B)),set(B),complete_Inf_Inf(set(B)),aa(set(C),set(set(B)),image(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_yh(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),F3),B3)),A3)) ) ) ) ) ).

% image_INT
tff(fact_6806_inj__on__iff__card__le,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(B),bool,finite_finite(B),B3))
       => ( ? [F6: fun(A,B)] :
              ( inj_on(A,B,F6,A3)
              & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F6),A3)),B3)) )
        <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B3))) ) ) ) ).

% inj_on_iff_card_le
tff(fact_6807_card__inj__on__le,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A3: set(A),B3: set(B)] :
      ( inj_on(A,B,F3,A3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),A3)),B3))
       => ( pp(aa(set(B),bool,finite_finite(B),B3))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B3))) ) ) ) ).

% card_inj_on_le
tff(fact_6808_card__le__inj,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(B),bool,finite_finite(B),B3))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(B),nat,finite_card(B),B3)))
         => ? [F4: fun(A,B)] :
              ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F4),A3)),B3))
              & inj_on(A,B,F4,A3) ) ) ) ) ).

% card_le_inj
tff(fact_6809_log__inj,axiom,
    ! [B2: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),one_one(real)),B2))
     => inj_on(real,real,log(B2),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real))) ) ).

% log_inj
tff(fact_6810_funpow__inj__finite,axiom,
    ! [A: $tType,P2: fun(A,A),X: A] :
      ( inj_on(A,A,P2,top_top(set(A)))
     => ( pp(aa(set(A),bool,finite_finite(A),collect(A,aa(A,fun(A,bool),aTP_Lamp_yi(fun(A,A),fun(A,fun(A,bool)),P2),X))))
       => ~ ! [N2: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N2))
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N2),P2),X) != X ) ) ) ) ).

% funpow_inj_finite
tff(fact_6811_pos__deriv__imp__strict__mono,axiom,
    ! [F3: fun(real,real),F10: fun(real,real)] :
      ( ! [X3: real] : has_field_derivative(real,F3,aa(real,real,F10,X3),topolo174197925503356063within(real,X3,top_top(set(real))))
     => ( ! [X3: real] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(real,real,F10,X3)))
       => order_strict_mono(real,real,F3) ) ) ).

% pos_deriv_imp_strict_mono
tff(fact_6812_Schroeder__Bernstein,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B),A3: set(A),B3: set(B),G: fun(B,A)] :
      ( inj_on(A,B,F3,A3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),A3)),B3))
       => ( inj_on(B,A,G,B3)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,G),B3)),A3))
           => ? [H3: fun(A,B)] : bij_betw(A,B,H3,A3,B3) ) ) ) ) ).

% Schroeder_Bernstein
tff(fact_6813_If__the__inv__into__in__Func,axiom,
    ! [B: $tType,A: $tType,G: fun(A,B),C5: set(A),B3: set(A),X: A] :
      ( inj_on(A,B,G,C5)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C5),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))))
       => pp(aa(set(fun(B,A)),bool,aa(fun(B,A),fun(set(fun(B,A)),bool),member(fun(B,A)),aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_yj(fun(A,B),fun(set(A),fun(A,fun(B,A))),G),C5),X)),bNF_Wellorder_Func(B,A,top_top(set(B)),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),B3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))))) ) ) ).

% If_the_inv_into_in_Func
tff(fact_6814_inj__on__diff__nat,axiom,
    ! [N3: set(nat),K: nat] :
      ( ! [N2: nat] :
          ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),N2),N3))
         => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),N2)) )
     => inj_on(nat,nat,aTP_Lamp_mg(nat,fun(nat,nat),K),N3) ) ).

% inj_on_diff_nat
tff(fact_6815_inj__on__Cons1,axiom,
    ! [A: $tType,X: A,A3: set(list(A))] : inj_on(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),A3) ).

% inj_on_Cons1
tff(fact_6816_inj__Suc,axiom,
    ! [N3: set(nat)] : inj_on(nat,nat,suc,N3) ).

% inj_Suc
tff(fact_6817_inj__on__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N3: set(nat)] : inj_on(nat,A,semiring_1_of_nat(A),N3) ) ).

% inj_on_of_nat
tff(fact_6818_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_6819_le__rel__bool__arg__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X6: fun(bool,A),Y6: fun(bool,A)] :
          ( pp(aa(fun(bool,A),bool,aa(fun(bool,A),fun(fun(bool,A),bool),ord_less_eq(fun(bool,A)),X6),Y6))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,X6,fFalse)),aa(bool,A,Y6,fFalse)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(bool,A,X6,fTrue)),aa(bool,A,Y6,fTrue))) ) ) ) ).

% le_rel_bool_arg_iff
tff(fact_6820_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ? [N2: nat,F4: fun(nat,A)] :
          ( ( A3 = aa(set(nat),set(A),image(nat,A,F4),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ea(nat,fun(nat,bool)),N2))) )
          & inj_on(nat,A,F4,collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ea(nat,fun(nat,bool)),N2))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_6821_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A3: set(A)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ? [F4: fun(A,nat),N2: nat] :
          ( ( aa(set(A),set(nat),image(A,nat,F4),A3) = collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_ea(nat,fun(nat,bool)),N2)) )
          & inj_on(A,nat,F4,A3) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_6822_inj__on__nth,axiom,
    ! [A: $tType,Xs: list(A),I5: set(nat)] :
      ( distinct(A,Xs)
     => ( ! [X3: nat] :
            ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),I5))
           => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X3),aa(list(A),nat,size_size(list(A)),Xs))) )
       => inj_on(nat,A,nth(A,Xs),I5) ) ) ).

% inj_on_nth
tff(fact_6823_infinite__countable__subset,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite(A),S2))
     => ? [F4: fun(nat,A)] :
          ( inj_on(nat,A,F4,top_top(set(nat)))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(nat),set(A),image(nat,A,F4),top_top(set(nat)))),S2)) ) ) ).

% infinite_countable_subset
tff(fact_6824_infinite__iff__countable__subset,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite(A),S2))
    <=> ? [F6: fun(nat,A)] :
          ( inj_on(nat,A,F6,top_top(set(nat)))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(nat),set(A),image(nat,A,F6),top_top(set(nat)))),S2)) ) ) ).

% infinite_iff_countable_subset
tff(fact_6825_inj__on__funpow__least,axiom,
    ! [A: $tType,N: nat,F3: 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)),N),F3),S) = S )
     => ( ! [M5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),M5))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M5),N))
             => ( aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M5),F3),S) != S ) ) )
       => inj_on(nat,A,aa(A,fun(nat,A),aTP_Lamp_yk(fun(A,A),fun(A,fun(nat,A)),F3),S),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ) ).

% inj_on_funpow_least
tff(fact_6826_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),aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,aa(num,num,bit0,one2))))))))))) ).

% inj_on_char_of_nat
tff(fact_6827_ex__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),S2: set(B),P: fun(set(A),bool)] :
      ( ? [T9: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T9),aa(set(B),set(A),image(B,A,F3),S2)))
          & pp(aa(set(A),bool,P,T9)) )
    <=> ? [T9: set(B)] :
          ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T9),S2))
          & inj_on(B,A,F3,T9)
          & pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F3),T9))) ) ) ).

% ex_subset_image_inj
tff(fact_6828_all__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),S2: set(B),P: fun(set(A),bool)] :
      ( ! [T9: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T9),aa(set(B),set(A),image(B,A,F3),S2)))
         => pp(aa(set(A),bool,P,T9)) )
    <=> ! [T9: set(B)] :
          ( ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T9),S2))
            & inj_on(B,A,F3,T9) )
         => pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F3),T9))) ) ) ).

% all_subset_image_inj
tff(fact_6829_Func__map__surj,axiom,
    ! [C: $tType,A: $tType,D: $tType,B: $tType,F1: fun(B,A),A13: set(B),B14: set(A),F2: fun(C,D),B23: set(C),A24: set(D)] :
      ( ( aa(set(B),set(A),image(B,A,F1),A13) = B14 )
     => ( inj_on(C,D,F2,B23)
       => ( pp(aa(set(D),bool,aa(set(D),fun(set(D),bool),ord_less_eq(set(D)),aa(set(C),set(D),image(C,D,F2),B23)),A24))
         => ( ( ( B23 = bot_bot(set(C)) )
             => ( A24 = bot_bot(set(D)) ) )
           => ( bNF_Wellorder_Func(C,A,B23,B14) = aa(set(fun(D,B)),set(fun(C,A)),image(fun(D,B),fun(C,A),bNF_We4925052301507509544nc_map(C,B,A,D,B23,F1,F2)),bNF_Wellorder_Func(D,B,A24,A13)) ) ) ) ) ) ).

% Func_map_surj
tff(fact_6830_suminf__reindex,axiom,
    ! [F3: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F3)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,F3,X3)))
         => ( ! [X3: nat] :
                ( ~ pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),X3),aa(set(nat),set(nat),image(nat,nat,G),top_top(set(nat)))))
               => ( aa(nat,real,F3,X3) = zero_zero(real) ) )
           => ( suminf(real,aa(fun(nat,nat),fun(nat,real),comp(nat,real,nat,F3),G)) = suminf(real,F3) ) ) ) ) ) ).

% suminf_reindex
tff(fact_6831_comp__funpow,axiom,
    ! [B: $tType,A: $tType,N: nat,F3: fun(A,A)] : aa(fun(fun(B,A),fun(B,A)),fun(fun(B,A),fun(B,A)),aa(nat,fun(fun(fun(B,A),fun(B,A)),fun(fun(B,A),fun(B,A))),compow(fun(fun(B,A),fun(B,A))),N),comp(A,A,B,F3)) = comp(A,A,B,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3)) ).

% comp_funpow
tff(fact_6832_funpow__add,axiom,
    ! [A: $tType,M: nat,N: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),F3) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),M),F3)),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3)) ).

% funpow_add
tff(fact_6833_funpow__Suc__right,axiom,
    ! [A: $tType,N: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F3) = aa(fun(A,A),fun(A,A),comp(A,A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3)),F3) ).

% funpow_Suc_right
tff(fact_6834_funpow_Osimps_I2_J,axiom,
    ! [A: $tType,N: nat,F3: fun(A,A)] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),aa(nat,nat,suc,N)),F3) = aa(fun(A,A),fun(A,A),comp(A,A,A,F3),aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),F3)) ).

% funpow.simps(2)
tff(fact_6835_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) )
         => ( ! [X3: B,Y4: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X3),Y4)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X3)),aa(B,A,H,Y4))
           => ( aa(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,groups7311177749621191930dd_sum(C,B,G),A3)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_6836_bij__betw__comp__iff2,axiom,
    ! [C: $tType,A: $tType,B: $tType,F10: fun(A,B),A8: set(A),A14: set(B),F3: fun(C,A),A3: set(C)] :
      ( bij_betw(A,B,F10,A8,A14)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,F3),A3)),A8))
       => ( bij_betw(C,A,F3,A3,A8)
        <=> bij_betw(C,B,aa(fun(C,A),fun(C,B),comp(A,B,C,F10),F3),A3,A14) ) ) ) ).

% bij_betw_comp_iff2
tff(fact_6837_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.atLeast_Suc_atMost_Suc_shift
tff(fact_6838_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.atLeast_Suc_lessThan_Suc_shift
tff(fact_6839_sum_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(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,M,N)) ) ).

% sum.atLeastAtMost_shift_bounds
tff(fact_6840_sum_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(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,M,N)) ) ).

% sum.atLeastLessThan_shift_bounds
tff(fact_6841_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: 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,M),aa(nat,nat,suc,N))) = 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,M,N)) ) ).

% prod.atLeast_Suc_atMost_Suc_shift
tff(fact_6842_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: 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,M),aa(nat,nat,suc,N))) = 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,M,N)) ) ).

% prod.atLeast_Suc_lessThan_Suc_shift
tff(fact_6843_prod_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: 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),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),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,M,N)) ) ).

% prod.atLeastAtMost_shift_bounds
tff(fact_6844_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: 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),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),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,M,N)) ) ).

% prod.atLeastLessThan_shift_bounds
tff(fact_6845_summable__inverse__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(nat,A),C2: A] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),comp(A,A,nat,inverse_inverse(A)),F3))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_yl(fun(nat,A),fun(A,fun(nat,A)),F3),C2)) ) ) ).

% summable_inverse_divide
tff(fact_6846_prod_Oreindex__nontrivial,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B),H: fun(B,C),G: fun(C,A)] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ! [X3: B,Y4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
               => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y4),A3))
                 => ( ( X3 != Y4 )
                   => ( ( aa(B,C,H,X3) = aa(B,C,H,Y4) )
                     => ( aa(C,A,G,aa(B,C,H,X3)) = one_one(A) ) ) ) ) )
           => ( aa(set(C),A,aa(fun(C,A),fun(set(C),A),groups7121269368397514597t_prod(C,A),G),aa(set(B),set(C),image(B,C,H),A3)) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),aa(fun(B,C),fun(B,A),comp(C,A,B,G),H)),A3) ) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_6847_DERIV__at__within__shift__lemma,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F3: fun(A,A),Y: A,Z: A,X: A,S2: set(A)] :
          ( has_field_derivative(A,F3,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),X),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),Z)),S2)))
         => has_field_derivative(A,aa(fun(A,A),fun(A,A),comp(A,A,A,F3),aa(A,fun(A,A),plus_plus(A),Z)),Y,topolo174197925503356063within(A,X,S2)) ) ) ).

% DERIV_at_within_shift_lemma
tff(fact_6848_sum__image__le,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(C),G: fun(A,B),F3: fun(C,A)] :
          ( pp(aa(set(C),bool,finite_finite(C),I5))
         => ( ! [I4: C] :
                ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),I4),I5))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),zero_zero(B)),aa(A,B,G,aa(C,A,F3,I4)))) )
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(C),set(A),image(C,A,F3),I5))),aa(set(C),B,groups7311177749621191930dd_sum(C,B,aa(fun(C,A),fun(C,B),comp(A,B,C,G),F3)),I5))) ) ) ) ).

% sum_image_le
tff(fact_6849_filterlim__shift__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(A,B),D2: A,F5: filter(B),A2: A] :
          ( filterlim(A,B,aa(fun(A,A),fun(A,B),comp(A,B,A,F3),aa(A,fun(A,A),plus_plus(A),D2)),F5,topolo174197925503356063within(A,aa(A,A,minus_minus(A,A2),D2),top_top(set(A))))
        <=> filterlim(A,B,F3,F5,topolo174197925503356063within(A,A2,top_top(set(A)))) ) ) ).

% filterlim_shift_iff
tff(fact_6850_filterlim__shift,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F3: fun(A,B),F5: filter(B),A2: A,D2: A] :
          ( filterlim(A,B,F3,F5,topolo174197925503356063within(A,A2,top_top(set(A))))
         => filterlim(A,B,aa(fun(A,A),fun(A,B),comp(A,B,A,F3),aa(A,fun(A,A),plus_plus(A),D2)),F5,topolo174197925503356063within(A,aa(A,A,minus_minus(A,A2),D2),top_top(set(A)))) ) ) ).

% filterlim_shift
tff(fact_6851_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_6852_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_6853_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: 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,N))) = 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),N))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_6854_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: 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,N))) = 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),N))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_6855_Func__map,axiom,
    ! [A: $tType,B: $tType,D: $tType,C: $tType,G: fun(A,B),A24: set(A),A13: set(B),F1: fun(B,C),B14: set(C),F2: fun(D,A),B23: set(D)] :
      ( pp(aa(set(fun(A,B)),bool,aa(fun(A,B),fun(set(fun(A,B)),bool),member(fun(A,B)),G),bNF_Wellorder_Func(A,B,A24,A13)))
     => ( pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),aa(set(B),set(C),image(B,C,F1),A13)),B14))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(D),set(A),image(D,A,F2),B23)),A24))
         => pp(aa(set(fun(D,C)),bool,aa(fun(D,C),fun(set(fun(D,C)),bool),member(fun(D,C)),aa(fun(A,B),fun(D,C),bNF_We4925052301507509544nc_map(D,B,C,A,B23,F1,F2),G)),bNF_Wellorder_Func(D,C,B23,B14))) ) ) ) ).

% Func_map
tff(fact_6856_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N)) = aa(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),M))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,N),M))) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_6857_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) = 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),M))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,N),M))) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_6858_sum_OatLeastAtMost__reindex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_add(A)
        & ord(B) )
     => ! [H: fun(nat,B),M: nat,N: nat,G: fun(B,A)] :
          ( bij_betw(nat,B,H,set_or1337092689740270186AtMost(nat,M,N),set_or1337092689740270186AtMost(B,aa(nat,B,H,M),aa(nat,B,H,N)))
         => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),set_or1337092689740270186AtMost(B,aa(nat,B,H,M),aa(nat,B,H,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,B),fun(nat,A),comp(B,A,nat,G),H)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ) ).

% sum.atLeastAtMost_reindex
tff(fact_6859_sum_OatLeastLessThan__reindex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_add(A)
        & ord(B) )
     => ! [H: fun(nat,B),M: nat,N: nat,G: fun(B,A)] :
          ( bij_betw(nat,B,H,set_or7035219750837199246ssThan(nat,M,N),set_or7035219750837199246ssThan(B,aa(nat,B,H,M),aa(nat,B,H,N)))
         => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),set_or7035219750837199246ssThan(B,aa(nat,B,H,M),aa(nat,B,H,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,B),fun(nat,A),comp(B,A,nat,G),H)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ) ).

% sum.atLeastLessThan_reindex
tff(fact_6860_prod_OatLeastAtMost__reindex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(A)
        & ord(B) )
     => ! [H: fun(nat,B),M: nat,N: nat,G: fun(B,A)] :
          ( bij_betw(nat,B,H,set_or1337092689740270186AtMost(nat,M,N),set_or1337092689740270186AtMost(B,aa(nat,B,H,M),aa(nat,B,H,N)))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),set_or1337092689740270186AtMost(B,aa(nat,B,H,M),aa(nat,B,H,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,B),fun(nat,A),comp(B,A,nat,G),H)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ) ).

% prod.atLeastAtMost_reindex
tff(fact_6861_prod_OatLeastLessThan__reindex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(A)
        & ord(B) )
     => ! [H: fun(nat,B),M: nat,N: nat,G: fun(B,A)] :
          ( bij_betw(nat,B,H,set_or7035219750837199246ssThan(nat,M,N),set_or7035219750837199246ssThan(B,aa(nat,B,H,M),aa(nat,B,H,N)))
         => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),set_or7035219750837199246ssThan(B,aa(nat,B,H,M),aa(nat,B,H,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),aa(fun(nat,B),fun(nat,A),comp(B,A,nat,G),H)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ) ).

% prod.atLeastLessThan_reindex
tff(fact_6862_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_ym(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_6863_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,nat),fun(nat,A),comp(nat,A,nat,G),aTP_Lamp_ym(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_6864_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: 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_ym(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_6865_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: 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_ym(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_6866_summable__reindex,axiom,
    ! [F3: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F3)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,F3,X3)))
         => summable(real,aa(fun(nat,nat),fun(nat,real),comp(nat,real,nat,F3),G)) ) ) ) ).

% summable_reindex
tff(fact_6867_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N)) = aa(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),M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,N),M))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_6868_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
         => ( aa(set(nat),A,aa(fun(nat,A),fun(set(nat),A),groups7121269368397514597t_prod(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) = 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),M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,N),M))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_6869_suminf__reindex__mono,axiom,
    ! [F3: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F3)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X3: nat] : pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(nat,real,F3,X3)))
         => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),suminf(real,aa(fun(nat,nat),fun(nat,real),comp(nat,real,nat,F3),G))),suminf(real,F3))) ) ) ) ).

% suminf_reindex_mono
tff(fact_6870_map__filter__on__comp,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),Y6: set(B),X6: set(A),F5: filter(B),F3: fun(A,C)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,G),Y6)),X6))
     => ( eventually(B,aTP_Lamp_yn(set(B),fun(B,bool),Y6),F5)
       => ( map_filter_on(A,C,X6,F3,map_filter_on(B,A,Y6,G,F5)) = map_filter_on(B,C,Y6,aa(fun(B,A),fun(B,C),comp(A,C,B,F3),G),F5) ) ) ) ).

% map_filter_on_comp
tff(fact_6871_rtrancl__finite__eq__relpow,axiom,
    ! [A: $tType,R: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,finite_finite(product_prod(A,A)),R))
     => ( transitive_rtrancl(A,R) = aa(set(set(product_prod(A,A))),set(product_prod(A,A)),complete_Sup_Sup(set(product_prod(A,A))),aa(set(nat),set(set(product_prod(A,A))),image(nat,set(product_prod(A,A)),aTP_Lamp_wd(set(product_prod(A,A)),fun(nat,set(product_prod(A,A))),R)),collect(nat,aTP_Lamp_wf(set(product_prod(A,A)),fun(nat,bool),R)))) ) ) ).

% rtrancl_finite_eq_relpow
tff(fact_6872_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_6873_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,X: A] : aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ma(A,A)),Y)),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ma(A,A)),X)) = aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ma(A,A)),X)),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ma(A,A)),Y)) ) ).

% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
tff(fact_6874_rtrancl__listrel1__ConsI2,axiom,
    ! [A: $tType,X: A,Y: A,R3: set(product_prod(A,A)),Xs: list(A),Ys: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X),Y)),transitive_rtrancl(A,R3)))
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),transitive_rtrancl(list(A),listrel1(A,R3))))
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))),transitive_rtrancl(list(A),listrel1(A,R3)))) ) ) ).

% rtrancl_listrel1_ConsI2
tff(fact_6875_rtrancl__Un__subset,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S2: set(product_prod(A,A))] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),transitive_rtrancl(A,R)),transitive_rtrancl(A,S2))),transitive_rtrancl(A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),sup_sup(set(product_prod(A,A))),R),S2)))) ).

% rtrancl_Un_subset
tff(fact_6876_rtrancl__mono,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_rtrancl(A,R3)),transitive_rtrancl(A,S))) ) ).

% rtrancl_mono
tff(fact_6877_rtrancl__subset,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),S2))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),S2),transitive_rtrancl(A,R)))
       => ( transitive_rtrancl(A,S2) = transitive_rtrancl(A,R) ) ) ) ).

% rtrancl_subset
tff(fact_6878_rtrancl__subset__rtrancl,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),transitive_rtrancl(A,S)))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_rtrancl(A,R3)),transitive_rtrancl(A,S))) ) ).

% rtrancl_subset_rtrancl
tff(fact_6879_listrel1__rtrancl__subset__rtrancl__listrel1,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,transitive_rtrancl(A,R3))),transitive_rtrancl(list(A),listrel1(A,R3)))) ).

% listrel1_rtrancl_subset_rtrancl_listrel1
tff(fact_6880_bit__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A2: A] : bit_se5641148757651400278ts_bit(A,aa(A,A,bit_se4197421643247451524op_bit(A,N),A2)) = aa(fun(nat,nat),fun(nat,bool),comp(nat,bool,nat,bit_se5641148757651400278ts_bit(A,A2)),aa(nat,fun(nat,nat),plus_plus(nat),N)) ) ).

% bit_drop_bit_eq
tff(fact_6881_rtrancl__listrel1__ConsI1,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A)),X: A] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),transitive_rtrancl(list(A),listrel1(A,R3))))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))),transitive_rtrancl(list(A),listrel1(A,R3)))) ) ).

% rtrancl_listrel1_ConsI1
tff(fact_6882_rtrancl__listrel1__eq__len,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),X),Y)),transitive_rtrancl(list(A),listrel1(A,R3))))
     => ( aa(list(A),nat,size_size(list(A)),X) = aa(list(A),nat,size_size(list(A)),Y) ) ) ).

% rtrancl_listrel1_eq_len
tff(fact_6883_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(set(product_prod(nat,nat)),bool,aa(product_prod(nat,nat),fun(set(product_prod(nat,nat)),bool),member(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,M),N)),transitive_rtrancl(nat,pred_nat)))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N)) ) ).

% pred_nat_trancl_eq_le
tff(fact_6884_prod_OUnion__comp,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [B3: set(set(B)),G: fun(B,A)] :
          ( ! [X3: set(B)] :
              ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),X3),B3))
             => pp(aa(set(B),bool,finite_finite(B),X3)) )
         => ( ! [A15: set(B)] :
                ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),A15),B3))
               => ! [A25: set(B)] :
                    ( pp(aa(set(set(B)),bool,aa(set(B),fun(set(set(B)),bool),member(set(B)),A25),B3))
                   => ( ( A15 != A25 )
                     => ! [X3: B] :
                          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A15))
                         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A25))
                           => ( aa(B,A,G,X3) = one_one(A) ) ) ) ) ) )
           => ( aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),G),aa(set(set(B)),set(B),complete_Sup_Sup(set(B)),B3)) = aa(set(set(B)),A,aa(fun(B,A),fun(set(set(B)),A),aa(fun(fun(B,A),fun(set(B),A)),fun(fun(B,A),fun(set(set(B)),A)),comp(fun(set(B),A),fun(set(set(B)),A),fun(B,A),groups7121269368397514597t_prod(set(B),A)),groups7121269368397514597t_prod(B,A)),G),B3) ) ) ) ) ).

% prod.Union_comp
tff(fact_6885_sum_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(int,A),M: nat,N: nat] : aa(set(int),A,groups7311177749621191930dd_sum(int,A,G),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = aa(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,M,N)) ) ).

% sum.atLeast_int_atMost_int_shift
tff(fact_6886_prod_OatLeast__int__atMost__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(int,A),M: nat,N: 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),M),aa(nat,int,semiring_1_of_nat(int),N))) = 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,M,N)) ) ).

% prod.atLeast_int_atMost_int_shift
tff(fact_6887_sum_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(int,A),M: nat,N: nat] : aa(set(int),A,groups7311177749621191930dd_sum(int,A,G),set_or7035219750837199246ssThan(int,aa(nat,int,semiring_1_of_nat(int),M),aa(nat,int,semiring_1_of_nat(int),N))) = aa(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,M,N)) ) ).

% sum.atLeast_int_lessThan_int_shift
tff(fact_6888_prod_OatLeast__int__lessThan__int__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(int,A),M: nat,N: 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),M),aa(nat,int,semiring_1_of_nat(int),N))) = 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,M,N)) ) ).

% prod.atLeast_int_lessThan_int_shift
tff(fact_6889_filterlim__at__top__iff__inverse__0,axiom,
    ! [A: $tType,F3: fun(A,real),F5: filter(A)] :
      ( eventually(A,aTP_Lamp_um(fun(A,real),fun(A,bool),F3),F5)
     => ( filterlim(A,real,F3,at_top(real),F5)
      <=> filterlim(A,real,aa(fun(A,real),fun(A,real),comp(real,real,A,inverse_inverse(real)),F3),topolo7230453075368039082e_nhds(real,zero_zero(real)),F5) ) ) ).

% filterlim_at_top_iff_inverse_0
tff(fact_6890_map__option__o__empty,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,B),X2: A] : aa(A,option(B),aa(fun(A,option(C)),fun(A,option(B)),comp(option(C),option(B),A,map_option(C,B,F3)),aTP_Lamp_yo(A,option(C))),X2) = none(B) ).

% map_option_o_empty
tff(fact_6891_set__list__bind,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),F3: fun(B,list(A))] : aa(list(A),set(A),set2(A),bind(B,A,Xs,F3)) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),aTP_Lamp_yp(fun(B,list(A)),fun(B,set(A)),F3)),aa(list(B),set(B),set2(B),Xs))) ).

% set_list_bind
tff(fact_6892_bind__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,list(A))] : bind(B,A,nil(B),F3) = nil(A) ).

% bind_simps(1)
tff(fact_6893_bind__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,X: B,Xs: list(B),F3: fun(B,list(A))] : bind(B,A,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs),F3) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(B,list(A),F3,X)),bind(B,A,Xs,F3)) ).

% bind_simps(2)
tff(fact_6894_list__bind__cong,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(A),F3: fun(A,list(B)),G: fun(A,list(B))] :
      ( ( Xs = Ys )
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
           => ( aa(A,list(B),F3,X3) = aa(A,list(B),G,X3) ) )
       => ( bind(A,B,Xs,F3) = bind(A,B,Ys,G) ) ) ) ).

% list_bind_cong
tff(fact_6895_at__right__eq,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Y))
         => ( topolo174197925503356063within(A,X,aa(A,set(A),set_ord_greaterThan(A),X)) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image(A,filter(A),aTP_Lamp_yq(A,fun(A,filter(A)),X)),aa(A,set(A),set_ord_greaterThan(A),X))) ) ) ) ).

% at_right_eq
tff(fact_6896_at__left__eq,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),X))
         => ( topolo174197925503356063within(A,X,aa(A,set(A),set_ord_lessThan(A),X)) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(A),set(filter(A)),image(A,filter(A),aTP_Lamp_yr(A,fun(A,filter(A)),X)),aa(A,set(A),set_ord_lessThan(A),X))) ) ) ) ).

% at_left_eq
tff(fact_6897_principal__le__iff,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),principal(A,A3)),principal(A,B3)))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3)) ) ).

% principal_le_iff
tff(fact_6898_le__principal,axiom,
    ! [A: $tType,F5: filter(A),A3: set(A)] :
      ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F5),principal(A,A3)))
    <=> eventually(A,aTP_Lamp_a(set(A),fun(A,bool),A3),F5) ) ).

% le_principal
tff(fact_6899_filterlim__base__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,I5: set(A),F5: fun(A,set(B)),F3: fun(B,C),G4: fun(D,set(C)),J4: set(D)] :
      ( ( I5 != bot_bot(set(A)) )
     => ( ! [I4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),I4),I5))
           => ! [J2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),J2),I5))
               => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F5,I4)),aa(A,set(B),F5,J2)))
                  | pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),F5,J2)),aa(A,set(B),F5,I4))) ) ) )
       => ( filterlim(B,C,F3,aa(set(filter(C)),filter(C),complete_Inf_Inf(filter(C)),aa(set(D),set(filter(C)),image(D,filter(C),aTP_Lamp_ys(fun(D,set(C)),fun(D,filter(C)),G4)),J4)),aa(set(filter(B)),filter(B),complete_Inf_Inf(filter(B)),aa(set(A),set(filter(B)),image(A,filter(B),aTP_Lamp_yt(fun(A,set(B)),fun(A,filter(B)),F5)),I5)))
        <=> ! [X4: D] :
              ( pp(aa(set(D),bool,aa(D,fun(set(D),bool),member(D),X4),J4))
             => ? [Xa4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),I5))
                  & ! [Xb4: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Xb4),aa(A,set(B),F5,Xa4)))
                     => pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),aa(B,C,F3,Xb4)),aa(D,set(C),G4,X4))) ) ) ) ) ) ) ).

% filterlim_base_iff
tff(fact_6900_at__infinity__def,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( at_infinity(A) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(real),set(filter(A)),image(real,filter(A),aTP_Lamp_yv(real,filter(A))),top_top(set(real)))) ) ) ).

% at_infinity_def
tff(fact_6901_nhds__metric,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [X: A] : topolo7230453075368039082e_nhds(A,X) = aa(set(filter(A)),filter(A),complete_Inf_Inf(filter(A)),aa(set(real),set(filter(A)),image(real,filter(A),aTP_Lamp_yx(A,fun(real,filter(A)),X)),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ).

% nhds_metric
tff(fact_6902_complete__uniform,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [S2: set(A)] :
          ( topolo2479028161051973599mplete(A,S2)
        <=> ! [F15: filter(A)] :
              ( pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F15),principal(A,S2)))
             => ( ( F15 != bot_bot(filter(A)) )
               => ( topolo6773858410816713723filter(A,F15)
                 => ? [X4: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
                      & pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),F15),topolo7230453075368039082e_nhds(A,X4))) ) ) ) ) ) ) ).

% complete_uniform
tff(fact_6903_uniformity__dist,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ( topolo7806501430040627800ormity(A) = aa(set(filter(product_prod(A,A))),filter(product_prod(A,A)),complete_Inf_Inf(filter(product_prod(A,A))),aa(set(real),set(filter(product_prod(A,A))),image(real,filter(product_prod(A,A)),aTP_Lamp_yz(real,filter(product_prod(A,A)))),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ) ) ).

% uniformity_dist
tff(fact_6904_Cauchy__uniform__iff,axiom,
    ! [A: $tType] :
      ( topolo7287701948861334536_space(A)
     => ! [X6: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X6)
        <=> ! [P6: fun(product_prod(A,A),bool)] :
              ( eventually(product_prod(A,A),P6,topolo7806501430040627800ormity(A))
             => ? [N6: nat] :
                ! [N5: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),N5))
                 => ! [M7: nat] :
                      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N6),M7))
                     => pp(aa(product_prod(A,A),bool,P6,aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,X6,N5)),aa(nat,A,X6,M7)))) ) ) ) ) ) ).

% Cauchy_uniform_iff
tff(fact_6905_uniformity__real__def,axiom,
    topolo7806501430040627800ormity(real) = aa(set(filter(product_prod(real,real))),filter(product_prod(real,real)),complete_Inf_Inf(filter(product_prod(real,real))),aa(set(real),set(filter(product_prod(real,real))),image(real,filter(product_prod(real,real)),aTP_Lamp_zb(real,filter(product_prod(real,real)))),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ).

% uniformity_real_def
tff(fact_6906_uniformity__complex__def,axiom,
    topolo7806501430040627800ormity(complex) = aa(set(filter(product_prod(complex,complex))),filter(product_prod(complex,complex)),complete_Inf_Inf(filter(product_prod(complex,complex))),aa(set(real),set(filter(product_prod(complex,complex))),image(real,filter(product_prod(complex,complex)),aTP_Lamp_zd(real,filter(product_prod(complex,complex)))),aa(real,set(real),set_ord_greaterThan(real),zero_zero(real)))) ).

% uniformity_complex_def
tff(fact_6907_eventually__uniformity__metric,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [P: fun(product_prod(A,A),bool)] :
          ( eventually(product_prod(A,A),P,topolo7806501430040627800ormity(A))
        <=> ? [E3: real] :
              ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),E3))
              & ! [X4: A,Y5: A] :
                  ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,X4,Y5)),E3))
                 => pp(aa(product_prod(A,A),bool,P,aa(A,product_prod(A,A),product_Pair(A,A,X4),Y5))) ) ) ) ) ).

% eventually_uniformity_metric
tff(fact_6908_these__insert__None,axiom,
    ! [A: $tType,A3: set(option(A))] : these(A,aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),A3)) = these(A,A3) ).

% these_insert_None
tff(fact_6909_Nats__altdef1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( semiring_1_Nats(A) = collect(A,aTP_Lamp_ze(A,bool)) ) ) ).

% Nats_altdef1
tff(fact_6910_Nats__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),semiring_1_Nats(A))) ) ).

% Nats_0
tff(fact_6911_Nats__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),semiring_1_Nats(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),semiring_1_Nats(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),semiring_1_Nats(A))) ) ) ) ).

% Nats_add
tff(fact_6912_Nats__1,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),one_one(A)),semiring_1_Nats(A))) ) ).

% Nats_1
tff(fact_6913_Nats__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),semiring_1_Nats(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),semiring_1_Nats(A)))
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),semiring_1_Nats(A))) ) ) ) ).

% Nats_mult
tff(fact_6914_Nats__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [W: num] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(num,A,numeral_numeral(A),W)),semiring_1_Nats(A))) ) ).

% Nats_numeral
tff(fact_6915_Nats__cases,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [X: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),semiring_1_Nats(A)))
         => ~ ! [N2: nat] : X != aa(nat,A,semiring_1_of_nat(A),N2) ) ) ).

% Nats_cases
tff(fact_6916_Nats__induct,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [X: A,P: fun(A,bool)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),semiring_1_Nats(A)))
         => ( ! [N2: nat] : pp(aa(A,bool,P,aa(nat,A,semiring_1_of_nat(A),N2)))
           => pp(aa(A,bool,P,X)) ) ) ) ).

% Nats_induct
tff(fact_6917_of__nat__in__Nats,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(nat,A,semiring_1_of_nat(A),N)),semiring_1_Nats(A))) ) ).

% of_nat_in_Nats
tff(fact_6918_Nats__diff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),semiring_1_Nats(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B2),semiring_1_Nats(A)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,minus_minus(A,A2),B2)),semiring_1_Nats(A))) ) ) ) ) ).

% Nats_diff
tff(fact_6919_Nats__subset__Ints,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),semiring_1_Nats(A)),ring_1_Ints(A))) ) ).

% Nats_subset_Ints
tff(fact_6920_Nats__subset__Rats,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),semiring_1_Nats(A)),field_char_0_Rats(A))) ) ).

% Nats_subset_Rats
tff(fact_6921_Nats__def,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( semiring_1_Nats(A) = aa(set(nat),set(A),image(nat,A,semiring_1_of_nat(A)),top_top(set(nat))) ) ) ).

% Nats_def
tff(fact_6922_Nats__altdef2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( semiring_1_Nats(A) = collect(A,aTP_Lamp_zf(A,bool)) ) ) ).

% Nats_altdef2
tff(fact_6923_these__not__empty__eq,axiom,
    ! [A: $tType,B3: set(option(A))] :
      ( ( these(A,B3) != bot_bot(set(A)) )
    <=> ( ( B3 != bot_bot(set(option(A))) )
        & ( B3 != aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_not_empty_eq
tff(fact_6924_these__empty__eq,axiom,
    ! [A: $tType,B3: set(option(A))] :
      ( ( these(A,B3) = bot_bot(set(A)) )
    <=> ( ( B3 = bot_bot(set(option(A))) )
        | ( B3 = aa(set(option(A)),set(option(A)),aa(option(A),fun(set(option(A)),set(option(A))),insert(option(A)),none(A)),bot_bot(set(option(A)))) ) ) ) ).

% these_empty_eq
tff(fact_6925_Some__image__these__eq,axiom,
    ! [A: $tType,A3: set(option(A))] : aa(set(A),set(option(A)),image(A,option(A),some(A)),these(A,A3)) = collect(option(A),aTP_Lamp_zg(set(option(A)),fun(option(A),bool),A3)) ).

% Some_image_these_eq
tff(fact_6926_rat__floor__lemma,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),ring_1_of_int(rat,divide_divide(int,A2,B2))),fract(A2,B2)))
      & pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),fract(A2,B2)),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_6927_sorted__key__list__of__set__def,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A)] : linord144544945434240204of_set(B,A,F3) = finite_folding_F(B,list(B),linorder_insort_key(B,A,F3),nil(B)) ) ).

% sorted_key_list_of_set_def
tff(fact_6928_less__rat,axiom,
    ! [B2: int,D2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D2 != zero_zero(int) )
       => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),fract(A2,B2)),fract(C2,D2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),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),D2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D2))),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),D2)))) ) ) ) ).

% less_rat
tff(fact_6929_le__rat,axiom,
    ! [B2: int,D2: int,A2: int,C2: int] :
      ( ( B2 != zero_zero(int) )
     => ( ( D2 != zero_zero(int) )
       => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),fract(A2,B2)),fract(C2,D2)))
        <=> pp(aa(int,bool,aa(int,fun(int,bool),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),D2)),aa(int,int,aa(int,fun(int,int),times_times(int),B2),D2))),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),D2)))) ) ) ) ).

% le_rat
tff(fact_6930_Rat__induct__pos,axiom,
    ! [P: fun(rat,bool),Q3: rat] :
      ( ! [A5: int,B5: int] :
          ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B5))
         => pp(aa(rat,bool,P,fract(A5,B5))) )
     => pp(aa(rat,bool,P,Q3)) ) ).

% Rat_induct_pos
tff(fact_6931_Fract__less__zero__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),fract(A2,B2)),zero_zero(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),zero_zero(int))) ) ) ).

% Fract_less_zero_iff
tff(fact_6932_zero__less__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),fract(A2,B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),A2)) ) ) ).

% zero_less_Fract_iff
tff(fact_6933_Fract__less__one__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),fract(A2,B2)),one_one(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),A2),B2)) ) ) ).

% Fract_less_one_iff
tff(fact_6934_one__less__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),one_one(rat)),fract(A2,B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),B2),A2)) ) ) ).

% one_less_Fract_iff
tff(fact_6935_Fract__le__zero__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),fract(A2,B2)),zero_zero(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),zero_zero(int))) ) ) ).

% Fract_le_zero_iff
tff(fact_6936_zero__le__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),fract(A2,B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),A2)) ) ) ).

% zero_le_Fract_iff
tff(fact_6937_Fract__le__one__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),fract(A2,B2)),one_one(rat)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),A2),B2)) ) ) ).

% Fract_le_one_iff
tff(fact_6938_one__le__Fract__iff,axiom,
    ! [B2: int,A2: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),B2))
     => ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),fract(A2,B2)))
      <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),B2),A2)) ) ) ).

% one_le_Fract_iff
tff(fact_6939_folding__on_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),X: A,A3: set(A),Z: B] :
      ( finite_folding_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)),S2))
       => ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( aa(set(A),B,finite_folding_F(A,B,F3,Z),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(B,B,aa(A,fun(B,B),F3,X),aa(set(A),B,finite_folding_F(A,B,F3,Z),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ).

% folding_on.insert_remove
tff(fact_6940_folding__on_Oremove,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),A3: set(A),X: A,Z: B] :
      ( finite_folding_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),S2))
       => ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( aa(set(A),B,finite_folding_F(A,B,F3,Z),A3) = aa(B,B,aa(A,fun(B,B),F3,X),aa(set(A),B,finite_folding_F(A,B,F3,Z),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% folding_on.remove
tff(fact_6941_card_Ofolding__on__axioms,axiom,
    ! [A: $tType] : finite_folding_on(A,nat,top_top(set(A)),aTP_Lamp_zh(A,fun(nat,nat))) ).

% card.folding_on_axioms
tff(fact_6942_sorted__list__of__set_Ofold__insort__key_Ofolding__on__axioms,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => finite_folding_on(A,list(A),top_top(set(A)),linorder_insort_key(A,A,aTP_Lamp_ma(A,A))) ) ).

% sorted_list_of_set.fold_insort_key.folding_on_axioms
tff(fact_6943_card__def,axiom,
    ! [B: $tType] : finite_card(B) = finite_folding_F(B,nat,aTP_Lamp_zi(B,fun(nat,nat)),zero_zero(nat)) ).

% card_def
tff(fact_6944_folding__on_Oinsert,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),X: A,A3: set(A),Z: B] :
      ( finite_folding_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)),S2))
       => ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( aa(set(A),B,finite_folding_F(A,B,F3,Z),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(B,B,aa(A,fun(B,B),F3,X),aa(set(A),B,finite_folding_F(A,B,F3,Z),A3)) ) ) ) ) ) ).

% folding_on.insert
tff(fact_6945_folding__idem__on_Oinsert__idem,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),X: A,A3: set(A),Z: B] :
      ( finite1890593828518410140dem_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)),S2))
       => ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( aa(set(A),B,finite_folding_F(A,B,F3,Z),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(B,B,aa(A,fun(B,B),F3,X),aa(set(A),B,finite_folding_F(A,B,F3,Z),A3)) ) ) ) ) ).

% folding_idem_on.insert_idem
tff(fact_6946_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S2: set(A),F3: fun(A,fun(B,B)),G: fun(C,A),R: set(C)] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,G),top_top(set(C)))),S2))
       => finite673082921795544331dem_on(C,B,R,aa(fun(C,A),fun(C,fun(B,B)),comp(A,fun(B,B),C,F3),G)) ) ) ).

% comp_fun_idem_on.comp_comp_fun_idem_on
tff(fact_6947_positive__rat,axiom,
    ! [A2: int,B2: int] :
      ( pp(aa(rat,bool,positive,fract(A2,B2)))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),A2),B2))) ) ).

% positive_rat
tff(fact_6948_nth__image,axiom,
    ! [A: $tType,L: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),L),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(set(nat),set(A),image(nat,A,nth(A,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),L)) = aa(list(A),set(A),set2(A),take(A,L,Xs)) ) ) ).

% nth_image
tff(fact_6949_take__Suc__Cons,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : take(A,aa(nat,nat,suc,N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),take(A,N,Xs)) ).

% take_Suc_Cons
tff(fact_6950_take__eq__Nil2,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( nil(A) = take(A,N,Xs) )
    <=> ( ( N = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil2
tff(fact_6951_take__eq__Nil,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( take(A,N,Xs) = nil(A) )
    <=> ( ( N = zero_zero(nat) )
        | ( Xs = nil(A) ) ) ) ).

% take_eq_Nil
tff(fact_6952_take0,axiom,
    ! [A: $tType,X2: list(A)] : take(A,zero_zero(nat),X2) = nil(A) ).

% take0
tff(fact_6953_take__all,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N))
     => ( take(A,N,Xs) = Xs ) ) ).

% take_all
tff(fact_6954_take__all__iff,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( take(A,N,Xs) = Xs )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).

% take_all_iff
tff(fact_6955_nth__take,axiom,
    ! [A: $tType,I2: nat,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),N))
     => ( aa(nat,A,nth(A,take(A,N,Xs)),I2) = aa(nat,A,nth(A,Xs),I2) ) ) ).

% nth_take
tff(fact_6956_take__update__cancel,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A),Y: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => ( take(A,N,list_update(A,Xs,M,Y)) = take(A,N,Xs) ) ) ).

% take_update_cancel
tff(fact_6957_nths__upt__eq__take,axiom,
    ! [A: $tType,L: list(A),N: nat] : nths(A,L,aa(nat,set(nat),set_ord_lessThan(nat),N)) = take(A,N,L) ).

% nths_upt_eq_take
tff(fact_6958_take__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] : take(A,N,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,N,Xs)),take(A,aa(nat,nat,minus_minus(nat,N),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).

% take_append
tff(fact_6959_take__Cons__numeral,axiom,
    ! [A: $tType,V: num,X: A,Xs: list(A)] : take(A,aa(num,nat,numeral_numeral(nat),V),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),take(A,aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V)),one_one(nat)),Xs)) ).

% take_Cons_numeral
tff(fact_6960_distinct__take,axiom,
    ! [A: $tType,Xs: list(A),I2: nat] :
      ( distinct(A,Xs)
     => distinct(A,take(A,I2,Xs)) ) ).

% distinct_take
tff(fact_6961_less__rat__def,axiom,
    ! [X: rat,Y: rat] :
      ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),X),Y))
    <=> pp(aa(rat,bool,positive,aa(rat,rat,minus_minus(rat,Y),X))) ) ).

% less_rat_def
tff(fact_6962_take__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ! [I4: nat] : take(A,I4,Xs) = take(A,I4,Ys)
     => ( Xs = Ys ) ) ).

% take_equalityI
tff(fact_6963_take__update__swap,axiom,
    ! [A: $tType,M: nat,Xs: list(A),N: nat,X: A] : take(A,M,list_update(A,Xs,N,X)) = list_update(A,take(A,M,Xs),N,X) ).

% take_update_swap
tff(fact_6964_take__0,axiom,
    ! [A: $tType,Xs: list(A)] : take(A,zero_zero(nat),Xs) = nil(A) ).

% take_0
tff(fact_6965_take__Nil,axiom,
    ! [A: $tType,N: nat] : take(A,N,nil(A)) = nil(A) ).

% take_Nil
tff(fact_6966_in__set__takeD,axiom,
    ! [A: $tType,X: A,N: nat,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),take(A,N,Xs))))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs))) ) ).

% in_set_takeD
tff(fact_6967_set__take__subset,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),take(A,N,Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% set_take_subset
tff(fact_6968_set__take__subset__set__take,axiom,
    ! [A: $tType,M: nat,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),take(A,M,Xs))),aa(list(A),set(A),set2(A),take(A,N,Xs)))) ) ).

% set_take_subset_set_take
tff(fact_6969_nth__take__lemma,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Ys: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),K),aa(list(A),nat,size_size(list(A)),Ys)))
       => ( ! [I4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),K))
             => ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Ys),I4) ) )
         => ( take(A,K,Xs) = take(A,K,Ys) ) ) ) ) ).

% nth_take_lemma
tff(fact_6970_take__Cons,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : take(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = case_nat(list(A),nil(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_zj(A,fun(list(A),fun(nat,list(A))),X),Xs),N) ).

% take_Cons
tff(fact_6971_take__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( ( ( N = zero_zero(nat) )
       => ( take(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = nil(A) ) )
      & ( ( N != zero_zero(nat) )
       => ( take(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),take(A,aa(nat,nat,minus_minus(nat,N),one_one(nat)),Xs)) ) ) ) ).

% take_Cons'
tff(fact_6972_lex__take__index,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),lex(A,R3)))
     => ~ ! [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Ys)))
             => ( ( take(A,I4,Xs) = take(A,I4,Ys) )
               => ~ pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,nth(A,Xs),I4)),aa(nat,A,nth(A,Ys),I4))),R3)) ) ) ) ) ).

% lex_take_index
tff(fact_6973_take__bit__horner__sum__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,Bs: list(bool)] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),groups4207007520872428315er_sum(bool,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),Bs)) = groups4207007520872428315er_sum(bool,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),take(bool,N,Bs)) ) ).

% take_bit_horner_sum_bit_eq
tff(fact_6974_take__Suc__conv__app__nth,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( take(A,aa(nat,nat,suc,I2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I2,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I2)),nil(A))) ) ) ).

% take_Suc_conv_app_nth
tff(fact_6975_nth__repl,axiom,
    ! [A: $tType,M: nat,Xs: list(A),N: nat,X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),M),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
       => ( ( M != N )
         => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,N,Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),drop(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),Xs)))),M) = aa(nat,A,nth(A,Xs),M) ) ) ) ) ).

% nth_repl
tff(fact_6976_pos__n__replace,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Y: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,N,Xs)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))),drop(A,aa(nat,nat,suc,N),Xs)))) ) ) ).

% pos_n_replace
tff(fact_6977_drop0,axiom,
    ! [A: $tType,X2: list(A)] : drop(A,zero_zero(nat),X2) = X2 ).

% drop0
tff(fact_6978_drop__drop,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A)] : drop(A,N,drop(A,M,Xs)) = drop(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M),Xs) ).

% drop_drop
tff(fact_6979_drop__Suc__Cons,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : drop(A,aa(nat,nat,suc,N),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = drop(A,N,Xs) ).

% drop_Suc_Cons
tff(fact_6980_length__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),drop(A,N,Xs)) = aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),N) ).

% length_drop
tff(fact_6981_append__take__drop__id,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,N,Xs)),drop(A,N,Xs)) = Xs ).

% append_take_drop_id
tff(fact_6982_drop__update__cancel,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),M))
     => ( drop(A,M,list_update(A,Xs,N,X)) = drop(A,M,Xs) ) ) ).

% drop_update_cancel
tff(fact_6983_drop__replicate,axiom,
    ! [A: $tType,I2: nat,K: nat,X: A] : drop(A,I2,replicate(A,K,X)) = replicate(A,aa(nat,nat,minus_minus(nat,K),I2),X) ).

% drop_replicate
tff(fact_6984_drop__eq__Nil2,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( nil(A) = drop(A,N,Xs) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).

% drop_eq_Nil2
tff(fact_6985_drop__eq__Nil,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( drop(A,N,Xs) = nil(A) )
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) ) ).

% drop_eq_Nil
tff(fact_6986_drop__all,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),N))
     => ( drop(A,N,Xs) = nil(A) ) ) ).

% drop_all
tff(fact_6987_drop__append,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] : drop(A,N,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,N,Xs)),drop(A,aa(nat,nat,minus_minus(nat,N),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).

% drop_append
tff(fact_6988_drop__Cons__numeral,axiom,
    ! [A: $tType,V: num,X: A,Xs: list(A)] : drop(A,aa(num,nat,numeral_numeral(nat),V),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = drop(A,aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V)),one_one(nat)),Xs) ).

% drop_Cons_numeral
tff(fact_6989_nth__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A),I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,drop(A,N,Xs)),I2) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),I2)) ) ) ).

% nth_drop
tff(fact_6990_drop__take,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A)] : drop(A,N,take(A,M,Xs)) = take(A,aa(nat,nat,minus_minus(nat,M),N),drop(A,N,Xs)) ).

% drop_take
tff(fact_6991_take__drop,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A)] : take(A,N,drop(A,M,Xs)) = drop(A,M,take(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M),Xs)) ).

% take_drop
tff(fact_6992_set__drop__subset,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),drop(A,N,Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% set_drop_subset
tff(fact_6993_drop__eq__nths,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : drop(A,N,Xs) = nths(A,Xs,collect(nat,aa(nat,fun(nat,bool),ord_less_eq(nat),N))) ).

% drop_eq_nths
tff(fact_6994_in__set__dropD,axiom,
    ! [A: $tType,X: A,N: nat,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),drop(A,N,Xs))))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs))) ) ).

% in_set_dropD
tff(fact_6995_drop__Nil,axiom,
    ! [A: $tType,N: nat] : drop(A,N,nil(A)) = nil(A) ).

% drop_Nil
tff(fact_6996_drop__0,axiom,
    ! [A: $tType,Xs: list(A)] : drop(A,zero_zero(nat),Xs) = Xs ).

% drop_0
tff(fact_6997_nth__via__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Y: A,Ys: list(A)] :
      ( ( drop(A,N,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
     => ( aa(nat,A,nth(A,Xs),N) = Y ) ) ).

% nth_via_drop
tff(fact_6998_distinct__drop,axiom,
    ! [A: $tType,Xs: list(A),I2: nat] :
      ( distinct(A,Xs)
     => distinct(A,drop(A,I2,Xs)) ) ).

% distinct_drop
tff(fact_6999_set__drop__subset__set__drop,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),M))
     => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),drop(A,M,Xs))),aa(list(A),set(A),set2(A),drop(A,N,Xs)))) ) ).

% set_drop_subset_set_drop
tff(fact_7000_take__add,axiom,
    ! [A: $tType,I2: nat,J: nat,Xs: list(A)] : take(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),J),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I2,Xs)),take(A,J,drop(A,I2,Xs))) ).

% take_add
tff(fact_7001_append__eq__conv__conj,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys) = Zs )
    <=> ( ( Xs = take(A,aa(list(A),nat,size_size(list(A)),Xs),Zs) )
        & ( Ys = drop(A,aa(list(A),nat,size_size(list(A)),Xs),Zs) ) ) ) ).

% append_eq_conv_conj
tff(fact_7002_drop__update__swap,axiom,
    ! [A: $tType,M: nat,N: nat,Xs: list(A),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( drop(A,M,list_update(A,Xs,N,X)) = list_update(A,drop(A,M,Xs),aa(nat,nat,minus_minus(nat,N),M),X) ) ) ).

% drop_update_swap
tff(fact_7003_drop__Cons,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] : drop(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = case_nat(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aTP_Lamp_zk(list(A),fun(nat,list(A)),Xs),N) ).

% drop_Cons
tff(fact_7004_nths__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A),I5: set(nat)] : nths(A,drop(A,N,Xs),I5) = nths(A,Xs,aa(set(nat),set(nat),image(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N)),I5)) ).

% nths_drop
tff(fact_7005_drop__Cons_H,axiom,
    ! [A: $tType,N: nat,X: A,Xs: list(A)] :
      ( ( ( N = zero_zero(nat) )
       => ( drop(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) )
      & ( ( N != zero_zero(nat) )
       => ( drop(A,N,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = drop(A,aa(nat,nat,minus_minus(nat,N),one_one(nat)),Xs) ) ) ) ).

% drop_Cons'
tff(fact_7006_append__eq__append__conv__if,axiom,
    ! [A: $tType,Xs_1: list(A),Xs_2: list(A),Ys_1: list(A),Ys_2: list(A)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs_1),Xs_2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys_1),Ys_2) )
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs_1)),aa(list(A),nat,size_size(list(A)),Ys_1)))
         => ( ( Xs_1 = take(A,aa(list(A),nat,size_size(list(A)),Xs_1),Ys_1) )
            & ( Xs_2 = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,aa(list(A),nat,size_size(list(A)),Xs_1),Ys_1)),Ys_2) ) ) )
        & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs_1)),aa(list(A),nat,size_size(list(A)),Ys_1)))
         => ( ( take(A,aa(list(A),nat,size_size(list(A)),Ys_1),Xs_1) = Ys_1 )
            & ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,aa(list(A),nat,size_size(list(A)),Ys_1),Xs_1)),Xs_2) = Ys_2 ) ) ) ) ) ).

% append_eq_append_conv_if
tff(fact_7007_Cons__nth__drop__Suc,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I2)),drop(A,aa(nat,nat,suc,I2),Xs)) = drop(A,I2,Xs) ) ) ).

% Cons_nth_drop_Suc
tff(fact_7008_set__take__disj__set__drop__if__distinct,axiom,
    ! [A: $tType,Vs: list(A),I2: nat,J: nat] :
      ( distinct(A,Vs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),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,I2,Vs))),aa(list(A),set(A),set2(A),drop(A,J,Vs))) = bot_bot(set(A)) ) ) ) ).

% set_take_disj_set_drop_if_distinct
tff(fact_7009_id__take__nth__drop,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I2,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,nth(A,Xs),I2)),drop(A,aa(nat,nat,suc,I2),Xs))) ) ) ).

% id_take_nth_drop
tff(fact_7010_upd__conv__take__nth__drop,axiom,
    ! [A: $tType,I2: nat,Xs: list(A),A2: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( list_update(A,Xs,I2,A2) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,I2,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),drop(A,aa(nat,nat,suc,I2),Xs))) ) ) ).

% upd_conv_take_nth_drop
tff(fact_7011_Rat_Opositive_Orep__eq,axiom,
    ! [X: rat] :
      ( pp(aa(rat,bool,positive,X))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),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,X))),aa(product_prod(int,int),int,product_snd(int,int),aa(rat,product_prod(int,int),rep_Rat,X))))) ) ).

% Rat.positive.rep_eq
tff(fact_7012_take__hd__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),take(A,N,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),drop(A,N,Xs))),nil(A))) = take(A,aa(nat,nat,suc,N),Xs) ) ) ).

% take_hd_drop
tff(fact_7013_hd__remdups__adj,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),A,hd(A),remdups_adj(A,Xs)) = aa(list(A),A,hd(A),Xs) ).

% hd_remdups_adj
tff(fact_7014_hd__append2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),A,hd(A),Xs) ) ) ).

% hd_append2
tff(fact_7015_hd__replicate,axiom,
    ! [A: $tType,N: nat,X: A] :
      ( ( N != zero_zero(nat) )
     => ( aa(list(A),A,hd(A),replicate(A,N,X)) = X ) ) ).

% hd_replicate
tff(fact_7016_hd__take,axiom,
    ! [A: $tType,J: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),J))
     => ( aa(list(A),A,hd(A),take(A,J,Xs)) = aa(list(A),A,hd(A),Xs) ) ) ).

% hd_take
tff(fact_7017_longest__common__prefix,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
    ? [Ps: list(A),Xs4: list(A),Ys5: list(A)] :
      ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ps),Xs4) )
      & ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ps),Ys5) )
      & ( ( Xs4 = nil(A) )
        | ( Ys5 = nil(A) )
        | ( aa(list(A),A,hd(A),Xs4) != aa(list(A),A,hd(A),Ys5) ) ) ) ).

% longest_common_prefix
tff(fact_7018_hd__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( ( Xs = nil(A) )
       => ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),A,hd(A),Ys) ) )
      & ( ( Xs != nil(A) )
       => ( aa(list(A),A,hd(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),A,hd(A),Xs) ) ) ) ).

% hd_append
tff(fact_7019_list_Osel_I1_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),A,hd(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = X21 ).

% list.sel(1)
tff(fact_7020_hd__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( ( Xs != nil(list(A)) )
     => ( ( aa(list(list(A)),list(A),hd(list(A)),Xs) != nil(A) )
       => ( aa(list(A),A,hd(A),concat(A,Xs)) = aa(list(A),A,hd(A),aa(list(list(A)),list(A),hd(list(A)),Xs)) ) ) ) ).

% hd_concat
tff(fact_7021_hd__in__set,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(list(A),A,hd(A),Xs)),aa(list(A),set(A),set2(A),Xs))) ) ).

% hd_in_set
tff(fact_7022_list_Oset__sel_I1_J,axiom,
    ! [A: $tType,A2: list(A)] :
      ( ( A2 != nil(A) )
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(list(A),A,hd(A),A2)),aa(list(A),set(A),set2(A),A2))) ) ).

% list.set_sel(1)
tff(fact_7023_hd__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),A,hd(A),Xs) = aa(nat,A,nth(A,Xs),zero_zero(nat)) ) ) ).

% hd_conv_nth
tff(fact_7024_hd__drop__conv__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),A,hd(A),drop(A,N,Xs)) = aa(nat,A,nth(A,Xs),N) ) ) ).

% hd_drop_conv_nth
tff(fact_7025_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),aa(list(A),A,hd(A),Xs)) ) ) ) ).

% remdups_adj_singleton_iff
tff(fact_7026_lexord__take__index__conv,axiom,
    ! [A: $tType,X: list(A),Y: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),X),Y)),lexord(A,R3)))
    <=> ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y)))
          & ( take(A,aa(list(A),nat,size_size(list(A)),X),Y) = X ) )
        | ? [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),X)),aa(list(A),nat,size_size(list(A)),Y))))
            & ( take(A,I3,X) = take(A,I3,Y) )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,aa(nat,A,nth(A,X),I3)),aa(nat,A,nth(A,Y),I3))),R3)) ) ) ) ).

% lexord_take_index_conv
tff(fact_7027_rotate__drop__take,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,N),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),drop(A,modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs)),Xs)),take(A,modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs)),Xs)) ).

% rotate_drop_take
tff(fact_7028_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb1
tff(fact_7029_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb2
tff(fact_7030_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2)) ) ) ) ).

% min.bounded_iff
tff(fact_7031_min__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),X))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Z),Y)) ) ) ) ).

% min_less_iff_conj
tff(fact_7032_min_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),A2))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ).

% min.absorb4
tff(fact_7033_min_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) ) ) ).

% min.absorb3
tff(fact_7034_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)) ).

% min_Suc_Suc
tff(fact_7035_min__0R,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),zero_zero(nat)) = zero_zero(nat) ).

% min_0R
tff(fact_7036_min__0L,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),zero_zero(nat)),N) = zero_zero(nat) ).

% min_0L
tff(fact_7037_rotate__is__Nil__conv,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( aa(list(A),list(A),rotate(A,N),Xs) = nil(A) )
    <=> ( Xs = nil(A) ) ) ).

% rotate_is_Nil_conv
tff(fact_7038_set__rotate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),rotate(A,N),Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% set_rotate
tff(fact_7039_length__rotate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),rotate(A,N),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_rotate
tff(fact_7040_take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)),A2) ) ).

% take_bit_take_bit
tff(fact_7041_take__take,axiom,
    ! [A: $tType,N: nat,M: nat,Xs: list(A)] : take(A,N,take(A,M,Xs)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),M),Xs) ).

% take_take
tff(fact_7042_distinct__rotate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),rotate(A,N),Xs))
    <=> distinct(A,Xs) ) ).

% distinct_rotate
tff(fact_7043_signed__take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,M),aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A2)) = aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)),A2) ) ).

% signed_take_bit_signed_take_bit
tff(fact_7044_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),U) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),V) ) ) ) ) ).

% min_number_of(1)
tff(fact_7045_min__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),zero_zero(A)),aa(num,A,numeral_numeral(A),X)) = zero_zero(A) ) ).

% min_0_1(3)
tff(fact_7046_min__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),X)),zero_zero(A)) = zero_zero(A) ) ).

% min_0_1(4)
tff(fact_7047_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_7048_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_7049_min__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),X)),one_one(A)) = one_one(A) ) ).

% min_0_1(6)
tff(fact_7050_min__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: num] : aa(A,A,aa(A,fun(A,A),ord_min(A),one_one(A)),aa(num,A,numeral_numeral(A),X)) = one_one(A) ) ).

% min_0_1(5)
tff(fact_7051_Int__atMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),A2)),aa(A,set(A),set_ord_atMost(A),B2)) = aa(A,set(A),set_ord_atMost(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)) ) ).

% Int_atMost
tff(fact_7052_length__take,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),take(A,N,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),N) ).

% length_take
tff(fact_7053_take__bit__of__mask,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),bit_se2239418461657761734s_mask(A,N)) = bit_se2239418461657761734s_mask(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)) ) ).

% take_bit_of_mask
tff(fact_7054_take__replicate,axiom,
    ! [A: $tType,I2: nat,K: nat,X: A] : take(A,I2,replicate(A,K,X)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I2),K),X) ).

% take_replicate
tff(fact_7055_rotate__Suc,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,aa(nat,nat,suc,N)),Xs) = aa(list(A),list(A),rotate1(A),aa(list(A),list(A),rotate(A,N),Xs)) ).

% rotate_Suc
tff(fact_7056_min__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)) ) ) ) ) ).

% min_number_of(4)
tff(fact_7057_min__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)) = aa(num,A,numeral_numeral(A),V) ) ) ) ) ).

% min_number_of(3)
tff(fact_7058_min__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(num,A,numeral_numeral(A),U) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)) ) ) ) ) ).

% min_number_of(2)
tff(fact_7059_Int__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D2)) ) ).

% Int_atLeastAtMost
tff(fact_7060_Int__atLeastAtMostR1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,D2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_atMost(A),B2)),set_or1337092689740270186AtMost(A,C2,D2)) = set_or1337092689740270186AtMost(A,C2,aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D2)) ) ).

% Int_atLeastAtMostR1
tff(fact_7061_Int__atLeastAtMostL1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,D2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),aa(A,set(A),set_ord_atMost(A),D2)) = set_or1337092689740270186AtMost(A,A2,aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D2)) ) ).

% Int_atLeastAtMostL1
tff(fact_7062_Int__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D2)) ) ).

% Int_atLeastLessThan
tff(fact_7063_Int__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or5935395276787703475ssThan(A,A2,B2)),set_or5935395276787703475ssThan(A,C2,D2)) = set_or5935395276787703475ssThan(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D2)) ) ).

% Int_greaterThanLessThan
tff(fact_7064_rotate__length01,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)))
     => ( aa(list(A),list(A),rotate(A,N),Xs) = Xs ) ) ).

% rotate_length01
tff(fact_7065_Int__greaterThanAtMost,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),set_or3652927894154168847AtMost(A,A2,B2)),set_or3652927894154168847AtMost(A,C2,D2)) = set_or3652927894154168847AtMost(A,aa(A,A,aa(A,fun(A,A),ord_max(A),A2),C2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),D2)) ) ).

% Int_greaterThanAtMost
tff(fact_7066_rotate__id,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs)) = zero_zero(nat) )
     => ( aa(list(A),list(A),rotate(A,N),Xs) = Xs ) ) ).

% rotate_id
tff(fact_7067_min__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),pred_numeral(K))) ).

% min_Suc_numeral
tff(fact_7068_min__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),pred_numeral(K)),N)) ).

% min_numeral_Suc
tff(fact_7069_rotate__def,axiom,
    ! [A: $tType,N: nat] : rotate(A,N) = 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))),N),rotate1(A)) ).

% rotate_def
tff(fact_7070_rotate__conv__mod,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,N),Xs) = aa(list(A),list(A),rotate(A,modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs))),Xs) ).

% rotate_conv_mod
tff(fact_7071_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Z))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),Z)) ) ) ) ).

% min_le_iff_disj
tff(fact_7072_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2)) ) ) ).

% min.coboundedI2
tff(fact_7073_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2)) ) ) ).

% min.coboundedI1
tff(fact_7074_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_7075_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_7076_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),B2)) ) ).

% min.cobounded2
tff(fact_7077_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),A2)) ) ).

% min.cobounded1
tff(fact_7078_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_7079_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2))) ) ) ) ).

% min.boundedI
tff(fact_7080_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2)) ) ) ) ).

% min.boundedE
tff(fact_7081_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) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2)) ) ) ).

% min.orderI
tff(fact_7082_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
         => ( A2 = aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) ) ) ) ).

% min.orderE
tff(fact_7083_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),C2))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),D2))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),aa(A,A,aa(A,fun(A,A),ord_min(A),C2),D2))) ) ) ) ).

% min.mono
tff(fact_7084_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X2: A,Xa2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Xa2))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X2),Xa2) = X2 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),Xa2))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X2),Xa2) = Xa2 ) ) ) ) ).

% min_def_raw
tff(fact_7085_min__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = A2 ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B2))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2) = B2 ) ) ) ) ).

% min_def
tff(fact_7086_min__absorb1,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Y: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y) = X ) ) ) ).

% min_absorb1
tff(fact_7087_min__absorb2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,X: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y),X))
         => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y) = Y ) ) ) ).

% min_absorb2
tff(fact_7088_greaterThan__Int__greaterThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(A,set(A),set_ord_lessThan(A),A2)),aa(A,set(A),set_ord_lessThan(A),B2)) = aa(A,set(A),set_ord_lessThan(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)) ) ).

% greaterThan_Int_greaterThan
tff(fact_7089_inf__nat__def,axiom,
    inf_inf(nat) = ord_min(nat) ).

% inf_nat_def
tff(fact_7090_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)),Q3) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q3)) ).

% nat_mult_min_left
tff(fact_7091_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),N),Q3)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q3)) ).

% nat_mult_min_right
tff(fact_7092_of__nat__min,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [X: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(nat,A,semiring_1_of_nat(A),X)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_min
tff(fact_7093_rotate__rotate,axiom,
    ! [A: $tType,M: nat,N: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,M),aa(list(A),list(A),rotate(A,N),Xs)) = aa(list(A),list(A),rotate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),Xs) ).

% rotate_rotate
tff(fact_7094_min__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ).

% min_add_distrib_left
tff(fact_7095_min__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),X),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),X),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),Z)) ) ).

% min_add_distrib_right
tff(fact_7096_rotate1__rotate__swap,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),rotate1(A),aa(list(A),list(A),rotate(A,N),Xs)) = aa(list(A),list(A),rotate(A,N),aa(list(A),list(A),rotate1(A),Xs)) ).

% rotate1_rotate_swap
tff(fact_7097_minus__min__eq__max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).

% minus_min_eq_max
tff(fact_7098_minus__max__eq__min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [X: A,Y: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,uminus_uminus(A),X)),aa(A,A,uminus_uminus(A),Y)) ) ).

% minus_max_eq_min
tff(fact_7099_min__less__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Z: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),Z))
            | pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y),Z)) ) ) ) ).

% min_less_iff_disj
tff(fact_7100_min_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(A,A,aa(A,fun(A,A),ord_min(A),B2),C2)))
         => ~ ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),B2))
             => ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2)) ) ) ) ).

% min.strict_boundedE
tff(fact_7101_min_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),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_7102_min_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2)) ) ) ).

% min.strict_coboundedI1
tff(fact_7103_min_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),B2),C2))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(A,A,aa(A,fun(A,A),ord_min(A),A2),B2)),C2)) ) ) ).

% min.strict_coboundedI2
tff(fact_7104_rotate__append,axiom,
    ! [A: $tType,L: list(A),Q3: list(A)] : aa(list(A),list(A),rotate(A,aa(list(A),nat,size_size(list(A)),L)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),L),Q3)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Q3),L) ).

% rotate_append
tff(fact_7105_min__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [X: A,Y: A,Z: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),Z) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,minus_minus(A,X),Z)),aa(A,A,minus_minus(A,Y),Z)) ) ).

% min_diff_distrib_left
tff(fact_7106_min__diff,axiom,
    ! [M: nat,I2: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,minus_minus(nat,M),I2)),aa(nat,nat,minus_minus(nat,N),I2)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N)),I2) ).

% min_diff
tff(fact_7107_concat__bit__assoc__sym,axiom,
    ! [M: nat,N: nat,K: int,L: int,R3: int] : aa(int,int,bit_concat_bit(M,aa(int,int,bit_concat_bit(N,K),L)),R3) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N),K),aa(int,int,bit_concat_bit(aa(nat,nat,minus_minus(nat,M),N),L),R3)) ).

% concat_bit_assoc_sym
tff(fact_7108_rotate__add,axiom,
    ! [A: $tType,M: nat,N: nat] : rotate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(fun(list(A),list(A)),fun(list(A),list(A)),comp(list(A),list(A),list(A),rotate(A,M)),rotate(A,N)) ).

% rotate_add
tff(fact_7109_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,M),aa(int,int,bit_concat_bit(N,K),L)) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N),K),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,minus_minus(nat,M),N)),L)) ).

% take_bit_concat_bit_eq
tff(fact_7110_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)) ) ) ) ) ).

% max_mult_distrib_left
tff(fact_7111_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),X)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)) ) ) ) ) ).

% min_mult_distrib_left
tff(fact_7112_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y)),P2) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)) ) ) ) ) ).

% max_mult_distrib_right
tff(fact_7113_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P2) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y)),P2) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,aa(A,fun(A,A),times_times(A),X),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)) ) ) ) ) ).

% min_mult_distrib_right
tff(fact_7114_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [P2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y),P2) = aa(A,A,aa(A,fun(A,A),ord_max(A),divide_divide(A,X,P2)),divide_divide(A,Y,P2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_max(A),X),Y),P2) = aa(A,A,aa(A,fun(A,A),ord_min(A),divide_divide(A,X,P2)),divide_divide(A,Y,P2)) ) ) ) ) ).

% max_divide_distrib_right
tff(fact_7115_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [P2: A,X: A,Y: A] :
          ( ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y),P2) = aa(A,A,aa(A,fun(A,A),ord_min(A),divide_divide(A,X,P2)),divide_divide(A,Y,P2)) ) )
          & ( ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),P2))
           => ( divide_divide(A,aa(A,A,aa(A,fun(A,A),ord_min(A),X),Y),P2) = aa(A,A,aa(A,fun(A,A),ord_max(A),divide_divide(A,X,P2)),divide_divide(A,Y,P2)) ) ) ) ) ).

% min_divide_distrib_right
tff(fact_7116_min__Suc1,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(nat,nat,suc,N)),M) = case_nat(nat,zero_zero(nat),aTP_Lamp_zl(nat,fun(nat,nat),N),M) ).

% min_Suc1
tff(fact_7117_min__Suc2,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),aa(nat,nat,suc,N)) = case_nat(nat,zero_zero(nat),aTP_Lamp_zm(nat,fun(nat,nat),N),M) ).

% min_Suc2
tff(fact_7118_mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,M: nat,N: nat] : modulo_modulo(A,modulo_modulo(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)) = modulo_modulo(A,A2,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N))) ) ).

% mod_exp_eq
tff(fact_7119_nth__rotate,axiom,
    ! [A: $tType,N: nat,Xs: list(A),M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),rotate(A,M),Xs)),N) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% nth_rotate
tff(fact_7120_mask__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat,M: nat] : modulo_modulo(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),N)),one_one(A)),aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),M)) = aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),M),N))),one_one(A)) ) ).

% mask_mod_exp
tff(fact_7121_hd__rotate__conv__nth,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),A,hd(A),aa(list(A),list(A),rotate(A,N),Xs)) = aa(nat,A,nth(A,Xs),modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% hd_rotate_conv_nth
tff(fact_7122_set__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)) = collect(product_prod(A,B),aa(list(B),fun(product_prod(A,B),bool),aTP_Lamp_zn(list(A),fun(list(B),fun(product_prod(A,B),bool)),Xs),Ys)) ).

% set_zip
tff(fact_7123_Nitpick_Osize__list__simp_I1_J,axiom,
    ! [A: $tType,Xs: list(A),F3: fun(A,nat)] :
      ( ( ( Xs = nil(A) )
       => ( aa(list(A),nat,size_list(A,F3),Xs) = zero_zero(nat) ) )
      & ( ( Xs != nil(A) )
       => ( aa(list(A),nat,size_list(A,F3),Xs) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F3,aa(list(A),A,hd(A),Xs))),aa(list(A),nat,size_list(A,F3),aa(list(A),list(A),tl(A),Xs)))) ) ) ) ).

% Nitpick.size_list_simp(1)
tff(fact_7124_min__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_min(extended_enat),Q3),zero_zero(extended_enat)) = zero_zero(extended_enat) ).

% min_enat_simps(2)
tff(fact_7125_min__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] : aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),ord_min(extended_enat),zero_zero(extended_enat)),Q3) = zero_zero(extended_enat) ).

% min_enat_simps(3)
tff(fact_7126_tl__append2,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),tl(A),Xs)),Ys) ) ) ).

% tl_append2
tff(fact_7127_remdups__adj__Cons__alt,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),tl(A),remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)))) = remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) ).

% remdups_adj_Cons_alt
tff(fact_7128_zip__eq__Nil__iff,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( zip(A,B,Xs,Ys) = nil(product_prod(A,B)) )
    <=> ( ( Xs = nil(A) )
        | ( Ys = nil(B) ) ) ) ).

% zip_eq_Nil_iff
tff(fact_7129_Nil__eq__zip__iff,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( nil(product_prod(A,B)) = zip(A,B,Xs,Ys) )
    <=> ( ( Xs = nil(A) )
        | ( Ys = nil(B) ) ) ) ).

% Nil_eq_zip_iff
tff(fact_7130_zip__Nil,axiom,
    ! [B: $tType,A: $tType,Ys: list(B)] : zip(A,B,nil(A),Ys) = nil(product_prod(A,B)) ).

% zip_Nil
tff(fact_7131_length__tl,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs)) = aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ).

% length_tl
tff(fact_7132_list_Ocollapse,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List != nil(A) )
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) = List ) ) ).

% list.collapse
tff(fact_7133_hd__Cons__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),Xs)),aa(list(A),list(A),tl(A),Xs)) = Xs ) ) ).

% hd_Cons_tl
tff(fact_7134_tl__replicate,axiom,
    ! [A: $tType,N: nat,X: A] : aa(list(A),list(A),tl(A),replicate(A,N,X)) = replicate(A,aa(nat,nat,minus_minus(nat,N),one_one(nat)),X) ).

% tl_replicate
tff(fact_7135_zip__replicate,axiom,
    ! [A: $tType,B: $tType,I2: nat,X: A,J: nat,Y: B] : zip(A,B,replicate(A,I2,X),replicate(B,J,Y)) = replicate(product_prod(A,B),aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),I2),J),aa(B,product_prod(A,B),product_Pair(A,B,X),Y)) ).

% zip_replicate
tff(fact_7136_zip__Cons__Cons,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,X),Y)),zip(A,B,Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_7137_zip__append,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Us: list(B),Ys: list(A),Vs: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Us) )
     => ( zip(A,B,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us),Vs)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),zip(A,B,Xs,Us)),zip(A,B,Ys,Vs)) ) ) ).

% zip_append
tff(fact_7138_length__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),zip(A,B,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) ).

% length_zip
tff(fact_7139_nth__zip,axiom,
    ! [A: $tType,B: $tType,I2: nat,Xs: list(A),Ys: list(B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),Ys)))
       => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),zip(A,B,Xs,Ys)),I2) = aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Xs),I2)),aa(nat,B,nth(B,Ys),I2)) ) ) ) ).

% nth_zip
tff(fact_7140_list_Oexpand,axiom,
    ! [A: $tType,List: list(A),List2: list(A)] :
      ( ( ( List = nil(A) )
      <=> ( List2 = nil(A) ) )
     => ( ( ( List != nil(A) )
         => ( ( List2 != nil(A) )
           => ( ( aa(list(A),A,hd(A),List) = aa(list(A),A,hd(A),List2) )
              & ( aa(list(A),list(A),tl(A),List) = aa(list(A),list(A),tl(A),List2) ) ) ) )
       => ( List = List2 ) ) ) ).

% list.expand
tff(fact_7141_take__zip,axiom,
    ! [A: $tType,B: $tType,N: nat,Xs: list(A),Ys: list(B)] : take(product_prod(A,B),N,zip(A,B,Xs,Ys)) = zip(A,B,take(A,N,Xs),take(B,N,Ys)) ).

% take_zip
tff(fact_7142_take__tl,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : take(A,N,aa(list(A),list(A),tl(A),Xs)) = aa(list(A),list(A),tl(A),take(A,aa(nat,nat,suc,N),Xs)) ).

% take_tl
tff(fact_7143_drop__Suc,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : drop(A,aa(nat,nat,suc,N),Xs) = drop(A,N,aa(list(A),list(A),tl(A),Xs)) ).

% drop_Suc
tff(fact_7144_tl__drop,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),tl(A),drop(A,N,Xs)) = drop(A,N,aa(list(A),list(A),tl(A),Xs)) ).

% tl_drop
tff(fact_7145_drop__zip,axiom,
    ! [A: $tType,B: $tType,N: nat,Xs: list(A),Ys: list(B)] : drop(product_prod(A,B),N,zip(A,B,Xs,Ys)) = zip(A,B,drop(A,N,Xs),drop(B,N,Ys)) ).

% drop_zip
tff(fact_7146_distinct__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => distinct(A,aa(list(A),list(A),tl(A),Xs)) ) ).

% distinct_tl
tff(fact_7147_list_Osel_I3_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = X222 ).

% list.sel(3)
tff(fact_7148_list_Osel_I2_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),tl(A),nil(A)) = nil(A) ).

% list.sel(2)
tff(fact_7149_zip_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType,Xs: list(A)] : zip(A,B,Xs,nil(B)) = nil(product_prod(A,B)) ).

% zip.simps(1)
tff(fact_7150_Nil__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),tl(A),Xs) )
    <=> ( ( Xs = nil(A) )
        | ? [X4: A] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)) ) ) ).

% Nil_tl
tff(fact_7151_tl__Nil,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),list(A),tl(A),Xs) = nil(A) )
    <=> ( ( Xs = nil(A) )
        | ? [X4: A] : Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),nil(A)) ) ) ).

% tl_Nil
tff(fact_7152_zip__update,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I2: nat,X: A,Ys: list(B),Y: B] : zip(A,B,list_update(A,Xs,I2,X),list_update(B,Ys,I2,Y)) = list_update(product_prod(A,B),zip(A,B,Xs,Ys),I2,aa(B,product_prod(A,B),product_Pair(A,B,X),Y)) ).

% zip_update
tff(fact_7153_list_Oset__sel_I2_J,axiom,
    ! [A: $tType,A2: list(A),X: A] :
      ( ( A2 != nil(A) )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),A2))))
       => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),A2))) ) ) ).

% list.set_sel(2)
tff(fact_7154_distinct__zipI2,axiom,
    ! [B: $tType,A: $tType,Ys: list(A),Xs: list(B)] :
      ( distinct(A,Ys)
     => distinct(product_prod(B,A),zip(B,A,Xs,Ys)) ) ).

% distinct_zipI2
tff(fact_7155_distinct__zipI1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( distinct(A,Xs)
     => distinct(product_prod(A,B),zip(A,B,Xs,Ys)) ) ).

% distinct_zipI1
tff(fact_7156_hd__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(B) )
       => ( aa(list(product_prod(A,B)),product_prod(A,B),hd(product_prod(A,B)),zip(A,B,Xs,Ys)) = aa(B,product_prod(A,B),product_Pair(A,B,aa(list(A),A,hd(A),Xs)),aa(list(B),B,hd(B),Ys)) ) ) ) ).

% hd_zip
tff(fact_7157_zip__same,axiom,
    ! [A: $tType,A2: A,B2: A,Xs: list(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),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))))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(list(A),set(A),set2(A),Xs)))
        & ( A2 = B2 ) ) ) ).

% zip_same
tff(fact_7158_in__set__zipE,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),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,Ys))))
     => ~ ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
         => ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y),aa(list(B),set(B),set2(B),Ys))) ) ) ).

% in_set_zipE
tff(fact_7159_set__zip__leftD,axiom,
    ! [B: $tType,A: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),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,Ys))))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs))) ) ).

% set_zip_leftD
tff(fact_7160_set__zip__rightD,axiom,
    ! [A: $tType,B: $tType,X: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),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,Ys))))
     => pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y),aa(list(B),set(B),set2(B),Ys))) ) ).

% set_zip_rightD
tff(fact_7161_update__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),I2: nat,Xy: product_prod(A,B)] : list_update(product_prod(A,B),zip(A,B,Xs,Ys),I2,Xy) = zip(A,B,list_update(A,Xs,I2,aa(product_prod(A,B),A,product_fst(A,B),Xy)),list_update(B,Ys,I2,aa(product_prod(A,B),B,product_snd(A,B),Xy))) ).

% update_zip
tff(fact_7162_zip__obtain__same__length,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(list(product_prod(A,B)),bool)] :
      ( ! [Zs2: list(A),Ws2: list(B),N2: nat] :
          ( ( aa(list(A),nat,size_size(list(A)),Zs2) = aa(list(B),nat,size_size(list(B)),Ws2) )
         => ( ( N2 = aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) )
           => ( ( Zs2 = take(A,N2,Xs) )
             => ( ( Ws2 = take(B,N2,Ys) )
               => pp(aa(list(product_prod(A,B)),bool,P,zip(A,B,Zs2,Ws2))) ) ) ) )
     => pp(aa(list(product_prod(A,B)),bool,P,zip(A,B,Xs,Ys))) ) ).

% zip_obtain_same_length
tff(fact_7163_zip__eq__ConsE,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Xy: product_prod(A,B),Xys: list(product_prod(A,B))] :
      ( ( zip(A,B,Xs,Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),Xy),Xys) )
     => ~ ! [X3: A,Xs4: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs4) )
           => ! [Y4: B,Ys5: list(B)] :
                ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys5) )
               => ( ( Xy = aa(B,product_prod(A,B),product_Pair(A,B,X3),Y4) )
                 => ( Xys != zip(A,B,Xs4,Ys5) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_7164_list_Oexhaust__sel,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List != nil(A) )
     => ( List = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) ) ) ).

% list.exhaust_sel
tff(fact_7165_tl__take,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),tl(A),take(A,N,Xs)) = take(A,aa(nat,nat,minus_minus(nat,N),one_one(nat)),aa(list(A),list(A),tl(A),Xs)) ).

% tl_take
tff(fact_7166_list__eq__iff__zip__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [X4: product_prod(A,A)] :
            ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),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,Ys))))
           => pp(aa(product_prod(A,A),bool,product_case_prod(A,A,bool,fequal(A)),X4)) ) ) ) ).

% list_eq_iff_zip_eq
tff(fact_7167_in__set__impl__in__set__zip2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Y: B] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y),aa(list(B),set(B),set2(B),Ys)))
       => ~ ! [X3: A] : ~ pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,X3),Y)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))) ) ) ).

% in_set_impl_in_set_zip2
tff(fact_7168_in__set__impl__in__set__zip1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ~ ! [Y4: B] : ~ pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,X),Y4)),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys)))) ) ) ).

% in_set_impl_in_set_zip1
tff(fact_7169_Nitpick_Osize__list__simp_I2_J,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( ( Xs = nil(A) )
       => ( aa(list(A),nat,size_size(list(A)),Xs) = zero_zero(nat) ) )
      & ( ( Xs != nil(A) )
       => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs))) ) ) ) ).

% Nitpick.size_list_simp(2)
tff(fact_7170_nth__tl,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs))))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N)) ) ) ).

% nth_tl
tff(fact_7171_remdups__adj__append,axiom,
    ! [A: $tType,Xs_1: list(A),X: A,Xs_2: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs_1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs_2))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs_1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))))),aa(list(A),list(A),tl(A),remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs_2)))) ).

% remdups_adj_append
tff(fact_7172_Cons__in__shuffles__iff,axiom,
    ! [A: $tType,Z: A,Zs: list(A),Xs: list(A),Ys: list(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z),Zs)),shuffles(A,Xs,Ys)))
    <=> ( ( ( Xs != nil(A) )
          & ( aa(list(A),A,hd(A),Xs) = Z )
          & pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,aa(list(A),list(A),tl(A),Xs),Ys))) )
        | ( ( Ys != nil(A) )
          & ( aa(list(A),A,hd(A),Ys) = Z )
          & pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,aa(list(A),list(A),tl(A),Ys)))) ) ) ) ).

% Cons_in_shuffles_iff
tff(fact_7173_concat__eq__concat__iff,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] :
      ( ! [X3: product_prod(list(A),list(A))] :
          ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),X3),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,Ys))))
         => pp(aa(product_prod(list(A),list(A)),bool,product_case_prod(list(A),list(A),bool,aTP_Lamp_zo(list(A),fun(list(A),bool))),X3)) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys) )
       => ( ( concat(A,Xs) = concat(A,Ys) )
        <=> ( Xs = Ys ) ) ) ) ).

% concat_eq_concat_iff
tff(fact_7174_concat__injective,axiom,
    ! [A: $tType,Xs: list(list(A)),Ys: list(list(A))] :
      ( ( concat(A,Xs) = concat(A,Ys) )
     => ( ( aa(list(list(A)),nat,size_size(list(list(A))),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Ys) )
       => ( ! [X3: product_prod(list(A),list(A))] :
              ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),X3),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,Ys))))
             => pp(aa(product_prod(list(A),list(A)),bool,product_case_prod(list(A),list(A),bool,aTP_Lamp_zo(list(A),fun(list(A),bool))),X3)) )
         => ( Xs = Ys ) ) ) ) ).

% concat_injective
tff(fact_7175_zip__append2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Zs: list(B)] : zip(A,B,Xs,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Zs)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),zip(A,B,take(A,aa(list(B),nat,size_size(list(B)),Ys),Xs),Ys)),zip(A,B,drop(A,aa(list(B),nat,size_size(list(B)),Ys),Xs),Zs)) ).

% zip_append2
tff(fact_7176_zip__append1,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(A),Zs: list(B)] : zip(A,B,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys),Zs) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),zip(A,B,Xs,take(B,aa(list(A),nat,size_size(list(A)),Xs),Zs))),zip(A,B,Ys,drop(B,aa(list(A),nat,size_size(list(A)),Xs),Zs))) ).

% zip_append1
tff(fact_7177_take__Suc,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( Xs != nil(A) )
     => ( take(A,aa(nat,nat,suc,N),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),Xs)),take(A,N,aa(list(A),list(A),tl(A),Xs))) ) ) ).

% take_Suc
tff(fact_7178_rotate1__hd__tl,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),list(A),rotate1(A),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),tl(A),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),Xs)),nil(A))) ) ) ).

% rotate1_hd_tl
tff(fact_7179_in__set__zip,axiom,
    ! [A: $tType,B: $tType,P2: product_prod(A,B),Xs: list(A),Ys: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),P2),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,Xs,Ys))))
    <=> ? [N5: nat] :
          ( ( aa(nat,A,nth(A,Xs),N5) = aa(product_prod(A,B),A,product_fst(A,B),P2) )
          & ( aa(nat,B,nth(B,Ys),N5) = aa(product_prod(A,B),B,product_snd(A,B),P2) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(B),nat,size_size(list(B)),Ys))) ) ) ).

% in_set_zip
tff(fact_7180_listrel__iff__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Xs),Ys)),listrel(A,B,R3)))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [X4: product_prod(A,B)] :
            ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),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,Ys))))
           => pp(aa(product_prod(A,B),bool,product_case_prod(A,B,bool,aTP_Lamp_ho(set(product_prod(A,B)),fun(A,fun(B,bool)),R3)),X4)) ) ) ) ).

% listrel_iff_zip
tff(fact_7181_zip__Cons1,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),Ys: list(B)] : zip(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys) = aa(list(B),list(product_prod(A,B)),case_list(list(product_prod(A,B)),B,nil(product_prod(A,B)),aa(list(A),fun(B,fun(list(B),list(product_prod(A,B)))),aTP_Lamp_zp(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),X),Xs)),Ys) ).

% zip_Cons1
tff(fact_7182_listrel__rtrancl__refl,axiom,
    ! [A: $tType,Xs: list(A),R3: set(product_prod(A,A))] : pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Xs)),listrel(A,A,transitive_rtrancl(A,R3)))) ).

% listrel_rtrancl_refl
tff(fact_7183_inf__enat__def,axiom,
    inf_inf(extended_enat) = ord_min(extended_enat) ).

% inf_enat_def
tff(fact_7184_listrel_ONil,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B))] : pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),nil(A)),nil(B))),listrel(A,B,R3))) ).

% listrel.Nil
tff(fact_7185_listrel__Nil1,axiom,
    ! [A: $tType,B: $tType,Xs: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),nil(A)),Xs)),listrel(A,B,R3)))
     => ( Xs = nil(B) ) ) ).

% listrel_Nil1
tff(fact_7186_listrel__Nil2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Xs),nil(B))),listrel(A,B,R3)))
     => ( Xs = nil(A) ) ) ).

% listrel_Nil2
tff(fact_7187_listrel__eq__len,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Xs),Ys)),listrel(A,B,R3)))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).

% listrel_eq_len
tff(fact_7188_list_Osimps_I4_J,axiom,
    ! [A: $tType,B: $tType,F1: B,F2: fun(A,fun(list(A),B))] : aa(list(A),B,case_list(B,A,F1,F2),nil(A)) = F1 ).

% list.simps(4)
tff(fact_7189_list_Ocase__distrib,axiom,
    ! [B: $tType,C: $tType,A: $tType,H: fun(B,C),F1: B,F2: fun(A,fun(list(A),B)),List: list(A)] : aa(B,C,H,aa(list(A),B,case_list(B,A,F1,F2),List)) = aa(list(A),C,case_list(C,A,aa(B,C,H,F1),aa(fun(A,fun(list(A),B)),fun(A,fun(list(A),C)),aTP_Lamp_zq(fun(B,C),fun(fun(A,fun(list(A),B)),fun(A,fun(list(A),C))),H),F2)),List) ).

% list.case_distrib
tff(fact_7190_list_Osimps_I5_J,axiom,
    ! [B: $tType,A: $tType,F1: B,F2: fun(A,fun(list(A),B)),X21: A,X222: list(A)] : aa(list(A),B,case_list(B,A,F1,F2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(list(A),B,aa(A,fun(list(A),B),F2,X21),X222) ).

% list.simps(5)
tff(fact_7191_listrel__rtrancl__trans,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A)),Zs: list(A)] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),listrel(A,A,transitive_rtrancl(A,R3))))
     => ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Ys),Zs)),listrel(A,A,transitive_rtrancl(A,R3))))
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Zs)),listrel(A,A,transitive_rtrancl(A,R3)))) ) ) ).

% listrel_rtrancl_trans
tff(fact_7192_listrel__mono,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R3),S))
     => pp(aa(set(product_prod(list(A),list(B))),bool,aa(set(product_prod(list(A),list(B))),fun(set(product_prod(list(A),list(B))),bool),ord_less_eq(set(product_prod(list(A),list(B)))),listrel(A,B,R3)),listrel(A,B,S))) ) ).

% listrel_mono
tff(fact_7193_tl__def,axiom,
    ! [A: $tType,List: list(A)] : aa(list(A),list(A),tl(A),List) = aa(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_zr(A,fun(list(A),list(A)))),List) ).

% tl_def
tff(fact_7194_tl__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),tl(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),case_list(list(A),A,aa(list(A),list(A),tl(A),Ys),aTP_Lamp_zs(list(A),fun(A,fun(list(A),list(A))),Ys)),Xs) ).

% tl_append
tff(fact_7195_listrel__Cons2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys))),listrel(A,B,R3)))
     => ~ ! [X3: A,Xs2: list(A)] :
            ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
           => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,X3),Y)),R3))
             => ~ pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Xs2),Ys)),listrel(A,B,R3))) ) ) ) ).

% listrel_Cons2
tff(fact_7196_listrel__Cons1,axiom,
    ! [B: $tType,A: $tType,Y: A,Ys: list(A),Xs: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),Xs)),listrel(A,B,R3)))
     => ~ ! [Y4: B,Ys3: list(B)] :
            ( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys3) )
           => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,Y),Y4)),R3))
             => ~ pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Ys),Ys3)),listrel(A,B,R3))) ) ) ) ).

% listrel_Cons1
tff(fact_7197_listrel_OCons,axiom,
    ! [B: $tType,A: $tType,X: A,Y: B,R3: set(product_prod(A,B)),Xs: list(A),Ys: list(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,X),Y)),R3))
     => ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Xs),Ys)),listrel(A,B,R3)))
       => pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys))),listrel(A,B,R3))) ) ) ).

% listrel.Cons
tff(fact_7198_listrel__reflcl__if__listrel1,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),listrel1(A,R3)))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),listrel(A,A,transitive_rtrancl(A,R3)))) ) ).

% listrel_reflcl_if_listrel1
tff(fact_7199_listrel__rtrancl__eq__rtrancl__listrel1,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : listrel(A,A,transitive_rtrancl(A,R3)) = transitive_rtrancl(list(A),listrel1(A,R3)) ).

% listrel_rtrancl_eq_rtrancl_listrel1
tff(fact_7200_rtrancl__listrel1__if__listrel,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),listrel(A,A,R3)))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),member(product_prod(list(A),list(A))),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Xs),Ys)),transitive_rtrancl(list(A),listrel1(A,R3)))) ) ).

% rtrancl_listrel1_if_listrel
tff(fact_7201_remdups__adj__Cons,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),case_list(list(A),A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)),aTP_Lamp_zt(A,fun(A,fun(list(A),list(A))),X)),remdups_adj(A,Xs)) ).

% remdups_adj_Cons
tff(fact_7202_listrel_Ocases,axiom,
    ! [B: $tType,A: $tType,A1: list(A),A22: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),A1),A22)),listrel(A,B,R3)))
     => ( ( ( A1 = nil(A) )
         => ( A22 != nil(B) ) )
       => ~ ! [X3: A,Y4: B,Xs2: list(A)] :
              ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
             => ! [Ys3: list(B)] :
                  ( ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys3) )
                 => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,X3),Y4)),R3))
                   => ~ pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Xs2),Ys3)),listrel(A,B,R3))) ) ) ) ) ) ).

% listrel.cases
tff(fact_7203_listrel_Osimps,axiom,
    ! [B: $tType,A: $tType,A1: list(A),A22: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),A1),A22)),listrel(A,B,R3)))
    <=> ( ( ( A1 = nil(A) )
          & ( A22 = nil(B) ) )
        | ? [X4: A,Y5: B,Xs3: list(A),Ys4: list(B)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
            & ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),Ys4) )
            & pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,X4),Y5)),R3))
            & pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Xs3),Ys4)),listrel(A,B,R3))) ) ) ) ).

% listrel.simps
tff(fact_7204_listrel__subset__rtrancl__listrel1,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel(A,A,R3)),transitive_rtrancl(list(A),listrel1(A,R3)))) ).

% listrel_subset_rtrancl_listrel1
tff(fact_7205_list_Ocase__eq__if,axiom,
    ! [B: $tType,A: $tType,List: list(A),F1: B,F2: fun(A,fun(list(A),B))] :
      ( ( ( List = nil(A) )
       => ( aa(list(A),B,case_list(B,A,F1,F2),List) = F1 ) )
      & ( ( List != nil(A) )
       => ( aa(list(A),B,case_list(B,A,F1,F2),List) = aa(list(A),B,aa(A,fun(list(A),B),F2,aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) ) ) ) ).

% list.case_eq_if
tff(fact_7206_listrel__iff__nth,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),R3: set(product_prod(A,B))] :
      ( pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),Xs),Ys)),listrel(A,B,R3)))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [N5: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N5),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,aa(nat,A,nth(A,Xs),N5)),aa(nat,B,nth(B,Ys),N5))),R3)) ) ) ) ).

% listrel_iff_nth
tff(fact_7207_list_Osplit__sel__asm,axiom,
    ! [B: $tType,A: $tType,P: fun(B,bool),F1: B,F2: fun(A,fun(list(A),B)),List: list(A)] :
      ( pp(aa(B,bool,P,aa(list(A),B,case_list(B,A,F1,F2),List)))
    <=> ~ ( ( ( List = nil(A) )
            & ~ pp(aa(B,bool,P,F1)) )
          | ( ( List = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) )
            & ~ pp(aa(B,bool,P,aa(list(A),B,aa(A,fun(list(A),B),F2,aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)))) ) ) ) ).

% list.split_sel_asm
tff(fact_7208_list_Osplit__sel,axiom,
    ! [B: $tType,A: $tType,P: fun(B,bool),F1: B,F2: fun(A,fun(list(A),B)),List: list(A)] :
      ( pp(aa(B,bool,P,aa(list(A),B,case_list(B,A,F1,F2),List)))
    <=> ( ( ( List = nil(A) )
         => pp(aa(B,bool,P,F1)) )
        & ( ( List = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)) )
         => pp(aa(B,bool,P,aa(list(A),B,aa(A,fun(list(A),B),F2,aa(list(A),A,hd(A),List)),aa(list(A),list(A),tl(A),List)))) ) ) ) ).

% list.split_sel
tff(fact_7209_zip__Cons,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Y: B,Ys: list(B)] : zip(A,B,Xs,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)) = aa(list(A),list(product_prod(A,B)),case_list(list(product_prod(A,B)),A,nil(product_prod(A,B)),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_zu(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Y),Ys)),Xs) ).

% zip_Cons
tff(fact_7210_min__list_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: A,Xs: list(A)] : min_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),A,case_list(A,A,X,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_zv(A,fun(list(A),fun(A,fun(list(A),A))),X),Xs)),Xs) ) ).

% min_list.simps
tff(fact_7211_listrel__def,axiom,
    ! [B: $tType,A: $tType,X2: set(product_prod(A,B))] : listrel(A,B,X2) = collect(product_prod(list(A),list(B)),product_case_prod(list(A),list(B),bool,listrelp(A,B,aTP_Lamp_ho(set(product_prod(A,B)),fun(A,fun(B,bool)),X2)))) ).

% listrel_def
tff(fact_7212_list_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List = nil(A) )
    <=> pp(aa(list(A),bool,case_list(bool,A,fTrue,aTP_Lamp_zw(A,fun(list(A),bool))),List)) ) ).

% list.disc_eq_case(1)
tff(fact_7213_list_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List != nil(A) )
    <=> pp(aa(list(A),bool,case_list(bool,A,fFalse,aTP_Lamp_zx(A,fun(list(A),bool))),List)) ) ).

% list.disc_eq_case(2)
tff(fact_7214_listrelp_OCons,axiom,
    ! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),X: A,Y: B,Xs: list(A),Ys: list(B)] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),R3,X),Y))
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R3),Xs),Ys))
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys))) ) ) ).

% listrelp.Cons
tff(fact_7215_listrelp_ONil,axiom,
    ! [A: $tType,B: $tType,R3: fun(A,fun(B,bool))] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R3),nil(A)),nil(B))) ).

% listrelp.Nil
tff(fact_7216_listrelp_Ocases,axiom,
    ! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A1: list(A),A22: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R3),A1),A22))
     => ( ( ( A1 = nil(A) )
         => ( A22 != nil(B) ) )
       => ~ ! [X3: A,Y4: B,Xs2: list(A)] :
              ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
             => ! [Ys3: list(B)] :
                  ( ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys3) )
                 => ( pp(aa(B,bool,aa(A,fun(B,bool),R3,X3),Y4))
                   => ~ pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R3),Xs2),Ys3)) ) ) ) ) ) ).

% listrelp.cases
tff(fact_7217_listrelp_Osimps,axiom,
    ! [A: $tType,B: $tType,R3: fun(A,fun(B,bool)),A1: list(A),A22: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R3),A1),A22))
    <=> ( ( ( A1 = nil(A) )
          & ( A22 = nil(B) ) )
        | ? [X4: A,Y5: B,Xs3: list(A),Ys4: list(B)] :
            ( ( A1 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
            & ( A22 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),Ys4) )
            & pp(aa(B,bool,aa(A,fun(B,bool),R3,X4),Y5))
            & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,R3),Xs3),Ys4)) ) ) ) ).

% listrelp.simps
tff(fact_7218_listrelp__listrel__eq,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),X2: list(A),Xa2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),listrelp(A,B,aTP_Lamp_ho(set(product_prod(A,B)),fun(A,fun(B,bool)),R3)),X2),Xa2))
    <=> pp(aa(set(product_prod(list(A),list(B))),bool,aa(product_prod(list(A),list(B)),fun(set(product_prod(list(A),list(B))),bool),member(product_prod(list(A),list(B))),aa(list(B),product_prod(list(A),list(B)),product_Pair(list(A),list(B),X2),Xa2)),listrel(A,B,R3))) ) ).

% listrelp_listrel_eq
tff(fact_7219_min__list_Oelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: list(A),Y: A] :
          ( ( min_list(A,X) = Y )
         => ( ! [X3: A,Xs2: list(A)] :
                ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
               => ( Y != aa(list(A),A,case_list(A,A,X3,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_zv(A,fun(list(A),fun(A,fun(list(A),A))),X3),Xs2)),Xs2) ) )
           => ~ ( ( X = nil(A) )
               => ( Y != undefined(A) ) ) ) ) ) ).

% min_list.elims
tff(fact_7220_map__of__zip__nth,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),I2: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( distinct(A,Xs)
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(B),nat,size_size(list(B)),Ys)))
         => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),aa(nat,A,nth(A,Xs),I2)) = aa(B,option(B),some(B),aa(nat,B,nth(B,Ys),I2)) ) ) ) ) ).

% map_of_zip_nth
tff(fact_7221_map__of__eq__empty__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] :
      ( ! [X4: A] : aa(A,option(B),map_of(A,B,Xys),X4) = none(B)
    <=> ( Xys = nil(product_prod(A,B)) ) ) ).

% map_of_eq_empty_iff
tff(fact_7222_empty__eq__map__of__iff,axiom,
    ! [B: $tType,A: $tType,Xys: list(product_prod(A,B))] :
      ( ( aTP_Lamp_zy(A,option(B)) = map_of(A,B,Xys) )
    <=> ( Xys = nil(product_prod(A,B)) ) ) ).

% empty_eq_map_of_iff
tff(fact_7223_map__of__zip__is__None,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),X) = none(B) )
      <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% map_of_zip_is_None
tff(fact_7224_map__of_Osimps_I1_J,axiom,
    ! [A: $tType,B: $tType,X2: A] : aa(A,option(B),map_of(A,B,nil(product_prod(A,B))),X2) = none(B) ).

% map_of.simps(1)
tff(fact_7225_map__of__Cons__code_I1_J,axiom,
    ! [B: $tType,A: $tType,K: B] : aa(B,option(A),map_of(B,A,nil(product_prod(B,A))),K) = none(A) ).

% map_of_Cons_code(1)
tff(fact_7226_hd__def,axiom,
    ! [A: $tType,List: list(A)] : aa(list(A),A,hd(A),List) = aa(list(A),A,case_list(A,A,undefined(A),aTP_Lamp_zz(A,fun(list(A),A))),List) ).

% hd_def
tff(fact_7227_map__of__zip__inject,axiom,
    ! [B: $tType,A: $tType,Ys: list(A),Xs: list(B),Zs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Ys) = aa(list(B),nat,size_size(list(B)),Xs) )
     => ( ( aa(list(A),nat,size_size(list(A)),Zs) = aa(list(B),nat,size_size(list(B)),Xs) )
       => ( distinct(B,Xs)
         => ( ( map_of(B,A,zip(B,A,Xs,Ys)) = map_of(B,A,zip(B,A,Xs,Zs)) )
           => ( Ys = Zs ) ) ) ) ) ).

% map_of_zip_inject
tff(fact_7228_map__of__zip__is__Some,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),X: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
      <=> ? [Y5: B] : aa(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),X) = aa(B,option(B),some(B),Y5) ) ) ).

% map_of_zip_is_Some
tff(fact_7229_map__of__eq__None__iff,axiom,
    ! [A: $tType,B: $tType,Xys: list(product_prod(B,A)),X: B] :
      ( ( aa(B,option(A),map_of(B,A,Xys),X) = none(A) )
    <=> ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),aa(set(product_prod(B,A)),set(B),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_7230_min__list_Opelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X: list(A),Y: A] :
          ( ( min_list(A,X) = Y )
         => ( pp(aa(list(A),bool,accp(list(A),min_list_rel(A)),X))
           => ( ! [X3: A,Xs2: list(A)] :
                  ( ( X = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2) )
                 => ( ( Y = aa(list(A),A,case_list(A,A,X3,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_zv(A,fun(list(A),fun(A,fun(list(A),A))),X3),Xs2)),Xs2) )
                   => ~ pp(aa(list(A),bool,accp(list(A),min_list_rel(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2))) ) )
             => ~ ( ( X = nil(A) )
                 => ( ( Y = undefined(A) )
                   => ~ pp(aa(list(A),bool,accp(list(A),min_list_rel(A)),nil(A))) ) ) ) ) ) ) ).

% min_list.pelims
tff(fact_7231_arg__min__list_Oelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [X: fun(A,B),Xa: list(A),Y: A] :
          ( ( arg_min_list(A,B,X,Xa) = Y )
         => ( ! [X3: A] :
                ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
               => ( Y != X3 ) )
           => ( ! [X3: A,Y4: A,Zs2: list(A)] :
                  ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2)) )
                 => ( Y != if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,X,X3)),aa(A,B,X,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2)))),X3,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2))) ) )
             => ~ ( ( Xa = nil(A) )
                 => ( Y != undefined(A) ) ) ) ) ) ) ).

% arg_min_list.elims
tff(fact_7232_arg__min__list_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),X: A] : arg_min_list(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) = X ) ).

% arg_min_list.simps(1)
tff(fact_7233_arg__min__list__in,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F3: fun(A,B)] :
          ( ( Xs != nil(A) )
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),arg_min_list(A,B,F3,Xs)),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% arg_min_list_in
tff(fact_7234_arg__min__list_Osimps_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),X: A,Y: A,Zs: list(A)] : arg_min_list(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs))) = if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X)),aa(A,B,F3,arg_min_list(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))),X,arg_min_list(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs))) ) ).

% arg_min_list.simps(2)
tff(fact_7235_arg__min__list_Opelims,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [X: fun(A,B),Xa: list(A),Y: A] :
          ( ( arg_min_list(A,B,X,Xa) = Y )
         => ( pp(aa(product_prod(fun(A,B),list(A)),bool,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),product_Pair(fun(A,B),list(A),X),Xa)))
           => ( ! [X3: A] :
                  ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A)) )
                 => ( ( Y = X3 )
                   => ~ pp(aa(product_prod(fun(A,B),list(A)),bool,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),product_Pair(fun(A,B),list(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),nil(A))))) ) )
             => ( ! [X3: A,Y4: A,Zs2: list(A)] :
                    ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2)) )
                   => ( ( Y = if(A,aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,X,X3)),aa(A,B,X,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2)))),X3,arg_min_list(A,B,X,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2))) )
                     => ~ pp(aa(product_prod(fun(A,B),list(A)),bool,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),product_Pair(fun(A,B),list(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y4),Zs2))))) ) )
               => ~ ( ( Xa = nil(A) )
                   => ( ( Y = undefined(A) )
                     => ~ pp(aa(product_prod(fun(A,B),list(A)),bool,accp(product_prod(fun(A,B),list(A)),arg_min_list_rel(A,B)),aa(list(A),product_prod(fun(A,B),list(A)),product_Pair(fun(A,B),list(A),X),nil(A)))) ) ) ) ) ) ) ) ).

% arg_min_list.pelims
tff(fact_7236_ran__map__of__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( distinct(A,Xs)
       => ( ran(A,B,map_of(A,B,zip(A,B,Xs,Ys))) = aa(list(B),set(B),set2(B),Ys) ) ) ) ).

% ran_map_of_zip
tff(fact_7237_ran__empty,axiom,
    ! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_aaa(B,option(A))) = bot_bot(set(A)) ).

% ran_empty
tff(fact_7238_ran__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : ran(A,B,M) = collect(B,aTP_Lamp_aab(fun(A,option(B)),fun(B,bool),M)) ).

% ran_def
tff(fact_7239_num__of__integer__code,axiom,
    ! [K: code_integer] :
      ( ( pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),one_one(code_integer)))
       => ( code_num_of_integer(K) = one2 ) )
      & ( ~ pp(aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less_eq(code_integer),K),one_one(code_integer)))
       => ( code_num_of_integer(K) = aa(product_prod(code_integer,code_integer),num,product_case_prod(code_integer,code_integer,num,aTP_Lamp_aac(code_integer,fun(code_integer,num))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit0,one2)))) ) ) ) ).

% num_of_integer_code
tff(fact_7240_dual__max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( max(A,aTP_Lamp_vr(A,fun(A,bool))) = ord_min(A) ) ) ).

% dual_max
tff(fact_7241_dual__min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( min(A,aTP_Lamp_vr(A,fun(A,bool))) = ord_max(A) ) ) ).

% dual_min
tff(fact_7242_listrel1__subset__listrel,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),R4: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),R4))
     => ( refl_on(A,top_top(set(A)),R4)
       => pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel1(A,R3)),listrel(A,A,R4))) ) ) ).

% listrel1_subset_listrel
tff(fact_7243_filterlim__finite__subsets__at__top,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,set(B)),A3: set(B),F5: filter(A)] :
      ( filterlim(A,set(B),F3,finite5375528669736107172at_top(B,A3),F5)
    <=> ! [X8: set(B)] :
          ( ( pp(aa(set(B),bool,finite_finite(B),X8))
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),X8),A3)) )
         => eventually(A,aa(set(B),fun(A,bool),aa(set(B),fun(set(B),fun(A,bool)),aTP_Lamp_aad(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,bool))),F3),A3),X8),F5) ) ) ).

% filterlim_finite_subsets_at_top
tff(fact_7244_card__Min__le__sum,axiom,
    ! [A: $tType,A3: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(nat),nat,lattic643756798350308766er_Min(nat),aa(set(A),set(nat),image(A,nat,F3),A3)))),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),A3))) ) ).

% card_Min_le_sum
tff(fact_7245_Min__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A] : aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% Min_singleton
tff(fact_7246_eventually__finite__subsets__at__top__weakI,axiom,
    ! [A: $tType,A3: set(A),P: fun(set(A),bool)] :
      ( ! [X7: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),X7))
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X7),A3))
           => pp(aa(set(A),bool,P,X7)) ) )
     => eventually(set(A),P,finite5375528669736107172at_top(A,A3)) ) ).

% eventually_finite_subsets_at_top_weakI
tff(fact_7247_Min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3)))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X4)) ) ) ) ) ) ).

% Min.bounded_iff
tff(fact_7248_Min__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3)))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X4)) ) ) ) ) ) ).

% Min_gr_iff
tff(fact_7249_Min__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [A3: set(B),C2: A] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ( A3 != bot_bot(set(B)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(B),set(A),image(B,A,aTP_Lamp_aae(A,fun(B,A),C2)),A3)) = C2 ) ) ) ) ).

% Min_const
tff(fact_7250_Min__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3)) ) ) ) ) ).

% Min_insert
tff(fact_7251_Min_Oin__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3)) = aa(set(A),A,lattic643756798350308766er_Min(A),A3) ) ) ) ) ).

% Min.in_idem
tff(fact_7252_Min__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),A3)) ) ) ) ).

% Min_in
tff(fact_7253_Min_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),A2)) ) ) ) ).

% Min.coboundedI
tff(fact_7254_Min__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ! [Y4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y4)) )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = X ) ) ) ) ) ).

% Min_eqI
tff(fact_7255_Min__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),X)) ) ) ) ).

% Min_le
tff(fact_7256_Min__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = M )
            <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M),A3))
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),X4)) ) ) ) ) ) ) ).

% Min_eq_iff
tff(fact_7257_Min__le__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),X))
            <=> ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),X)) ) ) ) ) ) ).

% Min_le_iff
tff(fact_7258_eq__Min__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( M = aa(set(A),A,lattic643756798350308766er_Min(A),A3) )
            <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M),A3))
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),M),X4)) ) ) ) ) ) ) ).

% eq_Min_iff
tff(fact_7259_Min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3)))
             => ! [A16: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A16),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A16)) ) ) ) ) ) ).

% Min.boundedE
tff(fact_7260_Min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A5)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3))) ) ) ) ) ).

% Min.boundedI
tff(fact_7261_Min__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),X))
            <=> ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),X)) ) ) ) ) ) ).

% Min_less_iff
tff(fact_7262_Min__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ! [B5: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B5),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),B5)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3)) = A2 ) ) ) ) ).

% Min_insert2
tff(fact_7263_Min__Inf,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = aa(set(A),A,complete_Inf_Inf(A),A3) ) ) ) ) ).

% Min_Inf
tff(fact_7264_min__list__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( ( Xs != nil(A) )
         => ( min_list(A,Xs) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(list(A),set(A),set2(A),Xs)) ) ) ) ).

% min_list_Min
tff(fact_7265_eventually__finite__subsets__at__top,axiom,
    ! [A: $tType,P: fun(set(A),bool),A3: set(A)] :
      ( eventually(set(A),P,finite5375528669736107172at_top(A,A3))
    <=> ? [X8: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),X8))
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X8),A3))
          & ! [Y7: set(A)] :
              ( ( pp(aa(set(A),bool,finite_finite(A),Y7))
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),X8),Y7))
                & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Y7),A3)) )
             => pp(aa(set(A),bool,P,Y7)) ) ) ) ).

% eventually_finite_subsets_at_top
tff(fact_7266_Min__antimono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M6: set(A),N3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),M6),N3))
         => ( ( M6 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite(A),N3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),N3)),aa(set(A),A,lattic643756798350308766er_Min(A),M6))) ) ) ) ) ).

% Min_antimono
tff(fact_7267_Min_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite(A),B3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798350308766er_Min(A),B3)),aa(set(A),A,lattic643756798350308766er_Min(A),A3))) ) ) ) ) ).

% Min.subset_imp
tff(fact_7268_hom__Min__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H: fun(A,A),N3: set(A)] :
          ( ! [X3: A,Y4: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Y4)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(A,A,H,X3)),aa(A,A,H,Y4))
         => ( pp(aa(set(A),bool,finite_finite(A),N3))
           => ( ( N3 != bot_bot(set(A)) )
             => ( aa(A,A,H,aa(set(A),A,lattic643756798350308766er_Min(A),N3)) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),image(A,A,H),N3)) ) ) ) ) ) ).

% hom_Min_commute
tff(fact_7269_Min_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( B3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
             => ( aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),B3)),aa(set(A),A,lattic643756798350308766er_Min(A),A3)) = aa(set(A),A,lattic643756798350308766er_Min(A),A3) ) ) ) ) ) ).

% Min.subset
tff(fact_7270_Min_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X3: A,Y4: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),ord_min(A),X3),Y4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y4),bot_bot(set(A))))))
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),A3)) ) ) ) ) ).

% Min.closed
tff(fact_7271_Min_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( ( A3 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),A3)) ) ) ) ) ) ).

% Min.insert_not_elem
tff(fact_7272_mono__Min__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F3: fun(A,B),A3: set(A)] :
          ( order_mono(A,B,F3)
         => ( pp(aa(set(A),bool,finite_finite(A),A3))
           => ( ( A3 != bot_bot(set(A)) )
             => ( aa(A,B,F3,aa(set(A),A,lattic643756798350308766er_Min(A),A3)) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image(A,B,F3),A3)) ) ) ) ) ) ).

% mono_Min_commute
tff(fact_7273_Min_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite(A),B3))
             => ( ( B3 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),aa(set(A),A,lattic643756798350308766er_Min(A),B3)) ) ) ) ) ) ) ).

% Min.union
tff(fact_7274_Min__add__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( linord4140545234300271783up_add(A)
     => ! [S2: set(B),F3: fun(B,A),K: A] :
          ( pp(aa(set(B),bool,finite_finite(B),S2))
         => ( ( S2 != bot_bot(set(B)) )
           => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_aaf(fun(B,A),fun(A,fun(B,A)),F3),K)),S2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(B),set(A),image(B,A,F3),S2))),K) ) ) ) ) ).

% Min_add_commute
tff(fact_7275_f__arg__min__list__f,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xs: list(A),F3: fun(A,B)] :
          ( ( Xs != nil(A) )
         => ( aa(A,B,F3,arg_min_list(A,B,F3,Xs)) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image(A,B,F3),aa(list(A),set(A),set2(A),Xs))) ) ) ) ).

% f_arg_min_list_f
tff(fact_7276_finite__subsets__at__top__def,axiom,
    ! [A: $tType,A3: set(A)] : finite5375528669736107172at_top(A,A3) = aa(set(filter(set(A))),filter(set(A)),complete_Inf_Inf(filter(set(A))),aa(set(set(A)),set(filter(set(A))),image(set(A),filter(set(A)),aTP_Lamp_aah(set(A),fun(set(A),filter(set(A))),A3)),collect(set(A),aTP_Lamp_aai(set(A),fun(set(A),bool),A3)))) ).

% finite_subsets_at_top_def
tff(fact_7277_Min_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( ( ( aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = X ) )
              & ( ( aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Min.remove
tff(fact_7278_Min_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( ( aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = X ) )
            & ( ( aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(A,A,aa(A,fun(A,A),ord_min(A),X),aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Min.insert_remove
tff(fact_7279_arg__min__SOME__Min,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S2: set(A),F3: fun(A,B)] :
          ( pp(aa(set(A),bool,finite_finite(A),S2))
         => ( lattic7623131987881927897min_on(A,B,F3,S2) = fChoice(A,aa(fun(A,B),fun(A,bool),aTP_Lamp_aaj(set(A),fun(fun(A,B),fun(A,bool)),S2),F3)) ) ) ) ).

% arg_min_SOME_Min
tff(fact_7280_sorted__list__of__set__nonempty,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),aa(set(A),list(A),linord4507533701916653071of_set(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),aa(set(A),A,lattic643756798350308766er_Min(A),A3)),bot_bot(set(A)))))) ) ) ) ) ).

% sorted_list_of_set_nonempty
tff(fact_7281_dual__Max,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Max(A,aTP_Lamp_vr(A,fun(A,bool))) = lattic643756798350308766er_Min(A) ) ) ).

% dual_Max
tff(fact_7282_minus__Max__eq__Min,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S2: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),S2))
         => ( ( S2 != bot_bot(set(A)) )
           => ( aa(A,A,uminus_uminus(A),aa(set(A),A,lattic643756798349783984er_Max(A),S2)) = aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),image(A,A,uminus_uminus(A)),S2)) ) ) ) ) ).

% minus_Max_eq_Min
tff(fact_7283_Max__singleton,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A] : aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% Max_singleton
tff(fact_7284_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),X))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),X)) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_7285_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),X))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),X)) ) ) ) ) ) ).

% Max_less_iff
tff(fact_7286_Max__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [A3: set(B),C2: A] :
          ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ( A3 != bot_bot(set(B)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(B),set(A),image(B,A,aTP_Lamp_aae(A,fun(B,A),C2)),A3)) = C2 ) ) ) ) ).

% Max_const
tff(fact_7287_Max__insert,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) ) ) ) ) ).

% Max_insert
tff(fact_7288_minus__Min__eq__Max,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [S2: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),S2))
         => ( ( S2 != bot_bot(set(A)) )
           => ( aa(A,A,uminus_uminus(A),aa(set(A),A,lattic643756798350308766er_Min(A),S2)) = aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),image(A,A,uminus_uminus(A)),S2)) ) ) ) ) ).

% minus_Min_eq_Max
tff(fact_7289_Max__ge,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A3))) ) ) ) ).

% Max_ge
tff(fact_7290_Max__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ! [Y4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y4),X)) )
           => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = X ) ) ) ) ) ).

% Max_eqI
tff(fact_7291_Max__eq__if,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,finite_finite(A),B3))
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                 => ? [Xa2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),B3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa2)) ) )
             => ( ! [X3: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),B3))
                   => ? [Xa2: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),A3))
                        & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Xa2)) ) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(set(A),A,lattic643756798349783984er_Max(A),B3) ) ) ) ) ) ) ).

% Max_eq_if
tff(fact_7292_Max_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(set(A),A,lattic643756798349783984er_Max(A),A3))) ) ) ) ).

% Max.coboundedI
tff(fact_7293_linorder_OMax_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : lattices_Max(A,Less_eq) = lattices_Max(A,Less_eq) ).

% linorder.Max.cong
tff(fact_7294_Max_Oin__idem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) = aa(set(A),A,lattic643756798349783984er_Max(A),A3) ) ) ) ) ).

% Max.in_idem
tff(fact_7295_Max__in,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),A3)) ) ) ) ).

% Max_in
tff(fact_7296_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),X)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),X)) ) ) ) ) ).

% Max.boundedI
tff(fact_7297_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),X))
             => ! [A16: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A16),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A16),X)) ) ) ) ) ) ).

% Max.boundedE
tff(fact_7298_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( M = aa(set(A),A,lattic643756798349783984er_Max(A),A3) )
            <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M),A3))
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),M)) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_7299_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A3)))
            <=> ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X4)) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_7300_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = M )
            <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),M),A3))
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),M)) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_7301_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A3)))
            <=> ? [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                  & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X4)) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_7302_Max__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ! [B5: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B5),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B5),A2)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),A3)) = A2 ) ) ) ) ).

% Max_insert2
tff(fact_7303_Max__Sup,axiom,
    ! [A: $tType] :
      ( comple5582772986160207858norder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(set(A),A,complete_Sup_Sup(A),A3) ) ) ) ) ).

% Max_Sup
tff(fact_7304_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite(A),B3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),aa(set(A),A,lattic643756798349783984er_Max(A),B3))) ) ) ) ) ).

% Max.subset_imp
tff(fact_7305_Max__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M6: set(A),N3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),M6),N3))
         => ( ( M6 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite(A),N3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic643756798349783984er_Max(A),M6)),aa(set(A),A,lattic643756798349783984er_Max(A),N3))) ) ) ) ) ).

% Max_mono
tff(fact_7306_hom__Max__commute,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [H: fun(A,A),N3: set(A)] :
          ( ! [X3: A,Y4: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Y4)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(A,A,H,X3)),aa(A,A,H,Y4))
         => ( pp(aa(set(A),bool,finite_finite(A),N3))
           => ( ( N3 != bot_bot(set(A)) )
             => ( aa(A,A,H,aa(set(A),A,lattic643756798349783984er_Max(A),N3)) = aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),image(A,A,H),N3)) ) ) ) ) ) ).

% hom_Max_commute
tff(fact_7307_Max_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( B3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
             => ( aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),B3)),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) = aa(set(A),A,lattic643756798349783984er_Max(A),A3) ) ) ) ) ) ).

% Max.subset
tff(fact_7308_Max_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( ( A3 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),A3)) ) ) ) ) ) ).

% Max.insert_not_elem
tff(fact_7309_Max_Oclosed,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X3: A,Y4: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),ord_max(A),X3),Y4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y4),bot_bot(set(A))))))
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),A3)) ) ) ) ) ).

% Max.closed
tff(fact_7310_mono__Max__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [F3: fun(A,B),A3: set(A)] :
          ( order_mono(A,B,F3)
         => ( pp(aa(set(A),bool,finite_finite(A),A3))
           => ( ( A3 != bot_bot(set(A)) )
             => ( aa(A,B,F3,aa(set(A),A,lattic643756798349783984er_Max(A),A3)) = aa(set(B),B,lattic643756798349783984er_Max(B),aa(set(A),set(B),image(A,B,F3),A3)) ) ) ) ) ) ).

% mono_Max_commute
tff(fact_7311_Max_Ounion,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite(A),B3))
             => ( ( B3 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),aa(set(A),A,lattic643756798349783984er_Max(A),A3)),aa(set(A),A,lattic643756798349783984er_Max(A),B3)) ) ) ) ) ) ) ).

% Max.union
tff(fact_7312_card__le__Suc__Max,axiom,
    ! [S2: set(nat)] :
      ( pp(aa(set(nat),bool,finite_finite(nat),S2))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(nat),nat,finite_card(nat),S2)),aa(nat,nat,suc,aa(set(nat),nat,lattic643756798349783984er_Max(nat),S2)))) ) ).

% card_le_Suc_Max
tff(fact_7313_divide__nat__def,axiom,
    ! [N: nat,M: nat] :
      ( ( ( N = zero_zero(nat) )
       => ( divide_divide(nat,M,N) = zero_zero(nat) ) )
      & ( ( N != zero_zero(nat) )
       => ( divide_divide(nat,M,N) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_aak(nat,fun(nat,fun(nat,bool)),N),M))) ) ) ) ).

% divide_nat_def
tff(fact_7314_Max__add__commute,axiom,
    ! [B: $tType,A: $tType] :
      ( linord4140545234300271783up_add(A)
     => ! [S2: set(B),F3: fun(B,A),K: A] :
          ( pp(aa(set(B),bool,finite_finite(B),S2))
         => ( ( S2 != bot_bot(set(B)) )
           => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_aaf(fun(B,A),fun(A,fun(B,A)),F3),K)),S2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(B),set(A),image(B,A,F3),S2))),K) ) ) ) ) ).

% Max_add_commute
tff(fact_7315_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),N))
     => ( aa(nat,nat,aa(nat,fun(nat,nat),gcd_gcd(nat),M),N) = aa(set(nat),nat,lattic643756798349783984er_Max(nat),collect(nat,aa(nat,fun(nat,bool),aTP_Lamp_aal(nat,fun(nat,fun(nat,bool)),N),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_7316_Max_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( ( aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = X ) )
            & ( ( aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Max.insert_remove
tff(fact_7317_Max_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( ( ( aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = X ) )
              & ( ( aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(A,A,aa(A,fun(A,A),ord_max(A),X),aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Max.remove
tff(fact_7318_sum__le__card__Max,axiom,
    ! [A: $tType,A3: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F3),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(A),nat,finite_card(A),A3)),aa(set(nat),nat,lattic643756798349783984er_Max(nat),aa(set(A),set(nat),image(A,nat,F3),A3))))) ) ).

% sum_le_card_Max
tff(fact_7319_dual__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( lattices_Min(A,aTP_Lamp_vr(A,fun(A,bool))) = lattic643756798349783984er_Max(A) ) ) ).

% dual_Min
tff(fact_7320_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N)))
       => ( upt(aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N)) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(num,nat,numeral_numeral(nat),M)),upt(aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N)))
       => ( upt(aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N)) = nil(nat) ) ) ) ).

% upt_rec_numeral
tff(fact_7321_remdups__upt,axiom,
    ! [M: nat,N: nat] : remdups(nat,upt(M,N)) = upt(M,N) ).

% remdups_upt
tff(fact_7322_tl__upt,axiom,
    ! [M: nat,N: nat] : aa(list(nat),list(nat),tl(nat),upt(M,N)) = upt(aa(nat,nat,suc,M),N) ).

% tl_upt
tff(fact_7323_hd__upt,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => ( aa(list(nat),nat,hd(nat),upt(I2,J)) = I2 ) ) ).

% hd_upt
tff(fact_7324_drop__upt,axiom,
    ! [M: nat,I2: nat,J: nat] : drop(nat,M,upt(I2,J)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),M),J) ).

% drop_upt
tff(fact_7325_length__upt,axiom,
    ! [I2: nat,J: nat] : aa(list(nat),nat,size_size(list(nat)),upt(I2,J)) = aa(nat,nat,minus_minus(nat,J),I2) ).

% length_upt
tff(fact_7326_take__upt,axiom,
    ! [I2: nat,M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),M)),N))
     => ( take(nat,M,upt(I2,N)) = upt(I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),M)) ) ) ).

% take_upt
tff(fact_7327_upt__conv__Nil,axiom,
    ! [J: nat,I2: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I2))
     => ( upt(I2,J) = nil(nat) ) ) ).

% upt_conv_Nil
tff(fact_7328_sorted__list__of__set__range,axiom,
    ! [M: nat,N: nat] : aa(set(nat),list(nat),linord4507533701916653071of_set(nat),set_or7035219750837199246ssThan(nat,M,N)) = upt(M,N) ).

% sorted_list_of_set_range
tff(fact_7329_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J: nat] :
      ( ( upt(I2,J) = nil(nat) )
    <=> ( ( J = zero_zero(nat) )
        | pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),I2)) ) ) ).

% upt_eq_Nil_conv
tff(fact_7330_nth__upt,axiom,
    ! [I2: nat,K: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K)),J))
     => ( aa(nat,nat,nth(nat,upt(I2,J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),K) ) ) ).

% nth_upt
tff(fact_7331_greaterThanLessThan__upt,axiom,
    ! [N: nat,M: nat] : set_or5935395276787703475ssThan(nat,N,M) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,N),M)) ).

% greaterThanLessThan_upt
tff(fact_7332_atLeastLessThan__upt,axiom,
    ! [I2: nat,J: nat] : set_or7035219750837199246ssThan(nat,I2,J) = aa(list(nat),set(nat),set2(nat),upt(I2,J)) ).

% atLeastLessThan_upt
tff(fact_7333_atLeastAtMost__upt,axiom,
    ! [N: nat,M: nat] : set_or1337092689740270186AtMost(nat,N,M) = aa(list(nat),set(nat),set2(nat),upt(N,aa(nat,nat,suc,M))) ).

% atLeastAtMost_upt
tff(fact_7334_greaterThanAtMost__upt,axiom,
    ! [N: nat,M: nat] : set_or3652927894154168847AtMost(nat,N,M) = aa(list(nat),set(nat),set2(nat),upt(aa(nat,nat,suc,N),aa(nat,nat,suc,M))) ).

% greaterThanAtMost_upt
tff(fact_7335_atLeast__upt,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_lessThan(nat),N) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),N)) ).

% atLeast_upt
tff(fact_7336_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list(nat),Q3: nat] :
      ( ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),M),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),N),Ns)) = upt(M,Q3) )
    <=> ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),N),Ns) = upt(aa(nat,nat,suc,M),Q3) ) ) ).

% upt_conv_Cons_Cons
tff(fact_7337_distinct__upt,axiom,
    ! [I2: nat,J: nat] : distinct(nat,upt(I2,J)) ).

% distinct_upt
tff(fact_7338_linorder_OMin_Ocong,axiom,
    ! [A: $tType,Less_eq: fun(A,fun(A,bool))] : lattices_Min(A,Less_eq) = lattices_Min(A,Less_eq) ).

% linorder.Min.cong
tff(fact_7339_upt__0,axiom,
    ! [I2: nat] : upt(I2,zero_zero(nat)) = nil(nat) ).

% upt_0
tff(fact_7340_atMost__upto,axiom,
    ! [N: nat] : aa(nat,set(nat),set_ord_atMost(nat),N) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),aa(nat,nat,suc,N))) ).

% atMost_upto
tff(fact_7341_upt__conv__Cons,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
     => ( upt(I2,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I2),upt(aa(nat,nat,suc,I2),J)) ) ) ).

% upt_conv_Cons
tff(fact_7342_enumerate__eq__zip,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : enumerate(A,N,Xs) = zip(nat,A,upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))),Xs) ).

% enumerate_eq_zip
tff(fact_7343_upt__add__eq__append,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( upt(I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I2,J)),upt(J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% upt_add_eq_append
tff(fact_7344_upt__eq__Cons__conv,axiom,
    ! [I2: nat,J: nat,X: nat,Xs: list(nat)] :
      ( ( upt(I2,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X),Xs) )
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
        & ( I2 = X )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat)),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_7345_upt__rec,axiom,
    ! [I2: nat,J: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
       => ( upt(I2,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I2),upt(aa(nat,nat,suc,I2),J)) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
       => ( upt(I2,J) = nil(nat) ) ) ) ).

% upt_rec
tff(fact_7346_upt__Suc,axiom,
    ! [I2: nat,J: nat] :
      ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
       => ( upt(I2,aa(nat,nat,suc,J)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I2,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) ) )
      & ( ~ pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
       => ( upt(I2,aa(nat,nat,suc,J)) = nil(nat) ) ) ) ).

% upt_Suc
tff(fact_7347_upt__Suc__append,axiom,
    ! [I2: nat,J: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
     => ( upt(I2,aa(nat,nat,suc,J)) = aa(list(nat),list(nat),aa(list(nat),fun(list(nat),list(nat)),append(nat),upt(I2,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_7348_horner__sum__bit__eq__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,N: nat] : groups4207007520872428315er_sum(bool,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(list(nat),list(bool),map(nat,bool,bit_se5641148757651400278ts_bit(A,A2)),upt(zero_zero(nat),N))) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A2) ) ).

% horner_sum_bit_eq_take_bit
tff(fact_7349_map__fst__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),list(nat),map(product_prod(nat,A),nat,product_fst(nat,A)),enumerate(A,N,Xs)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))) ).

% map_fst_enumerate
tff(fact_7350_map__ident,axiom,
    ! [A: $tType,X2: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_aam(A,A)),X2) = X2 ).

% map_ident
tff(fact_7351_list_Omap__disc__iff,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A2: list(A)] :
      ( ( aa(list(A),list(B),map(A,B,F3),A2) = nil(B) )
    <=> ( A2 = nil(A) ) ) ).

% list.map_disc_iff
tff(fact_7352_Nil__is__map__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] :
      ( ( nil(A) = aa(list(B),list(A),map(B,A,F3),Xs) )
    <=> ( Xs = nil(B) ) ) ).

% Nil_is_map_conv
tff(fact_7353_map__is__Nil__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = nil(A) )
    <=> ( Xs = nil(B) ) ) ).

% map_is_Nil_conv
tff(fact_7354_list_Omap__comp,axiom,
    ! [B: $tType,C: $tType,A: $tType,G: fun(B,C),F3: fun(A,B),V: list(A)] : aa(list(B),list(C),map(B,C,G),aa(list(A),list(B),map(A,B,F3),V)) = aa(list(A),list(C),map(A,C,aa(fun(A,B),fun(A,C),comp(B,C,A,G),F3)),V) ).

% list.map_comp
tff(fact_7355_List_Omap_Ocompositionality,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(B,C),G: fun(A,B),List: list(A)] : aa(list(B),list(C),map(B,C,F3),aa(list(A),list(B),map(A,B,G),List)) = aa(list(A),list(C),map(A,C,aa(fun(A,B),fun(A,C),comp(B,C,A,F3),G)),List) ).

% List.map.compositionality
tff(fact_7356_map__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,F3: fun(B,A),G: fun(C,B),Xs: list(C)] : aa(list(B),list(A),map(B,A,F3),aa(list(C),list(B),map(C,B,G),Xs)) = aa(list(C),list(A),map(C,A,aa(fun(C,B),fun(C,A),comp(B,A,C,F3),G)),Xs) ).

% map_map
tff(fact_7357_map__eq__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),G: fun(B,A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(B),list(A),map(B,A,G),Xs) )
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(list(B),set(B),set2(B),Xs)))
         => ( aa(B,A,F3,X4) = aa(B,A,G,X4) ) ) ) ).

% map_eq_conv
tff(fact_7358_length__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ).

% length_map
tff(fact_7359_map__append,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),Ys: list(B)] : aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(B),list(A),map(B,A,F3),Xs)),aa(list(B),list(A),map(B,A,F3),Ys)) ).

% map_append
tff(fact_7360_map__replicate,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),N: nat,X: B] : aa(list(B),list(A),map(B,A,F3),replicate(B,N,X)) = replicate(A,N,aa(B,A,F3,X)) ).

% map_replicate
tff(fact_7361_list_Oset__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),V: list(A)] : aa(list(B),set(B),set2(B),aa(list(A),list(B),map(A,B,F3),V)) = aa(set(A),set(B),image(A,B,F3),aa(list(A),set(A),set2(A),V)) ).

% list.set_map
tff(fact_7362_inj__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Xs: list(A),Ys: list(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( ( aa(list(A),list(B),map(A,B,F3),Xs) = aa(list(A),list(B),map(A,B,F3),Ys) )
      <=> ( Xs = Ys ) ) ) ).

% inj_map_eq_map
tff(fact_7363_map__comp__map,axiom,
    ! [B: $tType,C: $tType,A: $tType,F3: fun(C,B),G: fun(A,C)] : aa(fun(list(A),list(C)),fun(list(A),list(B)),comp(list(C),list(B),list(A),map(C,B,F3)),map(A,C,G)) = map(A,B,aa(fun(A,C),fun(A,B),comp(C,B,A,F3),G)) ).

% map_comp_map
tff(fact_7364_List_Omap_Ocomp,axiom,
    ! [C: $tType,B: $tType,A: $tType,F3: fun(B,C),G: fun(A,B)] : aa(fun(list(A),list(B)),fun(list(A),list(C)),comp(list(B),list(C),list(A),map(B,C,F3)),map(A,B,G)) = map(A,C,aa(fun(A,B),fun(A,C),comp(B,C,A,F3),G)) ).

% List.map.comp
tff(fact_7365_size__list__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,nat),G: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_list(A,F3),aa(list(B),list(A),map(B,A,G),Xs)) = aa(list(B),nat,size_list(B,aa(fun(B,A),fun(B,nat),comp(A,nat,B,F3),G)),Xs) ).

% size_list_map
tff(fact_7366_nth__map,axiom,
    ! [B: $tType,A: $tType,N: nat,Xs: list(A),F3: fun(A,B)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,B,nth(B,aa(list(A),list(B),map(A,B,F3),Xs)),N) = aa(A,B,F3,aa(nat,A,nth(A,Xs),N)) ) ) ).

% nth_map
tff(fact_7367_map__fst__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),zip(A,B,Xs,Ys)) = Xs ) ) ).

% map_fst_zip
tff(fact_7368_map__snd__zip,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),zip(A,B,Xs,Ys)) = Ys ) ) ).

% map_snd_zip
tff(fact_7369_concat__map__singleton,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : concat(A,aa(list(B),list(list(A)),map(B,list(A),aTP_Lamp_aan(fun(B,A),fun(B,list(A)),F3)),Xs)) = aa(list(B),list(A),map(B,A,F3),Xs) ).

% concat_map_singleton
tff(fact_7370_map__snd__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),list(A),map(product_prod(nat,A),A,product_snd(nat,A)),enumerate(A,N,Xs)) = Xs ).

% map_snd_enumerate
tff(fact_7371_inj__mapI,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => inj_on(list(A),list(B),map(A,B,F3),top_top(set(list(A)))) ) ).

% inj_mapI
tff(fact_7372_inj__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] :
      ( inj_on(list(A),list(B),map(A,B,F3),top_top(set(list(A))))
    <=> inj_on(A,B,F3,top_top(set(A))) ) ).

% inj_map
tff(fact_7373_enumerate__map__upt,axiom,
    ! [A: $tType,N: nat,F3: fun(nat,A),M: nat] : enumerate(A,N,aa(list(nat),list(A),map(nat,A,F3),upt(N,M))) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_aao(fun(nat,A),fun(nat,product_prod(nat,A)),F3)),upt(N,M)) ).

% enumerate_map_upt
tff(fact_7374_map__of__eqI,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),Ys: 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)),Ys)) )
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),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),X3) = aa(A,option(B),map_of(A,B,Ys),X3) ) )
       => ( map_of(A,B,Xs) = map_of(A,B,Ys) ) ) ) ).

% map_of_eqI
tff(fact_7375_zip__map1,axiom,
    ! [A: $tType,C: $tType,B: $tType,F3: fun(C,A),Xs: list(C),Ys: list(B)] : zip(A,B,aa(list(C),list(A),map(C,A,F3),Xs),Ys) = aa(list(product_prod(C,B)),list(product_prod(A,B)),map(product_prod(C,B),product_prod(A,B),product_case_prod(C,B,product_prod(A,B),aTP_Lamp_aap(fun(C,A),fun(C,fun(B,product_prod(A,B))),F3))),zip(C,B,Xs,Ys)) ).

% zip_map1
tff(fact_7376_zip__map2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),F3: fun(C,B),Ys: list(C)] : zip(A,B,Xs,aa(list(C),list(B),map(C,B,F3),Ys)) = aa(list(product_prod(A,C)),list(product_prod(A,B)),map(product_prod(A,C),product_prod(A,B),product_case_prod(A,C,product_prod(A,B),aTP_Lamp_aaq(fun(C,B),fun(A,fun(C,product_prod(A,B))),F3))),zip(A,C,Xs,Ys)) ).

% zip_map2
tff(fact_7377_map__zip__map,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,F3: fun(product_prod(B,C),A),G: fun(D,B),Xs: list(D),Ys: list(C)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F3),zip(B,C,aa(list(D),list(B),map(D,B,G),Xs),Ys)) = aa(list(product_prod(D,C)),list(A),map(product_prod(D,C),A,product_case_prod(D,C,A,aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_aar(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),F3),G))),zip(D,C,Xs,Ys)) ).

% map_zip_map
tff(fact_7378_zip__map__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,F3: fun(C,A),Xs: list(C),G: fun(D,B),Ys: list(D)] : zip(A,B,aa(list(C),list(A),map(C,A,F3),Xs),aa(list(D),list(B),map(D,B,G),Ys)) = aa(list(product_prod(C,D)),list(product_prod(A,B)),map(product_prod(C,D),product_prod(A,B),product_case_prod(C,D,product_prod(A,B),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_aas(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),F3),G))),zip(C,D,Xs,Ys)) ).

% zip_map_map
tff(fact_7379_map__zip__map2,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,F3: fun(product_prod(B,C),A),Xs: list(B),G: fun(D,C),Ys: list(D)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,F3),zip(B,C,Xs,aa(list(D),list(C),map(D,C,G),Ys))) = aa(list(product_prod(B,D)),list(A),map(product_prod(B,D),A,product_case_prod(B,D,A,aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_aat(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),F3),G))),zip(B,D,Xs,Ys)) ).

% map_zip_map2
tff(fact_7380_map2__map__map,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,H: fun(B,fun(C,A)),F3: fun(D,B),Xs: list(D),G: fun(D,C)] : aa(list(product_prod(B,C)),list(A),map(product_prod(B,C),A,product_case_prod(B,C,A,H)),zip(B,C,aa(list(D),list(B),map(D,B,F3),Xs),aa(list(D),list(C),map(D,C,G),Xs))) = aa(list(D),list(A),map(D,A,aa(fun(D,C),fun(D,A),aa(fun(D,B),fun(fun(D,C),fun(D,A)),aTP_Lamp_aau(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(D,C),fun(D,A))),H),F3),G)),Xs) ).

% map2_map_map
tff(fact_7381_remdups__adj__map__injective,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Xs: list(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( remdups_adj(B,aa(list(A),list(B),map(A,B,F3),Xs)) = aa(list(A),list(B),map(A,B,F3),remdups_adj(A,Xs)) ) ) ).

% remdups_adj_map_injective
tff(fact_7382_map__tl,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),tl(B),Xs)) = aa(list(A),list(A),tl(A),aa(list(B),list(A),map(B,A,F3),Xs)) ).

% map_tl
tff(fact_7383_list_Omap__sel_I2_J,axiom,
    ! [B: $tType,A: $tType,A2: list(A),F3: fun(A,B)] :
      ( ( A2 != nil(A) )
     => ( aa(list(B),list(B),tl(B),aa(list(A),list(B),map(A,B,F3),A2)) = aa(list(A),list(B),map(A,B,F3),aa(list(A),list(A),tl(A),A2)) ) ) ).

% list.map_sel(2)
tff(fact_7384_map__injective,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),Ys: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(B),list(A),map(B,A,F3),Ys) )
     => ( inj_on(B,A,F3,top_top(set(B)))
       => ( Xs = Ys ) ) ) ).

% map_injective
tff(fact_7385_list_Osimps_I8_J,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,B)] : aa(list(A),list(B),map(A,B,F3),nil(A)) = nil(B) ).

% list.simps(8)
tff(fact_7386_remdups__map__remdups,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : remdups(A,aa(list(B),list(A),map(B,A,F3),remdups(B,Xs))) = remdups(A,aa(list(B),list(A),map(B,A,F3),Xs)) ).

% remdups_map_remdups
tff(fact_7387_pair__list__eqI,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B))] :
      ( ( 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)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys) )
     => ( ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Xs) = aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Ys) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
tff(fact_7388_nths__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),I5: set(nat)] : nths(A,aa(list(B),list(A),map(B,A,F3),Xs),I5) = aa(list(B),list(A),map(B,A,F3),nths(B,Xs,I5)) ).

% nths_map
tff(fact_7389_map__concat,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(list(B))] : aa(list(B),list(A),map(B,A,F3),concat(B,Xs)) = concat(A,aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F3)),Xs)) ).

% map_concat
tff(fact_7390_list_Omap__ident,axiom,
    ! [A: $tType,T2: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_aam(A,A)),T2) = T2 ).

% list.map_ident
tff(fact_7391_rotate1__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : aa(list(A),list(A),rotate1(A),aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),rotate1(B),Xs)) ).

% rotate1_map
tff(fact_7392_map__eq__imp__length__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType,F3: fun(B,A),Xs: list(B),G: fun(C,A),Ys: list(C)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(C),list(A),map(C,A,G),Ys) )
     => ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_7393_list_Osize__gen__o__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,nat),G: fun(A,B)] : aa(fun(list(A),list(B)),fun(list(A),nat),comp(list(B),nat,list(A),size_list(B,F3)),map(A,B,G)) = size_list(A,aa(fun(A,B),fun(A,nat),comp(B,nat,A,F3),G)) ).

% list.size_gen_o_map
tff(fact_7394_rotate__map,axiom,
    ! [A: $tType,B: $tType,N: nat,F3: fun(B,A),Xs: list(B)] : aa(list(A),list(A),rotate(A,N),aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),rotate(B,N),Xs)) ).

% rotate_map
tff(fact_7395_hd__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F3: fun(A,B)] :
      ( ( Xs != nil(A) )
     => ( aa(list(B),B,hd(B),aa(list(A),list(B),map(A,B,F3),Xs)) = aa(A,B,F3,aa(list(A),A,hd(A),Xs)) ) ) ).

% hd_map
tff(fact_7396_list_Omap__sel_I1_J,axiom,
    ! [B: $tType,A: $tType,A2: list(A),F3: fun(A,B)] :
      ( ( A2 != nil(A) )
     => ( aa(list(B),B,hd(B),aa(list(A),list(B),map(A,B,F3),A2)) = aa(A,B,F3,aa(list(A),A,hd(A),A2)) ) ) ).

% list.map_sel(1)
tff(fact_7397_take__map,axiom,
    ! [A: $tType,B: $tType,N: nat,F3: fun(B,A),Xs: list(B)] : take(A,N,aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),take(B,N,Xs)) ).

% take_map
tff(fact_7398_drop__map,axiom,
    ! [A: $tType,B: $tType,N: nat,F3: fun(B,A),Xs: list(B)] : drop(A,N,aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),drop(B,N,Xs)) ).

% drop_map
tff(fact_7399_map__replicate__const,axiom,
    ! [B: $tType,A: $tType,K: A,Lst: list(B)] : aa(list(B),list(A),map(B,A,aTP_Lamp_aav(A,fun(B,A),K)),Lst) = replicate(A,aa(list(B),nat,size_size(list(B)),Lst),K) ).

% map_replicate_const
tff(fact_7400_append__eq__map__conv,axiom,
    ! [A: $tType,B: $tType,Ys: list(A),Zs: list(A),F3: fun(B,A),Xs: list(B)] :
      ( ( aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) = aa(list(B),list(A),map(B,A,F3),Xs) )
    <=> ? [Us2: list(B),Vs3: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),Vs3) )
          & ( Ys = aa(list(B),list(A),map(B,A,F3),Us2) )
          & ( Zs = aa(list(B),list(A),map(B,A,F3),Vs3) ) ) ) ).

% append_eq_map_conv
tff(fact_7401_map__eq__append__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Ys),Zs) )
    <=> ? [Us2: list(B),Vs3: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),Vs3) )
          & ( Ys = aa(list(B),list(A),map(B,A,F3),Us2) )
          & ( Zs = aa(list(B),list(A),map(B,A,F3),Vs3) ) ) ) ).

% map_eq_append_conv
tff(fact_7402_list_Osimps_I9_J,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),X21: A,X222: list(A)] : aa(list(A),list(B),map(A,B,F3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),aa(A,B,F3,X21)),aa(list(A),list(B),map(A,B,F3),X222)) ).

% list.simps(9)
tff(fact_7403_Cons__eq__map__D,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),F3: fun(B,A),Ys: list(B)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(B),list(A),map(B,A,F3),Ys) )
     => ? [Z2: B,Zs2: list(B)] :
          ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z2),Zs2) )
          & ( X = aa(B,A,F3,Z2) )
          & ( Xs = aa(list(B),list(A),map(B,A,F3),Zs2) ) ) ) ).

% Cons_eq_map_D
tff(fact_7404_map__eq__Cons__D,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),Xs: list(B),Y: A,Ys: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
     => ? [Z2: B,Zs2: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z2),Zs2) )
          & ( aa(B,A,F3,Z2) = Y )
          & ( aa(list(B),list(A),map(B,A,F3),Zs2) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_7405_Cons__eq__map__conv,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),F3: fun(B,A),Ys: list(B)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(B),list(A),map(B,A,F3),Ys) )
    <=> ? [Z5: B,Zs3: list(B)] :
          ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z5),Zs3) )
          & ( X = aa(B,A,F3,Z5) )
          & ( Xs = aa(list(B),list(A),map(B,A,F3),Zs3) ) ) ) ).

% Cons_eq_map_conv
tff(fact_7406_map__eq__Cons__conv,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,A),Xs: list(B),Y: A,Ys: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
    <=> ? [Z5: B,Zs3: list(B)] :
          ( ( Xs = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z5),Zs3) )
          & ( aa(B,A,F3,Z5) = Y )
          & ( aa(list(B),list(A),map(B,A,F3),Zs3) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_7407_map__update,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),K: nat,Y: B] : aa(list(B),list(A),map(B,A,F3),list_update(B,Xs,K,Y)) = list_update(A,aa(list(B),list(A),map(B,A,F3),Xs),K,aa(B,A,F3,Y)) ).

% map_update
tff(fact_7408_list_Omap__cong,axiom,
    ! [B: $tType,A: $tType,X: list(A),Ya: list(A),F3: fun(A,B),G: fun(A,B)] :
      ( ( X = Ya )
     => ( ! [Z2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z2),aa(list(A),set(A),set2(A),Ya)))
           => ( aa(A,B,F3,Z2) = aa(A,B,G,Z2) ) )
       => ( aa(list(A),list(B),map(A,B,F3),X) = aa(list(A),list(B),map(A,B,G),Ya) ) ) ) ).

% list.map_cong
tff(fact_7409_list_Omap__cong0,axiom,
    ! [B: $tType,A: $tType,X: list(A),F3: fun(A,B),G: fun(A,B)] :
      ( ! [Z2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z2),aa(list(A),set(A),set2(A),X)))
         => ( aa(A,B,F3,Z2) = aa(A,B,G,Z2) ) )
     => ( aa(list(A),list(B),map(A,B,F3),X) = aa(list(A),list(B),map(A,B,G),X) ) ) ).

% list.map_cong0
tff(fact_7410_list_Oinj__map__strong,axiom,
    ! [B: $tType,A: $tType,X: list(A),Xa: list(A),F3: fun(A,B),Fa: fun(A,B)] :
      ( ! [Z2: A,Za: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z2),aa(list(A),set(A),set2(A),X)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Za),aa(list(A),set(A),set2(A),Xa)))
           => ( ( aa(A,B,F3,Z2) = aa(A,B,Fa,Za) )
             => ( Z2 = Za ) ) ) )
     => ( ( aa(list(A),list(B),map(A,B,F3),X) = aa(list(A),list(B),map(A,B,Fa),Xa) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
tff(fact_7411_map__ext,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F3: fun(A,B),G: fun(A,B)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => ( aa(A,B,F3,X3) = aa(A,B,G,X3) ) )
     => ( aa(list(A),list(B),map(A,B,F3),Xs) = aa(list(A),list(B),map(A,B,G),Xs) ) ) ).

% map_ext
tff(fact_7412_map__idI,axiom,
    ! [A: $tType,Xs: list(A),F3: fun(A,A)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => ( aa(A,A,F3,X3) = X3 ) )
     => ( aa(list(A),list(A),map(A,A,F3),Xs) = Xs ) ) ).

% map_idI
tff(fact_7413_map__cong,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(A),F3: fun(A,B),G: fun(A,B)] :
      ( ( Xs = Ys )
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Ys)))
           => ( aa(A,B,F3,X3) = aa(A,B,G,X3) ) )
       => ( aa(list(A),list(B),map(A,B,F3),Xs) = aa(list(A),list(B),map(A,B,G),Ys) ) ) ) ).

% map_cong
tff(fact_7414_ex__map__conv,axiom,
    ! [B: $tType,A: $tType,Ys: list(B),F3: fun(A,B)] :
      ( ? [Xs3: list(A)] : Ys = aa(list(A),list(B),map(A,B,F3),Xs3)
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(list(B),set(B),set2(B),Ys)))
         => ? [Xa4: A] : X4 = aa(A,B,F3,Xa4) ) ) ).

% ex_map_conv
tff(fact_7415_distinct__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F3),Xs))
    <=> ( distinct(B,Xs)
        & inj_on(B,A,F3,aa(list(B),set(B),set2(B),Xs)) ) ) ).

% distinct_map
tff(fact_7416_map__inj__on,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),Ys: list(B)] :
      ( ( aa(list(B),list(A),map(B,A,F3),Xs) = aa(list(B),list(A),map(B,A,F3),Ys) )
     => ( inj_on(B,A,F3,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),Ys)))
       => ( Xs = Ys ) ) ) ).

% map_inj_on
tff(fact_7417_inj__on__map__eq__map,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Xs: list(A),Ys: list(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)))
     => ( ( aa(list(A),list(B),map(A,B,F3),Xs) = aa(list(A),list(B),map(A,B,F3),Ys) )
      <=> ( Xs = Ys ) ) ) ).

% inj_on_map_eq_map
tff(fact_7418_zip__map__fst__snd,axiom,
    ! [B: $tType,A: $tType,Zs: list(product_prod(A,B))] : zip(A,B,aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Zs),aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Zs)) = Zs ).

% zip_map_fst_snd
tff(fact_7419_image__set,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : aa(set(B),set(A),image(B,A,F3),aa(list(B),set(B),set2(B),Xs)) = aa(list(A),set(A),set2(A),aa(list(B),list(A),map(B,A,F3),Xs)) ).

% image_set
tff(fact_7420_map__removeAll__inj,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),X: A,Xs: list(A)] :
      ( inj_on(A,B,F3,top_top(set(A)))
     => ( aa(list(A),list(B),map(A,B,F3),aa(list(A),list(A),removeAll(A,X),Xs)) = aa(list(B),list(B),removeAll(B,aa(A,B,F3,X)),aa(list(A),list(B),map(A,B,F3),Xs)) ) ) ).

% map_removeAll_inj
tff(fact_7421_map__replicate__trivial,axiom,
    ! [A: $tType,X: A,I2: nat] : aa(list(nat),list(A),map(nat,A,aTP_Lamp_aaw(A,fun(nat,A),X)),upt(zero_zero(nat),I2)) = replicate(A,I2,X) ).

% map_replicate_trivial
tff(fact_7422_zip__eq__conv,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),Zs: list(product_prod(A,B))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( zip(A,B,Xs,Ys) = Zs )
      <=> ( ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Zs) = Xs )
          & ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Zs) = Ys ) ) ) ) ).

% zip_eq_conv
tff(fact_7423_distinct__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),X: B,Xs: list(B)] :
          ( distinct(A,aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),Xs)))
        <=> ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(B,A,F3,X)),aa(set(B),set(A),image(B,A,F3),aa(list(B),set(B),set2(B),Xs))))
            & distinct(A,aa(list(B),list(A),map(B,A,F3),Xs)) ) ) ) ).

% distinct_insort_key
tff(fact_7424_map__removeAll__inj__on,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),X: A,Xs: list(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(B),map(A,B,F3),aa(list(A),list(A),removeAll(A,X),Xs)) = aa(list(B),list(B),removeAll(B,aa(A,B,F3,X)),aa(list(A),list(B),map(A,B,F3),Xs)) ) ) ).

% map_removeAll_inj_on
tff(fact_7425_map__upt__Suc,axiom,
    ! [A: $tType,F3: fun(nat,A),N: nat] : aa(list(nat),list(A),map(nat,A,F3),upt(zero_zero(nat),aa(nat,nat,suc,N))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,F3,zero_zero(nat))),aa(list(nat),list(A),map(nat,A,aTP_Lamp_rx(fun(nat,A),fun(nat,A),F3)),upt(zero_zero(nat),N))) ).

% map_upt_Suc
tff(fact_7426_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_7427_inj__mapD,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B)] :
      ( inj_on(list(A),list(B),map(A,B,F3),top_top(set(list(A))))
     => inj_on(A,B,F3,top_top(set(A))) ) ).

% inj_mapD
tff(fact_7428_nth__map__upt,axiom,
    ! [A: $tType,I2: nat,N: nat,M: nat,F3: fun(nat,A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(nat,nat,minus_minus(nat,N),M)))
     => ( aa(nat,A,nth(A,aa(list(nat),list(A),map(nat,A,F3),upt(M,N))),I2) = aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I2)) ) ) ).

% nth_map_upt
tff(fact_7429_map__of__zip__map,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),F3: fun(A,B),X2: A] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,aa(list(A),list(B),map(A,B,F3),Xs))),X2) = aa(B,option(B),some(B),aa(A,B,F3,X2)) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
       => ( aa(A,option(B),map_of(A,B,zip(A,B,Xs,aa(list(A),list(B),map(A,B,F3),Xs))),X2) = none(B) ) ) ) ).

% map_of_zip_map
tff(fact_7430_map__fst__zip__take,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),zip(A,B,Xs,Ys)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)),Xs) ).

% map_fst_zip_take
tff(fact_7431_map__snd__zip__take,axiom,
    ! [B: $tType,A: $tType,Xs: list(B),Ys: list(A)] : aa(list(product_prod(B,A)),list(A),map(product_prod(B,A),A,product_snd(B,A)),zip(B,A,Xs,Ys)) = take(A,aa(nat,nat,aa(nat,fun(nat,nat),ord_min(nat),aa(list(B),nat,size_size(list(B)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)),Ys) ).

% map_snd_zip_take
tff(fact_7432_inj__on__mapI,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A3: set(list(A))] :
      ( inj_on(A,B,F3,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(list(A)),set(set(A)),image(list(A),set(A),set2(A)),A3)))
     => inj_on(list(A),list(B),map(A,B,F3),A3) ) ).

% inj_on_mapI
tff(fact_7433_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),N: nat,M: nat,F3: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,minus_minus(nat,N),M) )
     => ( ! [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
           => ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,F3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I4)) ) )
       => ( aa(list(nat),list(A),map(nat,A,F3),upt(M,N)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_7434_transpose__rectangle,axiom,
    ! [A: $tType,Xs: list(list(A)),N: nat] :
      ( ( ( Xs = nil(list(A)) )
       => ( N = zero_zero(nat) ) )
     => ( ! [I4: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),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),I4)) = N ) )
       => ( transpose(A,Xs) = aa(list(nat),list(list(A)),map(nat,list(A),aTP_Lamp_aay(list(list(A)),fun(nat,list(A)),Xs)),upt(zero_zero(nat),N)) ) ) ) ).

% transpose_rectangle
tff(fact_7435_sum__list__map__eq__sum__count2,axiom,
    ! [A: $tType,Xs: list(A),X6: set(A),F3: fun(A,nat)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),X6))
     => ( pp(aa(set(A),bool,finite_finite(A),X6))
       => ( groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),Xs)) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_aaz(list(A),fun(fun(A,nat),fun(A,nat)),Xs),F3)),X6) ) ) ) ).

% sum_list_map_eq_sum_count2
tff(fact_7436_sum__list__eq__0__iff,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Ns: list(A)] :
          ( ( groups8242544230860333062m_list(A,Ns) = zero_zero(A) )
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Ns)))
             => ( X4 = zero_zero(A) ) ) ) ) ).

% sum_list_eq_0_iff
tff(fact_7437_sum__list_OCons,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [X: A,Xs: list(A)] : groups8242544230860333062m_list(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),X),groups8242544230860333062m_list(A,Xs)) ) ).

% sum_list.Cons
tff(fact_7438_sum__list__append,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A),Ys: list(A)] : groups8242544230860333062m_list(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,Xs)),groups8242544230860333062m_list(A,Ys)) ) ).

% sum_list_append
tff(fact_7439_sum__list__upt,axiom,
    ! [M: nat,N: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),M),N))
     => ( groups8242544230860333062m_list(nat,upt(M,N)) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_fu(nat,nat)),set_or7035219750837199246ssThan(nat,M,N)) ) ) ).

% sum_list_upt
tff(fact_7440_map__Suc__upt,axiom,
    ! [M: nat,N: nat] : aa(list(nat),list(nat),map(nat,nat,suc),upt(M,N)) = upt(aa(nat,nat,suc,M),aa(nat,nat,suc,N)) ).

% map_Suc_upt
tff(fact_7441_zip__assoc,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys,Zs)) = aa(list(product_prod(product_prod(A,B),C)),list(product_prod(A,product_prod(B,C))),map(product_prod(product_prod(A,B),C),product_prod(A,product_prod(B,C)),product_case_prod(product_prod(A,B),C,product_prod(A,product_prod(B,C)),product_case_prod(A,B,fun(C,product_prod(A,product_prod(B,C))),aTP_Lamp_aba(A,fun(B,fun(C,product_prod(A,product_prod(B,C)))))))),zip(product_prod(A,B),C,zip(A,B,Xs,Ys),Zs)) ).

% zip_assoc
tff(fact_7442_product__concat__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product(A,B,Xs,Ys) = concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_abb(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs)) ).

% product_concat_map
tff(fact_7443_transpose__map__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(list(B))] : transpose(A,aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F3)),Xs)) = aa(list(list(B)),list(list(A)),map(list(B),list(A),map(B,A,F3)),transpose(B,Xs)) ).

% transpose_map_map
tff(fact_7444_sum__list__addf,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [F3: fun(B,A),G: fun(B,A),Xs: list(B)] : groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_fb(fun(B,A),fun(fun(B,A),fun(B,A)),F3),G)),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F3),Xs))),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,G),Xs))) ) ).

% sum_list_addf
tff(fact_7445_length__concat,axiom,
    ! [B: $tType,Xss: list(list(B))] : aa(list(B),nat,size_size(list(B)),concat(B,Xss)) = groups8242544230860333062m_list(nat,aa(list(list(B)),list(nat),map(list(B),nat,size_size(list(B))),Xss)) ).

% length_concat
tff(fact_7446_List_Obind__def,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F3: fun(A,list(B))] : bind(A,B,Xs,F3) = concat(B,aa(list(A),list(list(B)),map(A,list(B),F3),Xs)) ).

% List.bind_def
tff(fact_7447_n__lists_Osimps_I2_J,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : n_lists(A,aa(nat,nat,suc,N),Xs) = concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),aTP_Lamp_abd(list(A),fun(list(A),list(list(A))),Xs)),n_lists(A,N,Xs))) ).

% n_lists.simps(2)
tff(fact_7448_product__lists_Osimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] : product_lists(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),Xss)) = concat(list(A),aa(list(A),list(list(list(A))),map(A,list(list(A)),aTP_Lamp_abe(list(list(A)),fun(A,list(list(A))),Xss)),Xs)) ).

% product_lists.simps(2)
tff(fact_7449_member__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [X: A,Xs: list(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),groups8242544230860333062m_list(A,Xs))) ) ) ).

% member_le_sum_list
tff(fact_7450_zip__same__conv__map,axiom,
    ! [A: $tType,Xs: list(A)] : zip(A,A,Xs,Xs) = aa(list(A),list(product_prod(A,A)),map(A,product_prod(A,A),aTP_Lamp_abf(A,product_prod(A,A))),Xs) ).

% zip_same_conv_map
tff(fact_7451_transpose_Oelims,axiom,
    ! [A: $tType,X: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,X) = Y )
     => ( ( ( X = nil(list(A)) )
         => ( Y != nil(list(A)) ) )
       => ( ! [Xss2: list(list(A))] :
              ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2) )
             => ( Y != transpose(A,Xss2) ) )
         => ~ ! [X3: A,Xs2: list(A),Xss2: list(list(A))] :
                ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2) )
               => ( Y != aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_abg(A,fun(list(A),list(A))))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_abh(A,fun(list(A),list(list(A)))))),Xss2))))) ) ) ) ) ) ).

% transpose.elims
tff(fact_7452_transpose_Osimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Xss: list(list(A))] : transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_abg(A,fun(list(A),list(A))))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_abh(A,fun(list(A),list(list(A)))))),Xss))))) ).

% transpose.simps(3)
tff(fact_7453_transpose_Osimps_I2_J,axiom,
    ! [A: $tType,Xss: list(list(A))] : transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss)) = transpose(A,Xss) ).

% transpose.simps(2)
tff(fact_7454_transpose_Osimps_I1_J,axiom,
    ! [A: $tType] : transpose(A,nil(list(A))) = nil(list(A)) ).

% transpose.simps(1)
tff(fact_7455_inj__split__Cons,axiom,
    ! [A: $tType,X6: set(product_prod(list(A),A))] : inj_on(product_prod(list(A),A),list(A),product_case_prod(list(A),A,list(A),aTP_Lamp_abc(list(A),fun(A,list(A)))),X6) ).

% inj_split_Cons
tff(fact_7456_Groups__List_Osum__list__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X3)) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),groups8242544230860333062m_list(A,Xs))) ) ) ).

% Groups_List.sum_list_nonneg
tff(fact_7457_sum__list__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),X3)) )
         => ( ( groups8242544230860333062m_list(A,Xs) = zero_zero(A) )
          <=> ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
               => ( X4 = zero_zero(A) ) ) ) ) ) ).

% sum_list_nonneg_eq_0_iff
tff(fact_7458_sum__list__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xs: list(A)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),zero_zero(A))) )
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),groups8242544230860333062m_list(A,Xs)),zero_zero(A))) ) ) ).

% sum_list_nonpos
tff(fact_7459_sum__list__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Xs: list(A)] : pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),groups8242544230860333062m_list(A,Xs))),groups8242544230860333062m_list(A,aa(list(A),list(A),map(A,A,abs_abs(A)),Xs)))) ) ).

% sum_list_abs
tff(fact_7460_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_7461_map__add__upt,axiom,
    ! [N: nat,M: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_abi(nat,fun(nat,nat),N)),upt(zero_zero(nat),M)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% map_add_upt
tff(fact_7462_zip__left__commute,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C)] : zip(A,product_prod(B,C),Xs,zip(B,C,Ys,Zs)) = aa(list(product_prod(B,product_prod(A,C))),list(product_prod(A,product_prod(B,C))),map(product_prod(B,product_prod(A,C)),product_prod(A,product_prod(B,C)),product_case_prod(B,product_prod(A,C),product_prod(A,product_prod(B,C)),aTP_Lamp_abk(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))))),zip(B,product_prod(A,C),Ys,zip(A,C,Xs,Zs))) ).

% zip_left_commute
tff(fact_7463_zip__commute,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : zip(A,B,Xs,Ys) = aa(list(product_prod(B,A)),list(product_prod(A,B)),map(product_prod(B,A),product_prod(A,B),product_case_prod(B,A,product_prod(A,B),aTP_Lamp_abl(B,fun(A,product_prod(A,B))))),zip(B,A,Ys,Xs)) ).

% zip_commute
tff(fact_7464_sum__list__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & ordere6658533253407199908up_add(B) )
     => ! [Xs: list(A),F3: fun(A,B),G: fun(A,B)] :
          ( ! [X3: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,X3)),aa(A,B,G,X3))) )
         => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),Xs))),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G),Xs)))) ) ) ).

% sum_list_mono
tff(fact_7465_distinct__sum__list__conv__Sum,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs: list(A)] :
          ( distinct(A,Xs)
         => ( groups8242544230860333062m_list(A,Xs) = aa(set(A),A,groups7311177749621191930dd_sum(A,A,aTP_Lamp_abm(A,A)),aa(list(A),set(A),set2(A),Xs)) ) ) ) ).

% distinct_sum_list_conv_Sum
tff(fact_7466_zip__replicate1,axiom,
    ! [A: $tType,B: $tType,N: nat,X: A,Ys: list(B)] : zip(A,B,replicate(A,N,X),Ys) = aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),product_Pair(A,B,X)),take(B,N,Ys)) ).

% zip_replicate1
tff(fact_7467_subseqs_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : subseqs(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(list(A)),list(list(A)),aa(list(list(A)),fun(list(list(A)),list(list(A))),append(list(A)),aa(list(list(A)),list(list(A)),map(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X)),subseqs(A,Xs))),subseqs(A,Xs)) ).

% subseqs.simps(2)
tff(fact_7468_map__decr__upt,axiom,
    ! [M: nat,N: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_ym(nat,nat)),upt(aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = upt(M,N) ).

% map_decr_upt
tff(fact_7469_sum__list__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( monoid_add(B)
        & strict9044650504122735259up_add(B) )
     => ! [Xs: list(A),F3: fun(A,B),G: fun(A,B)] :
          ( ( Xs != nil(A) )
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
               => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,F3,X3)),aa(A,B,G,X3))) )
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,F3),Xs))),groups8242544230860333062m_list(B,aa(list(A),list(B),map(A,B,G),Xs)))) ) ) ) ).

% sum_list_strict_mono
tff(fact_7470_elem__le__sum__list,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [K: nat,Ns: list(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Ns)))
         => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Ns),K)),groups8242544230860333062m_list(A,Ns))) ) ) ).

% elem_le_sum_list
tff(fact_7471_transpose__empty,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( ( transpose(A,Xs) = nil(list(A)) )
    <=> ! [X4: list(A)] :
          ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),X4),aa(list(list(A)),set(list(A)),set2(list(A)),Xs)))
         => ( X4 = nil(A) ) ) ) ).

% transpose_empty
tff(fact_7472_zip__replicate2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),N: nat,Y: B] : zip(A,B,Xs,replicate(B,N,Y)) = aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_abl(B,fun(A,product_prod(A,B))),Y)),take(A,N,Xs)) ).

% zip_replicate2
tff(fact_7473_sum_Odistinct__set__conv__list,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Xs: list(B),G: fun(B,A)] :
          ( distinct(B,Xs)
         => ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(list(B),set(B),set2(B),Xs)) = groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,G),Xs)) ) ) ) ).

% sum.distinct_set_conv_list
tff(fact_7474_sum__list__distinct__conv__sum__set,axiom,
    ! [C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Xs: list(B),F3: fun(B,C)] :
          ( distinct(B,Xs)
         => ( groups8242544230860333062m_list(C,aa(list(B),list(C),map(B,C,F3),Xs)) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,F3),aa(list(B),set(B),set2(B),Xs)) ) ) ) ).

% sum_list_distinct_conv_sum_set
tff(fact_7475_sum__list__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [X: B,Xs: list(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),aa(list(B),set(B),set2(B),Xs)))
         => ( groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F3),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F3,X)),groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F3),remove1(B,X,Xs)))) ) ) ) ).

% sum_list_map_remove1
tff(fact_7476_sum__code,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),Xs: list(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),aa(list(B),set(B),set2(B),Xs)) = groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,G),remdups(B,Xs))) ) ).

% sum_code
tff(fact_7477_size__list__conv__sum__list,axiom,
    ! [B: $tType,F3: fun(B,nat),Xs: list(B)] : aa(list(B),nat,size_list(B,F3),Xs) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,aa(list(B),list(nat),map(B,nat,F3),Xs))),aa(list(B),nat,size_size(list(B)),Xs)) ).

% size_list_conv_sum_list
tff(fact_7478_product_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,X: A,Xs: list(A),Ys: list(B)] : product(A,B,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs),Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(list(product_prod(A,B)),fun(list(product_prod(A,B)),list(product_prod(A,B))),append(product_prod(A,B)),aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),product_Pair(A,B,X)),Ys)),product(A,B,Xs,Ys)) ).

% product.simps(2)
tff(fact_7479_sum__list__triv,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [R3: B,Xs: list(C)] : groups8242544230860333062m_list(B,aa(list(C),list(B),map(C,B,aTP_Lamp_abn(B,fun(C,B),R3)),Xs)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),aa(list(C),nat,size_size(list(C)),Xs))),R3) ) ).

% sum_list_triv
tff(fact_7480_sum__list__Suc,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A)] : groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,aTP_Lamp_kg(fun(A,nat),fun(A,nat),F3)),Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% sum_list_Suc
tff(fact_7481_extract__Cons__code,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
      ( ( pp(aa(A,bool,P,X))
       => ( extract(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),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)),nil(A)),aa(list(A),product_prod(A,list(A)),product_Pair(A,list(A),X),Xs))) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( extract(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = case_option(option(product_prod(list(A),product_prod(A,list(A)))),product_prod(list(A),product_prod(A,list(A))),none(product_prod(list(A),product_prod(A,list(A)))),product_case_prod(list(A),product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))),aTP_Lamp_abp(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),X)),extract(A,P,Xs)) ) ) ) ).

% extract_Cons_code
tff(fact_7482_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_abq(int,fun(int,int))),Qr) ).

% Divides.adjust_div_def
tff(fact_7483_enumerate__replicate__eq,axiom,
    ! [A: $tType,N: nat,M: nat,A2: A] : enumerate(A,N,replicate(A,M,A2)) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_abr(A,fun(nat,product_prod(nat,A)),A2)),upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))) ).

% enumerate_replicate_eq
tff(fact_7484_sum__list__sum__nth,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Xs: list(B)] : groups8242544230860333062m_list(B,Xs) = aa(set(nat),B,groups7311177749621191930dd_sum(nat,B,nth(B,Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% sum_list_sum_nth
tff(fact_7485_card__length__sum__list__rec,axiom,
    ! [M: nat,N3: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),one_one(nat)),M))
     => ( aa(set(list(nat)),nat,finite_card(list(nat)),collect(list(nat),aa(nat,fun(list(nat),bool),aTP_Lamp_abs(nat,fun(nat,fun(list(nat),bool)),M),N3))) = 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),bool),aTP_Lamp_abt(nat,fun(nat,fun(list(nat),bool)),M),N3)))),aa(set(list(nat)),nat,finite_card(list(nat)),collect(list(nat),aa(nat,fun(list(nat),bool),aTP_Lamp_abu(nat,fun(nat,fun(list(nat),bool)),M),N3)))) ) ) ).

% card_length_sum_list_rec
tff(fact_7486_card__length__sum__list,axiom,
    ! [M: nat,N3: nat] : aa(set(list(nat)),nat,finite_card(list(nat)),collect(list(nat),aa(nat,fun(list(nat),bool),aTP_Lamp_abs(nat,fun(nat,fun(list(nat),bool)),M),N3))) = aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),M)),one_one(nat))),N3) ).

% card_length_sum_list
tff(fact_7487_sum__list__map__eq__sum__count,axiom,
    ! [A: $tType,F3: fun(A,nat),Xs: list(A)] : groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),Xs)) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_abv(fun(A,nat),fun(list(A),fun(A,nat)),F3),Xs)),aa(list(A),set(A),set2(A),Xs)) ).

% sum_list_map_eq_sum_count
tff(fact_7488_sum__list__update,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [K: nat,Xs: list(A),X: A] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs)))
         => ( groups8242544230860333062m_list(A,list_update(A,Xs,K,X)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),groups8242544230860333062m_list(A,Xs)),X)),aa(nat,A,nth(A,Xs),K)) ) ) ) ).

% sum_list_update
tff(fact_7489_transpose_Opsimps_I3_J,axiom,
    ! [A: $tType,X: A,Xs: list(A),Xss: list(list(A))] :
      ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss)))
     => ( transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Xss)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_abg(A,fun(list(A),list(A))))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_abh(A,fun(list(A),list(list(A)))))),Xss))))) ) ) ).

% transpose.psimps(3)
tff(fact_7490_transpose_Opelims,axiom,
    ! [A: $tType,X: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,X) = Y )
     => ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),X))
       => ( ( ( X = nil(list(A)) )
           => ( ( Y = nil(list(A)) )
             => ~ pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),nil(list(A)))) ) )
         => ( ! [Xss2: list(list(A))] :
                ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2) )
               => ( ( Y = transpose(A,Xss2) )
                 => ~ pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2))) ) )
           => ~ ! [X3: A,Xs2: list(A),Xss2: list(list(A))] :
                  ( ( X = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2) )
                 => ( ( Y = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_abg(A,fun(list(A),list(A))))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_abh(A,fun(list(A),list(list(A)))))),Xss2))))) )
                   => ~ pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2))) ) ) ) ) ) ) ).

% transpose.pelims
tff(fact_7491_transpose_Opsimps_I1_J,axiom,
    ! [A: $tType] :
      ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),nil(list(A))))
     => ( transpose(A,nil(list(A))) = nil(list(A)) ) ) ).

% transpose.psimps(1)
tff(fact_7492_transpose_Opsimps_I2_J,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss)))
     => ( transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss)) = transpose(A,Xss) ) ) ).

% transpose.psimps(2)
tff(fact_7493_transpose_Opinduct,axiom,
    ! [A: $tType,A0: list(list(A)),P: fun(list(list(A)),bool)] :
      ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),A0))
     => ( ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),nil(list(A))))
         => pp(aa(list(list(A)),bool,P,nil(list(A)))) )
       => ( ! [Xss2: list(list(A))] :
              ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2)))
             => ( pp(aa(list(list(A)),bool,P,Xss2))
               => pp(aa(list(list(A)),bool,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2))) ) )
         => ( ! [X3: A,Xs2: list(A),Xss2: list(list(A))] :
                ( pp(aa(list(list(A)),bool,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2)))
               => ( pp(aa(list(list(A)),bool,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_abh(A,fun(list(A),list(list(A)))))),Xss2)))))
                 => pp(aa(list(list(A)),bool,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),Xss2))) ) )
           => pp(aa(list(list(A)),bool,P,A0)) ) ) ) ) ).

% transpose.pinduct
tff(fact_7494_product__code,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product_product(A,B,aa(list(A),set(A),set2(A),Xs),aa(list(B),set(B),set2(B),Ys)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_abb(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs))) ).

% product_code
tff(fact_7495_transpose__aux__filter__tail,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_abh(A,fun(list(A),list(list(A)))))),Xss)) = aa(list(list(A)),list(list(A)),map(list(A),list(A),tl(A)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_abw(list(A),bool)),Xss)) ).

% transpose_aux_filter_tail
tff(fact_7496_filter__filter,axiom,
    ! [A: $tType,P: fun(A,bool),Q: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),filter2(A,Q),Xs)) = aa(list(A),list(A),filter2(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_abx(fun(A,bool),fun(fun(A,bool),fun(A,bool)),P),Q)),Xs) ).

% filter_filter
tff(fact_7497_filter__True,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X3)) )
     => ( aa(list(A),list(A),filter2(A,P),Xs) = Xs ) ) ).

% filter_True
tff(fact_7498_filter__append,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Ys: list(A)] : aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),filter2(A,P),Xs)),aa(list(A),list(A),filter2(A,P),Ys)) ).

% filter_append
tff(fact_7499_remove1__filter__not,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
      ( ~ pp(aa(A,bool,P,X))
     => ( remove1(A,X,aa(list(A),list(A),filter2(A,P),Xs)) = aa(list(A),list(A),filter2(A,P),Xs) ) ) ).

% remove1_filter_not
tff(fact_7500_removeAll__filter__not,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
      ( ~ pp(aa(A,bool,P,X))
     => ( aa(list(A),list(A),removeAll(A,X),aa(list(A),list(A),filter2(A,P),Xs)) = aa(list(A),list(A),filter2(A,P),Xs) ) ) ).

% removeAll_filter_not
tff(fact_7501_set__filter,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,P),Xs)) = collect(A,aa(list(A),fun(A,bool),aTP_Lamp_aby(fun(A,bool),fun(list(A),fun(A,bool)),P),Xs)) ).

% set_filter
tff(fact_7502_filter__False,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => ~ pp(aa(A,bool,P,X3)) )
     => ( aa(list(A),list(A),filter2(A,P),Xs) = nil(A) ) ) ).

% filter_False
tff(fact_7503_length__filter__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F3: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),aa(list(B),list(A),map(B,A,F3),Xs))) = aa(list(B),nat,size_size(list(B)),aa(list(B),list(B),filter2(B,aa(fun(B,A),fun(B,bool),comp(A,bool,B,P),F3)),Xs)) ).

% length_filter_map
tff(fact_7504_filter__concat,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(list(A))] : aa(list(A),list(A),filter2(A,P2),concat(A,Xs)) = concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),filter2(A,P2)),Xs)) ).

% filter_concat
tff(fact_7505_distinct__map__filter,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B),P: fun(B,bool)] :
      ( distinct(A,aa(list(B),list(A),map(B,A,F3),Xs))
     => distinct(A,aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,P),Xs))) ) ).

% distinct_map_filter
tff(fact_7506_filter__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F3: fun(B,A),Xs: list(B)] : aa(list(A),list(A),filter2(A,P),aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,aa(fun(B,A),fun(B,bool),comp(A,bool,B,P),F3)),Xs)) ).

% filter_map
tff(fact_7507_inj__on__filter__key__eq,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Y: A,Xs: list(A)] :
      ( inj_on(A,B,F3,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(A),filter2(A,aa(A,fun(A,bool),aTP_Lamp_abz(fun(A,B),fun(A,fun(A,bool)),F3),Y)),Xs) = aa(list(A),list(A),filter2(A,aa(A,fun(A,bool),fequal(A),Y)),Xs) ) ) ).

% inj_on_filter_key_eq
tff(fact_7508_distinct__filter,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( distinct(A,Xs)
     => distinct(A,aa(list(A),list(A),filter2(A,P),Xs)) ) ).

% distinct_filter
tff(fact_7509_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),aa(list(A),list(A),filter2(A,aTP_Lamp_a(set(A),fun(A,bool),A3)),Xs)) ).

% inter_set_filter
tff(fact_7510_length__filter__le,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_filter_le
tff(fact_7511_remdups__filter,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : remdups(A,aa(list(A),list(A),filter2(A,P),Xs)) = aa(list(A),list(A),filter2(A,P),remdups(A,Xs)) ).

% remdups_filter
tff(fact_7512_sum__length__filter__compl,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aTP_Lamp_aca(fun(A,bool),fun(A,bool),P)),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ).

% sum_length_filter_compl
tff(fact_7513_length__filter__less,axiom,
    ! [A: $tType,X: A,Xs: list(A),P: fun(A,bool)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( ~ pp(aa(A,bool,P,X))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% length_filter_less
tff(fact_7514_replicate__length__filter,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : replicate(A,aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,aa(A,fun(A,bool),fequal(A),X)),Xs)),X) = aa(list(A),list(A),filter2(A,aa(A,fun(A,bool),fequal(A),X)),Xs) ).

% replicate_length_filter
tff(fact_7515_filter__cong,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),P: fun(A,bool),Q: fun(A,bool)] :
      ( ( Xs = Ys )
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Ys)))
           => ( pp(aa(A,bool,P,X3))
            <=> pp(aa(A,bool,Q,X3)) ) )
       => ( aa(list(A),list(A),filter2(A,P),Xs) = aa(list(A),list(A),filter2(A,Q),Ys) ) ) ) ).

% filter_cong
tff(fact_7516_filter__id__conv,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P),Xs) = Xs )
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X4)) ) ) ).

% filter_id_conv
tff(fact_7517_empty__filter__conv,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),filter2(A,P),Xs) )
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => ~ pp(aa(A,bool,P,X4)) ) ) ).

% empty_filter_conv
tff(fact_7518_filter__empty__conv,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P),Xs) = nil(A) )
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => ~ pp(aa(A,bool,P,X4)) ) ) ).

% filter_empty_conv
tff(fact_7519_filter__eq__Cons__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Ys: list(A),X: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
    <=> ? [Us2: list(A),Vs3: list(A)] :
          ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs3)) )
          & ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Us2)))
             => ~ pp(aa(A,bool,P,X4)) )
          & pp(aa(A,bool,P,X))
          & ( Xs = aa(list(A),list(A),filter2(A,P),Vs3) ) ) ) ).

% filter_eq_Cons_iff
tff(fact_7520_Cons__eq__filter__iff,axiom,
    ! [A: $tType,X: A,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(A),list(A),filter2(A,P),Ys) )
    <=> ? [Us2: list(A),Vs3: list(A)] :
          ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs3)) )
          & ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Us2)))
             => ~ pp(aa(A,bool,P,X4)) )
          & pp(aa(A,bool,P,X))
          & ( Xs = aa(list(A),list(A),filter2(A,P),Vs3) ) ) ) ).

% Cons_eq_filter_iff
tff(fact_7521_filter__eq__ConsD,axiom,
    ! [A: $tType,P: fun(A,bool),Ys: list(A),X: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),filter2(A,P),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) )
     => ? [Us3: list(A),Vs2: list(A)] :
          ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs2)) )
          & ! [X2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Us3)))
             => ~ pp(aa(A,bool,P,X2)) )
          & pp(aa(A,bool,P,X))
          & ( Xs = aa(list(A),list(A),filter2(A,P),Vs2) ) ) ) ).

% filter_eq_ConsD
tff(fact_7522_Cons__eq__filterD,axiom,
    ! [A: $tType,X: A,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) = aa(list(A),list(A),filter2(A,P),Ys) )
     => ? [Us3: list(A),Vs2: list(A)] :
          ( ( Ys = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Vs2)) )
          & ! [X2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Us3)))
             => ~ pp(aa(A,bool,P,X2)) )
          & pp(aa(A,bool,P,X))
          & ( Xs = aa(list(A),list(A),filter2(A,P),Vs2) ) ) ) ).

% Cons_eq_filterD
tff(fact_7523_filter__is__subset,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,P),Xs))),aa(list(A),set(A),set2(A),Xs))) ).

% filter_is_subset
tff(fact_7524_filter__shuffles,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Ys: list(A)] : aa(set(list(A)),set(list(A)),image(list(A),list(A),filter2(A,P)),shuffles(A,Xs,Ys)) = shuffles(A,aa(list(A),list(A),filter2(A,P),Xs),aa(list(A),list(A),filter2(A,P),Ys)) ).

% filter_shuffles
tff(fact_7525_filter__replicate,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,N: nat] :
      ( ( pp(aa(A,bool,P,X))
       => ( aa(list(A),list(A),filter2(A,P),replicate(A,N,X)) = replicate(A,N,X) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( aa(list(A),list(A),filter2(A,P),replicate(A,N,X)) = nil(A) ) ) ) ).

% filter_replicate
tff(fact_7526_filter_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
      ( ( pp(aa(A,bool,P,X))
       => ( aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),filter2(A,P),Xs)) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),filter2(A,P),Xs) ) ) ) ).

% filter.simps(2)
tff(fact_7527_partition__in__shuffles,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] : pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Xs),shuffles(A,aa(list(A),list(A),filter2(A,P),Xs),aa(list(A),list(A),filter2(A,aTP_Lamp_aca(fun(A,bool),fun(A,bool),P)),Xs)))) ).

% partition_in_shuffles
tff(fact_7528_filter__insort__triv,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [P: fun(B,bool),X: B,F3: fun(B,A),Xs: list(B)] :
          ( ~ pp(aa(B,bool,P,X))
         => ( aa(list(B),list(B),filter2(B,P),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),Xs)) = aa(list(B),list(B),filter2(B,P),Xs) ) ) ) ).

% filter_insort_triv
tff(fact_7529_filter__remove1,axiom,
    ! [A: $tType,Q: fun(A,bool),X: A,Xs: list(A)] : aa(list(A),list(A),filter2(A,Q),remove1(A,X,Xs)) = remove1(A,X,aa(list(A),list(A),filter2(A,Q),Xs)) ).

% filter_remove1
tff(fact_7530_removeAll__filter__not__eq,axiom,
    ! [A: $tType,X: A] : removeAll(A,X) = filter2(A,aTP_Lamp_acb(A,fun(A,bool),X)) ).

% removeAll_filter_not_eq
tff(fact_7531_filter_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,bool)] : aa(list(A),list(A),filter2(A,P),nil(A)) = nil(A) ).

% filter.simps(1)
tff(fact_7532_sum__list__filter__le__nat,axiom,
    ! [A: $tType,F3: fun(A,nat),P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),aa(list(A),list(A),filter2(A,P),Xs)))),groups8242544230860333062m_list(nat,aa(list(A),list(nat),map(A,nat,F3),Xs)))) ).

% sum_list_filter_le_nat
tff(fact_7533_filter__in__nths,axiom,
    ! [A: $tType,Xs: list(A),S: set(nat)] :
      ( distinct(A,Xs)
     => ( aa(list(A),list(A),filter2(A,aa(set(nat),fun(A,bool),aTP_Lamp_acc(list(A),fun(set(nat),fun(A,bool)),Xs),S)),Xs) = nths(A,Xs,S) ) ) ).

% filter_in_nths
tff(fact_7534_sum__list__map__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(B),P: fun(B,bool),F3: fun(B,A)] :
          ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(list(B),set(B),set2(B),Xs)))
             => ( ~ pp(aa(B,bool,P,X3))
               => ( aa(B,A,F3,X3) = zero_zero(A) ) ) )
         => ( groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,P),Xs))) = groups8242544230860333062m_list(A,aa(list(B),list(A),map(B,A,F3),Xs)) ) ) ) ).

% sum_list_map_filter
tff(fact_7535_set__minus__filter__out,axiom,
    ! [A: $tType,Xs: list(A),Y: A] : aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y),bot_bot(set(A)))) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aTP_Lamp_acd(A,fun(A,bool),Y)),Xs)) ).

% set_minus_filter_out
tff(fact_7536_filter__shuffles__disjoint1_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_ace(list(A),fun(A,bool),Xs)),Zs) = Ys ) ) ) ).

% filter_shuffles_disjoint1(2)
tff(fact_7537_filter__shuffles__disjoint1_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_acf(list(A),fun(A,bool),Xs)),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint1(1)
tff(fact_7538_filter__shuffles__disjoint2_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_ace(list(A),fun(A,bool),Ys)),Zs) = Xs ) ) ) ).

% filter_shuffles_disjoint2(2)
tff(fact_7539_filter__shuffles__disjoint2_I1_J,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),Ys)) = bot_bot(set(A)) )
     => ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Zs),shuffles(A,Xs,Ys)))
       => ( aa(list(A),list(A),filter2(A,aTP_Lamp_acf(list(A),fun(A,bool),Ys)),Zs) = Ys ) ) ) ).

% filter_shuffles_disjoint2(1)
tff(fact_7540_filter__eq__nths,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),filter2(A,P),Xs) = nths(A,Xs,collect(nat,aa(list(A),fun(nat,bool),aTP_Lamp_acg(fun(A,bool),fun(list(A),fun(nat,bool)),P),Xs))) ).

% filter_eq_nths
tff(fact_7541_length__filter__conv__card,axiom,
    ! [A: $tType,P2: fun(A,bool),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),filter2(A,P2),Xs)) = aa(set(nat),nat,finite_card(nat),collect(nat,aa(list(A),fun(nat,bool),aTP_Lamp_acg(fun(A,bool),fun(list(A),fun(nat,bool)),P2),Xs))) ).

% length_filter_conv_card
tff(fact_7542_distinct__length__filter,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( distinct(A,Xs)
     => ( aa(list(A),nat,size_size(list(A)),aa(list(A),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_7543_transpose__aux__filter__head,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_abg(A,fun(list(A),list(A))))),Xss)) = aa(list(list(A)),list(A),map(list(A),A,hd(A)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_abw(list(A),bool)),Xss)) ).

% transpose_aux_filter_head
tff(fact_7544_nth__transpose,axiom,
    ! [A: $tType,I2: nat,Xs: list(list(A))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs))))
     => ( aa(nat,list(A),nth(list(A),transpose(A,Xs)),I2) = aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_ach(nat,fun(list(A),A),I2)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_aci(nat,fun(list(A),bool),I2)),Xs)) ) ) ).

% nth_transpose
tff(fact_7545_transpose__max__length,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(nat,nat,foldr(list(A),nat,aTP_Lamp_acj(list(A),fun(nat,nat)),transpose(A,Xs)),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_abw(list(A),bool)),Xs)) ).

% transpose_max_length
tff(fact_7546_map__filter__map__filter,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),P: fun(B,bool),Xs: list(B)] : aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,P),Xs)) = map_filter(B,A,aa(fun(B,bool),fun(B,option(A)),aTP_Lamp_ack(fun(B,A),fun(fun(B,bool),fun(B,option(A))),F3),P),Xs) ).

% map_filter_map_filter
tff(fact_7547_foldr__append,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,fun(A,A)),Xs: list(B),Ys: list(B),A2: A] : aa(A,A,foldr(B,A,F3,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)),A2) = aa(A,A,foldr(B,A,F3,Xs),aa(A,A,foldr(B,A,F3,Ys),A2)) ).

% foldr_append
tff(fact_7548_foldr__replicate,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),N: nat,X: B] : foldr(B,A,F3,replicate(B,N,X)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),aa(B,fun(A,A),F3,X)) ).

% foldr_replicate
tff(fact_7549_foldr__Cons,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B)),X: A,Xs: list(A)] : foldr(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X)),foldr(A,B,F3,Xs)) ).

% foldr_Cons
tff(fact_7550_map__filter__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,option(A))] : map_filter(B,A,F3,nil(B)) = nil(A) ).

% map_filter_simps(2)
tff(fact_7551_foldr__cong,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: A,L: list(B),K: list(B),F3: fun(B,fun(A,A)),G: fun(B,fun(A,A))] :
      ( ( A2 = B2 )
     => ( ( L = K )
       => ( ! [A5: A,X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(list(B),set(B),set2(B),L)))
             => ( aa(A,A,aa(B,fun(A,A),F3,X3),A5) = aa(A,A,aa(B,fun(A,A),G,X3),A5) ) )
         => ( aa(A,A,foldr(B,A,F3,L),A2) = aa(A,A,foldr(B,A,G,K),B2) ) ) ) ) ).

% foldr_cong
tff(fact_7552_foldr__map,axiom,
    ! [C: $tType,B: $tType,A: $tType,G: fun(B,fun(A,A)),F3: fun(C,B),Xs: list(C),A2: A] : aa(A,A,foldr(B,A,G,aa(list(C),list(B),map(C,B,F3),Xs)),A2) = aa(A,A,foldr(C,A,aa(fun(C,B),fun(C,fun(A,A)),comp(B,fun(A,A),C,G),F3),Xs),A2) ).

% foldr_map
tff(fact_7553_map__filter__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),X: B,Xs: list(B)] : map_filter(B,A,F3,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = case_option(list(A),A,map_filter(B,A,F3,Xs),aa(list(B),fun(A,list(A)),aTP_Lamp_acl(fun(B,option(A)),fun(list(B),fun(A,list(A))),F3),Xs),aa(B,option(A),F3,X)) ).

% map_filter_simps(1)
tff(fact_7554_sum__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [Xs: list(A)] : groups8242544230860333062m_list(A,Xs) = aa(A,A,foldr(A,A,plus_plus(A),Xs),zero_zero(A)) ) ).

% sum_list.eq_foldr
tff(fact_7555_nths__shift__lemma__Suc,axiom,
    ! [A: $tType,P: fun(nat,bool),Xs: list(A),Is: list(nat)] : aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_acm(fun(nat,bool),fun(product_prod(A,nat),bool),P)),zip(A,nat,Xs,Is))) = aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_acn(fun(nat,bool),fun(product_prod(A,nat),bool),P)),zip(A,nat,Xs,aa(list(nat),list(nat),map(nat,nat,suc),Is)))) ).

% nths_shift_lemma_Suc
tff(fact_7556_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F3: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F3,A2,Xs) = aa(A,A,foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_aco(fun(B,A),fun(A,fun(B,fun(A,A))),F3),A2),Xs),zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_7557_nths__shift__lemma,axiom,
    ! [A: $tType,A3: set(nat),Xs: list(A),I2: nat] : aa(list(product_prod(A,nat)),list(A),map(product_prod(A,nat),A,product_fst(A,nat)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_acp(set(nat),fun(product_prod(A,nat),bool),A3)),zip(A,nat,Xs,upt(I2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),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)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aa(nat,fun(product_prod(A,nat),bool),aTP_Lamp_acq(set(nat),fun(nat,fun(product_prod(A,nat),bool)),A3),I2)),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_shift_lemma
tff(fact_7558_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)),aa(list(product_prod(A,nat)),list(product_prod(A,nat)),filter2(product_prod(A,nat),aTP_Lamp_acp(set(nat),fun(product_prod(A,nat),bool),A3)),zip(A,nat,Xs,upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))))) ).

% nths_def
tff(fact_7559_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_acj(list(A),fun(nat,nat)),Xs),zero_zero(nat)) ).

% length_transpose
tff(fact_7560_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_acr(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_acs(list(B),fun(nat,nat)),aa(list(list(B)),list(list(B)),filter2(list(B),aTP_Lamp_act(list(B),bool)),Xss)),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_7561_sorted__wrt__less__sum__mono__lowerbound,axiom,
    ! [B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [F3: fun(nat,B),Ns: list(nat)] :
          ( ! [X3: nat,Y4: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X3),Y4))
             => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(nat,B,F3,X3)),aa(nat,B,F3,Y4))) )
         => ( sorted_wrt(nat,ord_less(nat),Ns)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(nat),B,groups7311177749621191930dd_sum(nat,B,F3),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(nat),nat,size_size(list(nat)),Ns)))),groups8242544230860333062m_list(B,aa(list(nat),list(B),map(nat,B,F3),Ns)))) ) ) ) ).

% sorted_wrt_less_sum_mono_lowerbound
tff(fact_7562_length__product__lists,axiom,
    ! [B: $tType,Xss: list(list(B))] : aa(list(list(B)),nat,size_size(list(list(B))),product_lists(B,Xss)) = aa(nat,nat,foldr(nat,nat,times_times(nat),aa(list(list(B)),list(nat),map(list(B),nat,size_size(list(B))),Xss)),one_one(nat)) ).

% length_product_lists
tff(fact_7563_sorted__map__same,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),G: fun(list(B),A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,aa(list(B),fun(B,bool),aa(fun(list(B),A),fun(list(B),fun(B,bool)),aTP_Lamp_acu(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,bool))),F3),G),Xs)),Xs))) ) ).

% sorted_map_same
tff(fact_7564_sorted__filter,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),P: fun(B,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),filter2(B,P),Xs))) ) ) ).

% sorted_filter
tff(fact_7565_sorted__wrt__filter,axiom,
    ! [A: $tType,F3: fun(A,fun(A,bool)),Xs: list(A),P: fun(A,bool)] :
      ( sorted_wrt(A,F3,Xs)
     => sorted_wrt(A,F3,aa(list(A),list(A),filter2(A,P),Xs)) ) ).

% sorted_wrt_filter
tff(fact_7566_sorted__same,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [G: fun(list(A),A),Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),filter2(A,aa(list(A),fun(A,bool),aTP_Lamp_acv(fun(list(A),A),fun(list(A),fun(A,bool)),G),Xs)),Xs)) ) ).

% sorted_same
tff(fact_7567_sorted__wrt__map,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(A,bool)),F3: fun(B,A),Xs: list(B)] :
      ( sorted_wrt(A,R,aa(list(B),list(A),map(B,A,F3),Xs))
    <=> sorted_wrt(B,aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_acw(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),R),F3),Xs) ) ).

% sorted_wrt_map
tff(fact_7568_sorted__map,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
        <=> sorted_wrt(B,aTP_Lamp_acx(fun(B,A),fun(B,fun(B,bool)),F3),Xs) ) ) ).

% sorted_map
tff(fact_7569_sorted__map__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),X: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
         => sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),remove1(B,X,Xs))) ) ) ).

% sorted_map_remove1
tff(fact_7570_sorted__insort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),X: B,Xs: list(B)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),Xs)))
        <=> sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs)) ) ) ).

% sorted_insort_key
tff(fact_7571_sorted__distinct__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( distinct(A,Xs)
           => ( sorted_wrt(A,ord_less_eq(A),Ys)
             => ( distinct(A,Ys)
               => ( ( aa(list(A),set(A),set2(A),Xs) = aa(list(A),set(A),set2(A),Ys) )
                 => ( Xs = Ys ) ) ) ) ) ) ) ).

% sorted_distinct_set_unique
tff(fact_7572_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_7573_sorted__tl,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),tl(A),Xs)) ) ) ).

% sorted_tl
tff(fact_7574_sorted__wrt01,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,fun(A,bool))] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_7575_sorted__remdups,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remdups(A,Xs)) ) ) ).

% sorted_remdups
tff(fact_7576_sorted__wrt__take,axiom,
    ! [A: $tType,F3: fun(A,fun(A,bool)),Xs: list(A),N: nat] :
      ( sorted_wrt(A,F3,Xs)
     => sorted_wrt(A,F3,take(A,N,Xs)) ) ).

% sorted_wrt_take
tff(fact_7577_sorted__take,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),N: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),take(A,N,Xs)) ) ) ).

% sorted_take
tff(fact_7578_sorted__drop,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),N: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),drop(A,N,Xs)) ) ) ).

% sorted_drop
tff(fact_7579_sorted__wrt__drop,axiom,
    ! [A: $tType,F3: fun(A,fun(A,bool)),Xs: list(A),N: nat] :
      ( sorted_wrt(A,F3,Xs)
     => sorted_wrt(A,F3,drop(A,N,Xs)) ) ).

% sorted_wrt_drop
tff(fact_7580_sorted__wrt_Oelims_I3_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A)] :
      ( ~ sorted_wrt(A,X,Xa)
     => ~ ! [X3: A,Ys3: list(A)] :
            ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3) )
           => ( ! [Xa3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),aa(list(A),set(A),set2(A),Ys3)))
                 => pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Xa3)) )
              & sorted_wrt(A,X,Ys3) ) ) ) ).

% sorted_wrt.elims(3)
tff(fact_7581_sorted__wrt_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),X: A,Ys: list(A)] :
      ( sorted_wrt(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))
    <=> ( ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Ys)))
           => pp(aa(A,bool,aa(A,fun(A,bool),P,X),X4)) )
        & sorted_wrt(A,P,Ys) ) ) ).

% sorted_wrt.simps(2)
tff(fact_7582_sorted__wrt__append,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
      ( sorted_wrt(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
    <=> ( sorted_wrt(A,P,Xs)
        & sorted_wrt(A,P,Ys)
        & ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
           => ! [Xa4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),aa(list(A),set(A),set2(A),Ys)))
               => pp(aa(A,bool,aa(A,fun(A,bool),P,X4),Xa4)) ) ) ) ) ).

% sorted_wrt_append
tff(fact_7583_strict__sorted__equal,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less(A),Xs)
         => ( sorted_wrt(A,ord_less(A),Ys)
           => ( ( aa(list(A),set(A),set2(A),Ys) = aa(list(A),set(A),set2(A),Xs) )
             => ( Ys = Xs ) ) ) ) ) ).

% strict_sorted_equal
tff(fact_7584_sorted__wrt__mono__rel,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,fun(A,bool)),Q: fun(A,fun(A,bool))] :
      ( ! [X3: A,Y4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),aa(list(A),set(A),set2(A),Xs)))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),P,X3),Y4))
             => pp(aa(A,bool,aa(A,fun(A,bool),Q,X3),Y4)) ) ) )
     => ( sorted_wrt(A,P,Xs)
       => sorted_wrt(A,Q,Xs) ) ) ).

% sorted_wrt_mono_rel
tff(fact_7585_strict__sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Ys: list(A)] :
          ( sorted_wrt(A,ord_less(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))
        <=> ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Ys)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X),X4)) )
            & sorted_wrt(A,ord_less(A),Ys) ) ) ) ).

% strict_sorted_simps(2)
tff(fact_7586_sorted__append,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))
        <=> ( sorted_wrt(A,ord_less_eq(A),Xs)
            & sorted_wrt(A,ord_less_eq(A),Ys)
            & ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
               => ! [Xa4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),aa(list(A),set(A),set2(A),Ys)))
                   => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Xa4)) ) ) ) ) ) ).

% sorted_append
tff(fact_7587_sorted__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Ys: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys))
        <=> ( ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Ys)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X4)) )
            & sorted_wrt(A,ord_less_eq(A),Ys) ) ) ) ).

% sorted_simps(2)
tff(fact_7588_sorted__replicate,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [N: nat,X: A] : sorted_wrt(A,ord_less_eq(A),replicate(A,N,X)) ) ).

% sorted_replicate
tff(fact_7589_sorted2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Y: A,Zs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))
        <=> ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),Y))
            & sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) ) ) ) ).

% sorted2
tff(fact_7590_sorted1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A] : sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ) ).

% sorted1
tff(fact_7591_sorted__wrt1,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),X: A] : sorted_wrt(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ).

% sorted_wrt1
tff(fact_7592_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : sorted_wrt(A,ord_less(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
tff(fact_7593_sorted__wrt__true,axiom,
    ! [A: $tType,Xs: list(A)] : sorted_wrt(A,aTP_Lamp_acy(A,fun(A,bool)),Xs) ).

% sorted_wrt_true
tff(fact_7594_strict__sorted__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less(A),nil(A)) ) ).

% strict_sorted_simps(1)
tff(fact_7595_sorted__wrt_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool))] : sorted_wrt(A,P,nil(A)) ).

% sorted_wrt.simps(1)
tff(fact_7596_sorted__remdups__adj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remdups_adj(A,Xs)) ) ) ).

% sorted_remdups_adj
tff(fact_7597_sorted__nths,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I5: set(nat)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),nths(A,Xs,I5)) ) ) ).

% sorted_nths
tff(fact_7598_sorted__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A2: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),remove1(A,A2,Xs)) ) ) ).

% sorted_remove1
tff(fact_7599_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : sorted_wrt(A,ord_less_eq(A),aa(set(A),list(A),linord4507533701916653071of_set(A),A3)) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
tff(fact_7600_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_7601_sorted__insort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ma(A,A)),X),Xs))
        <=> sorted_wrt(A,ord_less_eq(A),Xs) ) ) ).

% sorted_insort
tff(fact_7602_sorted0,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => sorted_wrt(A,ord_less_eq(A),nil(A)) ) ).

% sorted0
tff(fact_7603_sorted__upt,axiom,
    ! [M: nat,N: nat] : sorted_wrt(nat,ord_less_eq(nat),upt(M,N)) ).

% sorted_upt
tff(fact_7604_sorted__wrt__upt,axiom,
    ! [M: nat,N: nat] : sorted_wrt(nat,ord_less(nat),upt(M,N)) ).

% sorted_wrt_upt
tff(fact_7605_sorted__wrt__iff__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
      ( sorted_wrt(A,P,Xs)
    <=> ! [I3: nat,J3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J3))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Xs),J3))) ) ) ) ).

% sorted_wrt_iff_nth_less
tff(fact_7606_sorted__wrt__nth__less,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A),I2: nat,J: nat] :
      ( sorted_wrt(A,P,Xs)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),J))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
         => pp(aa(A,bool,aa(A,fun(A,bool),P,aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Xs),J))) ) ) ) ).

% sorted_wrt_nth_less
tff(fact_7607_sorted01,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_7608_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] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),J3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),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_7609_sorted__wrt_Oelims_I2_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A)] :
      ( sorted_wrt(A,X,Xa)
     => ( ( Xa != nil(A) )
       => ~ ! [X3: A,Ys3: list(A)] :
              ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3) )
             => ~ ( ! [Xa2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),aa(list(A),set(A),set2(A),Ys3)))
                     => pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Xa2)) )
                  & sorted_wrt(A,X,Ys3) ) ) ) ) ).

% sorted_wrt.elims(2)
tff(fact_7610_sorted__wrt_Oelims_I1_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A),Y: bool] :
      ( ( sorted_wrt(A,X,Xa)
      <=> pp(Y) )
     => ( ( ( Xa = nil(A) )
         => ~ pp(Y) )
       => ~ ! [X3: A,Ys3: list(A)] :
              ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3) )
             => ( pp(Y)
              <=> ~ ( ! [Xa4: A] :
                        ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),aa(list(A),set(A),set2(A),Ys3)))
                       => pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Xa4)) )
                    & sorted_wrt(A,X,Ys3) ) ) ) ) ) ).

% sorted_wrt.elims(1)
tff(fact_7611_finite__sorted__distinct__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ? [X3: list(A)] :
              ( ( aa(list(A),set(A),set2(A),X3) = A3 )
              & sorted_wrt(A,ord_less_eq(A),X3)
              & distinct(A,X3)
              & ! [Y3: list(A)] :
                  ( ( ( aa(list(A),set(A),set2(A),Y3) = A3 )
                    & sorted_wrt(A,ord_less_eq(A),Y3)
                    & distinct(A,Y3) )
                 => ( Y3 = X3 ) ) ) ) ) ).

% finite_sorted_distinct_unique
tff(fact_7612_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)
           => ( aa(set(A),list(A),linord4507533701916653071of_set(A),aa(list(A),set(A),set2(A),Xs)) = Xs ) ) ) ) ).

% sorted_list_of_set.idem_if_sorted_distinct
tff(fact_7613_filter__insort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),P: fun(B,bool),X: B] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
         => ( pp(aa(B,bool,P,X))
           => ( aa(list(B),list(B),filter2(B,P),aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),aa(list(B),list(B),filter2(B,P),Xs)) ) ) ) ) ).

% filter_insort
tff(fact_7614_concat__conv__foldr,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(A,Xss) = aa(list(A),list(A),foldr(list(A),list(A),append(A),Xss),nil(A)) ).

% concat_conv_foldr
tff(fact_7615_insort__remove1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,Xs: list(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),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_ma(A,A)),A2),remove1(A,A2,Xs)) = Xs ) ) ) ) ).

% insort_remove1
tff(fact_7616_sorted__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I3)),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),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_7617_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ~ ! [L4: list(A)] :
                ( sorted_wrt(A,ord_less(A),L4)
               => ( ( aa(list(A),set(A),set2(A),L4) = A3 )
                 => ( aa(list(A),nat,size_size(list(A)),L4) != aa(set(A),nat,finite_card(A),A3) ) ) ) ) ) ).

% sorted_list_of_set.finite_set_strict_sorted
tff(fact_7618_sorted__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
        <=> ! [I3: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),J3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I3)),aa(nat,A,nth(A,Xs),J3))) ) ) ) ) ).

% sorted_iff_nth_mono
tff(fact_7619_sorted__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I2: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),I2)),aa(nat,A,nth(A,Xs),J))) ) ) ) ) ).

% sorted_nth_mono
tff(fact_7620_sorted__wrt__less__idx,axiom,
    ! [Ns: list(nat),I2: nat] :
      ( sorted_wrt(nat,ord_less(nat),Ns)
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(nat),nat,size_size(list(nat)),Ns)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),aa(nat,nat,nth(nat,Ns),I2))) ) ) ).

% sorted_wrt_less_idx
tff(fact_7621_sorted__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : sorted_wrt(nat,ord_less_eq(nat),aa(list(product_prod(nat,A)),list(nat),map(product_prod(nat,A),nat,product_fst(nat,A)),enumerate(A,N,Xs))) ).

% sorted_enumerate
tff(fact_7622_map__sorted__distinct__set__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),Ys: list(B)] :
          ( inj_on(B,A,F3,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),Ys)))
         => ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
           => ( distinct(A,aa(list(B),list(A),map(B,A,F3),Xs))
             => ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Ys))
               => ( distinct(A,aa(list(B),list(A),map(B,A,F3),Ys))
                 => ( ( aa(list(B),set(B),set2(B),Xs) = aa(list(B),set(B),set2(B),Ys) )
                   => ( Xs = Ys ) ) ) ) ) ) ) ) ).

% map_sorted_distinct_set_unique
tff(fact_7623_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),L: list(A)] :
          ( pp(aa(set(A),bool,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) ) )
          <=> ( aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = L ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_unique
tff(fact_7624_sorted__insort__is__snoc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),A2: A] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),A2)) )
           => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),linorder_insort_key(A,A,aTP_Lamp_ma(A,A)),A2),Xs) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),nil(A))) ) ) ) ) ).

% sorted_insort_is_snoc
tff(fact_7625_insort__key__remove1,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [A2: B,Xs: list(B),F3: fun(B,A)] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),A2),aa(list(B),set(B),set2(B),Xs)))
         => ( sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),Xs))
           => ( ( aa(list(B),B,hd(B),aa(list(B),list(B),filter2(B,aa(fun(B,A),fun(B,bool),aTP_Lamp_acz(B,fun(fun(B,A),fun(B,bool)),A2),F3)),Xs)) = A2 )
             => ( aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),A2),remove1(B,A2,Xs)) = Xs ) ) ) ) ) ).

% insort_key_remove1
tff(fact_7626_nth__nth__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A)),I2: nat,J: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs))))
       => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_aci(nat,fun(list(A),bool),I2)),Xs))))
         => ( aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),transpose(A,Xs)),I2)),J) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Xs),J)),I2) ) ) ) ) ).

% nth_nth_transpose_sorted
tff(fact_7627_transpose__column,axiom,
    ! [A: $tType,Xs: list(list(A)),I2: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
       => ( aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_ach(nat,fun(list(A),A),I2)),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_aci(nat,fun(list(A),bool),I2)),transpose(A,Xs))) = aa(nat,list(A),nth(list(A),Xs),I2) ) ) ) ).

% transpose_column
tff(fact_7628_rev__rev__ident,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),rev(A),Xs)) = Xs ).

% rev_rev_ident
tff(fact_7629_rev__is__rev__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),rev(A),Xs) = aa(list(A),list(A),rev(A),Ys) )
    <=> ( Xs = Ys ) ) ).

% rev_is_rev_conv
tff(fact_7630_Nil__is__rev__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( nil(A) = aa(list(A),list(A),rev(A),Xs) )
    <=> ( Xs = nil(A) ) ) ).

% Nil_is_rev_conv
tff(fact_7631_rev__is__Nil__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),list(A),rev(A),Xs) = nil(A) )
    <=> ( Xs = nil(A) ) ) ).

% rev_is_Nil_conv
tff(fact_7632_set__rev,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% set_rev
tff(fact_7633_length__rev,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_rev
tff(fact_7634_rev__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Ys)),aa(list(A),list(A),rev(A),Xs)) ).

% rev_append
tff(fact_7635_distinct__rev,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,aa(list(A),list(A),rev(A),Xs))
    <=> distinct(A,Xs) ) ).

% distinct_rev
tff(fact_7636_rev__replicate,axiom,
    ! [A: $tType,N: nat,X: A] : aa(list(A),list(A),rev(A),replicate(A,N,X)) = replicate(A,N,X) ).

% rev_replicate
tff(fact_7637_remdups__adj__rev,axiom,
    ! [A: $tType,Xs: list(A)] : remdups_adj(A,aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),remdups_adj(A,Xs)) ).

% remdups_adj_rev
tff(fact_7638_inj__on__rev,axiom,
    ! [A: $tType,A3: set(list(A))] : inj_on(list(A),list(A),rev(A),A3) ).

% inj_on_rev
tff(fact_7639_singleton__rev__conv,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) = aa(list(A),list(A),rev(A),Xs) )
    <=> ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) = Xs ) ) ).

% singleton_rev_conv
tff(fact_7640_rev__singleton__conv,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( ( aa(list(A),list(A),rev(A),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) )
    <=> ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)) ) ) ).

% rev_singleton_conv
tff(fact_7641_rev__eq__Cons__iff,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys: list(A)] :
      ( ( aa(list(A),list(A),rev(A),Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
    <=> ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Ys)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))) ) ) ).

% rev_eq_Cons_iff
tff(fact_7642_rev__filter,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),filter2(A,P),Xs)) = aa(list(A),list(A),filter2(A,P),aa(list(A),list(A),rev(A),Xs)) ).

% rev_filter
tff(fact_7643_rev__concat,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(A),list(A),rev(A),concat(A,Xs)) = concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),rev(A)),aa(list(list(A)),list(list(A)),rev(list(A)),Xs))) ).

% rev_concat
tff(fact_7644_rev__map,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : aa(list(A),list(A),rev(A),aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),rev(B),Xs)) ).

% rev_map
tff(fact_7645_zip__rev,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( zip(A,B,aa(list(A),list(A),rev(A),Xs),aa(list(B),list(B),rev(B),Ys)) = aa(list(product_prod(A,B)),list(product_prod(A,B)),rev(product_prod(A,B)),zip(A,B,Xs,Ys)) ) ) ).

% zip_rev
tff(fact_7646_sorted__wrt__rev,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
      ( sorted_wrt(A,P,aa(list(A),list(A),rev(A),Xs))
    <=> sorted_wrt(A,aTP_Lamp_ada(fun(A,fun(A,bool)),fun(A,fun(A,bool)),P),Xs) ) ).

% sorted_wrt_rev
tff(fact_7647_sorted__wrt__upto,axiom,
    ! [I2: int,J: int] : sorted_wrt(int,ord_less(int),upto(I2,J)) ).

% sorted_wrt_upto
tff(fact_7648_sorted__upto,axiom,
    ! [M: int,N: int] : sorted_wrt(int,ord_less_eq(int),upto(M,N)) ).

% sorted_upto
tff(fact_7649_rev_Osimps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),rev(A),nil(A)) = nil(A) ).

% rev.simps(1)
tff(fact_7650_rev__swap,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),list(A),rev(A),Xs) = Ys )
    <=> ( Xs = aa(list(A),list(A),rev(A),Ys) ) ) ).

% rev_swap
tff(fact_7651_rev_Osimps_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(list(A),list(A),rev(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ).

% rev.simps(2)
tff(fact_7652_take__rev,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : take(A,N,aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),drop(A,aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),N),Xs)) ).

% take_rev
tff(fact_7653_rev__take,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] : aa(list(A),list(A),rev(A),take(A,I2,Xs)) = drop(A,aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),I2),aa(list(A),list(A),rev(A),Xs)) ).

% rev_take
tff(fact_7654_rev__drop,axiom,
    ! [A: $tType,I2: nat,Xs: list(A)] : aa(list(A),list(A),rev(A),drop(A,I2,Xs)) = take(A,aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),I2),aa(list(A),list(A),rev(A),Xs)) ).

% rev_drop
tff(fact_7655_drop__rev,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : drop(A,N,aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),take(A,aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),N),Xs)) ).

% drop_rev
tff(fact_7656_rotate__rev,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(A),list(A),rotate(A,N),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),rotate(A,aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),modulo_modulo(nat,N,aa(list(A),nat,size_size(list(A)),Xs)))),Xs)) ).

% rotate_rev
tff(fact_7657_rev__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),rev(A),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,nat,suc,N))) ) ) ).

% rev_nth
tff(fact_7658_rev__update,axiom,
    ! [A: $tType,K: nat,Xs: list(A),Y: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),K),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),list(A),rev(A),list_update(A,Xs,K,Y)) = list_update(A,aa(list(A),list(A),rev(A),Xs),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),K)),one_one(nat)),Y) ) ) ).

% rev_update
tff(fact_7659_sorted__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),transpose(A,Xs)))) ).

% sorted_transpose
tff(fact_7660_sorted__rev__iff__nth__Suc,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
        <=> ! [I3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,suc,I3)),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),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_7661_sorted__rev__iff__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
        <=> ! [I3: nat,J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I3),J3))
             => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),aa(list(A),nat,size_size(list(A)),Xs)))
               => pp(aa(A,bool,aa(A,fun(A,bool),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_7662_sorted__rev__nth__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),I2: nat,J: nat] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),I2),J))
           => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(nat,A,nth(A,Xs),J)),aa(nat,A,nth(A,Xs),I2))) ) ) ) ) ).

% sorted_rev_nth_mono
tff(fact_7663_foldr__max__sorted,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),Y: A] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),Xs))
         => ( ( ( Xs = nil(A) )
             => ( aa(A,A,foldr(A,A,ord_max(A),Xs),Y) = Y ) )
            & ( ( Xs != nil(A) )
             => ( aa(A,A,foldr(A,A,ord_max(A),Xs),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_7664_length__transpose__sorted,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( ( ( Xs = nil(list(A)) )
         => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = zero_zero(nat) ) )
        & ( ( Xs != nil(list(A)) )
         => ( aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),zero_zero(nat))) ) ) ) ) ).

% length_transpose_sorted
tff(fact_7665_transpose__column__length,axiom,
    ! [A: $tType,Xs: list(list(A)),I2: nat] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I2),aa(list(list(A)),nat,size_size(list(list(A))),Xs)))
       => ( aa(list(list(A)),nat,size_size(list(list(A))),aa(list(list(A)),list(list(A)),filter2(list(A),aTP_Lamp_aci(nat,fun(list(A),bool),I2)),transpose(A,Xs))) = aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I2)) ) ) ) ).

% transpose_column_length
tff(fact_7666_sorted__wrt_Opelims_I1_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A),Y: bool] :
      ( ( sorted_wrt(A,X,Xa)
      <=> pp(Y) )
     => ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),product_Pair(fun(A,fun(A,bool)),list(A),X),Xa)))
       => ( ( ( Xa = nil(A) )
           => ( pp(Y)
             => ~ pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),product_Pair(fun(A,fun(A,bool)),list(A),X),nil(A)))) ) )
         => ~ ! [X3: A,Ys3: list(A)] :
                ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3) )
               => ( ( pp(Y)
                  <=> ( ! [Xa4: A] :
                          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa4),aa(list(A),set(A),set2(A),Ys3)))
                         => pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Xa4)) )
                      & sorted_wrt(A,X,Ys3) ) )
                 => ~ pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),product_Pair(fun(A,fun(A,bool)),list(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3)))) ) ) ) ) ) ).

% sorted_wrt.pelims(1)
tff(fact_7667_sorted__wrt_Opelims_I2_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A)] :
      ( sorted_wrt(A,X,Xa)
     => ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),product_Pair(fun(A,fun(A,bool)),list(A),X),Xa)))
       => ( ( ( Xa = nil(A) )
           => ~ pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),product_Pair(fun(A,fun(A,bool)),list(A),X),nil(A)))) )
         => ~ ! [X3: A,Ys3: list(A)] :
                ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3) )
               => ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),product_Pair(fun(A,fun(A,bool)),list(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3))))
                 => ~ ( ! [Xa2: A] :
                          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa2),aa(list(A),set(A),set2(A),Ys3)))
                         => pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Xa2)) )
                      & sorted_wrt(A,X,Ys3) ) ) ) ) ) ) ).

% sorted_wrt.pelims(2)
tff(fact_7668_length__concat__rev,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(A),nat,size_size(list(A)),concat(A,aa(list(list(A)),list(list(A)),rev(list(A)),Xs))) = aa(list(A),nat,size_size(list(A)),concat(A,Xs)) ).

% length_concat_rev
tff(fact_7669_sorted__wrt_Opelims_I3_J,axiom,
    ! [A: $tType,X: fun(A,fun(A,bool)),Xa: list(A)] :
      ( ~ sorted_wrt(A,X,Xa)
     => ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),product_Pair(fun(A,fun(A,bool)),list(A),X),Xa)))
       => ~ ! [X3: A,Ys3: list(A)] :
              ( ( Xa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3) )
             => ( pp(aa(product_prod(fun(A,fun(A,bool)),list(A)),bool,accp(product_prod(fun(A,fun(A,bool)),list(A)),sorted_wrt_rel(A)),aa(list(A),product_prod(fun(A,fun(A,bool)),list(A)),product_Pair(fun(A,fun(A,bool)),list(A),X),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Ys3))))
               => ( ! [Xa3: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Xa3),aa(list(A),set(A),set2(A),Ys3)))
                     => pp(aa(A,bool,aa(A,fun(A,bool),X,X3),Xa3)) )
                  & sorted_wrt(A,X,Ys3) ) ) ) ) ) ).

% sorted_wrt.pelims(3)
tff(fact_7670_folding__insort__key_Ofinite__set__strict__sorted,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F3: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S2))
       => ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ~ ! [L4: list(B)] :
                ( sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F3),L4))
               => ( ( aa(list(B),set(B),set2(B),L4) = A3 )
                 => ( aa(list(B),nat,size_size(list(B)),L4) != aa(set(B),nat,finite_card(B),A3) ) ) ) ) ) ) ).

% folding_insort_key.finite_set_strict_sorted
tff(fact_7671_transpose__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( sorted_wrt(nat,ord_less_eq(nat),aa(list(nat),list(nat),rev(nat),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xs)))
     => ( transpose(A,transpose(A,Xs)) = takeWhile(list(A),aTP_Lamp_abw(list(A),bool),Xs) ) ) ).

% transpose_transpose
tff(fact_7672_takeWhile__idem,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : takeWhile(A,P,takeWhile(A,P,Xs)) = takeWhile(A,P,Xs) ).

% takeWhile_idem
tff(fact_7673_takeWhile__eq__all__conv,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( takeWhile(A,P,Xs) = Xs )
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X4)) ) ) ).

% takeWhile_eq_all_conv
tff(fact_7674_takeWhile__append1,axiom,
    ! [A: $tType,X: A,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( ~ pp(aa(A,bool,P,X))
       => ( takeWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = takeWhile(A,P,Xs) ) ) ) ).

% takeWhile_append1
tff(fact_7675_takeWhile__append2,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X3)) )
     => ( takeWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),takeWhile(A,P,Ys)) ) ) ).

% takeWhile_append2
tff(fact_7676_takeWhile__replicate,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,N: nat] :
      ( ( pp(aa(A,bool,P,X))
       => ( takeWhile(A,P,replicate(A,N,X)) = replicate(A,N,X) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( takeWhile(A,P,replicate(A,N,X)) = nil(A) ) ) ) ).

% takeWhile_replicate
tff(fact_7677_sorted__takeWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),takeWhile(A,P,Xs)) ) ) ).

% sorted_takeWhile
tff(fact_7678_folding__insort__key_Odistinct__if__distinct__map,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F3: fun(B,A),Xs: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( distinct(A,aa(list(B),list(A),map(B,A,F3),Xs))
       => distinct(B,Xs) ) ) ).

% folding_insort_key.distinct_if_distinct_map
tff(fact_7679_zip__takeWhile__fst,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),Xs: list(A),Ys: list(B)] : zip(A,B,takeWhile(A,P,Xs),Ys) = takeWhile(product_prod(A,B),aa(fun(product_prod(A,B),A),fun(product_prod(A,B),bool),comp(A,bool,product_prod(A,B),P),product_fst(A,B)),zip(A,B,Xs,Ys)) ).

% zip_takeWhile_fst
tff(fact_7680_zip__takeWhile__snd,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),P: fun(B,bool),Ys: list(B)] : zip(A,B,Xs,takeWhile(B,P,Ys)) = takeWhile(product_prod(A,B),aa(fun(product_prod(A,B),B),fun(product_prod(A,B),bool),comp(B,bool,product_prod(A,B),P),product_snd(A,B)),zip(A,B,Xs,Ys)) ).

% zip_takeWhile_snd
tff(fact_7681_takeWhile__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F3: fun(B,A),Xs: list(B)] : takeWhile(A,P,aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),takeWhile(B,aa(fun(B,A),fun(B,bool),comp(A,bool,B,P),F3),Xs)) ).

% takeWhile_map
tff(fact_7682_takeWhile_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,bool)] : takeWhile(A,P,nil(A)) = nil(A) ).

% takeWhile.simps(1)
tff(fact_7683_takeWhile__tail,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A),L: list(A)] :
      ( ~ pp(aa(A,bool,P,X))
     => ( takeWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),L))) = takeWhile(A,P,Xs) ) ) ).

% takeWhile_tail
tff(fact_7684_takeWhile_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
      ( ( pp(aa(A,bool,P,X))
       => ( takeWhile(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),takeWhile(A,P,Xs)) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( takeWhile(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = nil(A) ) ) ) ).

% takeWhile.simps(2)
tff(fact_7685_takeWhile__cong,axiom,
    ! [A: $tType,L: list(A),K: list(A),P: fun(A,bool),Q: fun(A,bool)] :
      ( ( L = K )
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),L)))
           => ( pp(aa(A,bool,P,X3))
            <=> pp(aa(A,bool,Q,X3)) ) )
       => ( takeWhile(A,P,L) = takeWhile(A,Q,K) ) ) ) ).

% takeWhile_cong
tff(fact_7686_set__takeWhileD,axiom,
    ! [A: $tType,X: A,P: fun(A,bool),Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),takeWhile(A,P,Xs))))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
        & pp(aa(A,bool,P,X)) ) ) ).

% set_takeWhileD
tff(fact_7687_takeWhile__eq__take,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : takeWhile(A,P,Xs) = take(A,aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)),Xs) ).

% takeWhile_eq_take
tff(fact_7688_takeWhile__eq__Nil__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( takeWhile(A,P,Xs) = nil(A) )
    <=> ( ( Xs = nil(A) )
        | ~ pp(aa(A,bool,P,aa(list(A),A,hd(A),Xs))) ) ) ).

% takeWhile_eq_Nil_iff
tff(fact_7689_length__takeWhile__le,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_takeWhile_le
tff(fact_7690_distinct__takeWhile,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( distinct(A,Xs)
     => distinct(A,takeWhile(A,P,Xs)) ) ).

% distinct_takeWhile
tff(fact_7691_folding__insort__key_Oinj__on,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F3: fun(B,A)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => inj_on(B,A,F3,S2) ) ).

% folding_insort_key.inj_on
tff(fact_7692_nth__length__takeWhile,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs)))
     => ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs))))) ) ).

% nth_length_takeWhile
tff(fact_7693_takeWhile__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_7694_takeWhile__append,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
      ( ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
           => pp(aa(A,bool,P,X3)) )
       => ( takeWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),takeWhile(A,P,Ys)) ) )
      & ( ~ ! [X2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,P,X2)) )
       => ( takeWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = takeWhile(A,P,Xs) ) ) ) ).

% takeWhile_append
tff(fact_7695_length__takeWhile__less__P__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,bool),Xs: list(A)] :
      ( ! [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),J))
         => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I4))) )
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),Xs)))
       => pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),J),aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)))) ) ) ).

% length_takeWhile_less_P_nth
tff(fact_7696_takeWhile__eq__take__P__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A),P: fun(A,bool)] :
      ( ! [I4: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),N))
         => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I4),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I4))) ) )
     => ( ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
         => ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),N))) )
       => ( takeWhile(A,P,Xs) = take(A,N,Xs) ) ) ) ).

% takeWhile_eq_take_P_nth
tff(fact_7697_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_ma(A,A)) ) ).

% sorted_list_of_set.folding_insort_key_axioms
tff(fact_7698_filter__equals__takeWhile__sorted__rev,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B),T2: A] :
          ( sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),rev(A),aa(list(B),list(A),map(B,A,F3),Xs)))
         => ( aa(list(B),list(B),filter2(B,aa(A,fun(B,bool),aTP_Lamp_adb(fun(B,A),fun(A,fun(B,bool)),F3),T2)),Xs) = takeWhile(B,aa(A,fun(B,bool),aTP_Lamp_adb(fun(B,A),fun(A,fun(B,bool)),F3),T2),Xs) ) ) ) ).

% filter_equals_takeWhile_sorted_rev
tff(fact_7699_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F3: fun(B,A),A3: set(B),L: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S2))
       => ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ( sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F3),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) ) )
          <=> ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A3) = L ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_unique
tff(fact_7700_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F3: fun(B,A),X: B,A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A3)),S2))
       => ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),aa(set(B),set(B),minus_minus(set(B),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),bot_bot(set(B))))) = remove1(B,X,aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A3)) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_remove
tff(fact_7701_linorder_Osorted__key__list__of__set_Ocong,axiom,
    ! [B: $tType,A: $tType,Less_eq: fun(A,fun(A,bool))] : sorted8670434370408473282of_set(A,B,Less_eq) = sorted8670434370408473282of_set(A,B,Less_eq) ).

% linorder.sorted_key_list_of_set.cong
tff(fact_7702_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F3: fun(B,A),A3: set(B),B3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S2))
       => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),S2))
         => ( ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A3) = aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),B3) )
           => ( pp(aa(set(B),bool,finite_finite(B),A3))
             => ( pp(aa(set(B),bool,finite_finite(B),B3))
               => ( A3 = B3 ) ) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_inject
tff(fact_7703_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F3: fun(B,A)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),bot_bot(set(B))) = nil(B) ) ) ).

% folding_insort_key.sorted_key_list_of_set_empty
tff(fact_7704_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F3: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S2))
       => ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( aa(list(B),set(B),set2(B),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A3)) = A3 ) ) ) ) ).

% folding_insort_key.set_sorted_key_list_of_set
tff(fact_7705_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F3: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S2))
       => ( aa(list(B),nat,size_size(list(B)),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A3)) = aa(set(B),nat,finite_card(B),A3) ) ) ) ).

% folding_insort_key.length_sorted_key_list_of_set
tff(fact_7706_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F3: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S2))
       => distinct(A,aa(list(B),list(A),map(B,A,F3),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A3))) ) ) ).

% folding_insort_key.distinct_sorted_key_list_of_set
tff(fact_7707_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F3: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S2))
       => sorted_wrt(A,Less_eq,aa(list(B),list(A),map(B,A,F3),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A3))) ) ) ).

% folding_insort_key.sorted_sorted_key_list_of_set
tff(fact_7708_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F3: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S2))
       => sorted_wrt(A,Less,aa(list(B),list(A),map(B,A,F3),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A3))) ) ) ).

% folding_insort_key.strict_sorted_key_list_of_set
tff(fact_7709_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F3: fun(B,A),A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),S2))
       => ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A3) = nil(B) )
          <=> ( A3 = bot_bot(set(B)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
tff(fact_7710_folding__insort__key_Oidem__if__sorted__distinct,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F3: fun(B,A),Xs: list(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(list(B),set(B),set2(B),Xs)),S2))
       => ( sorted_wrt(A,Less_eq,aa(list(B),list(A),map(B,A,F3),Xs))
         => ( distinct(B,Xs)
           => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),aa(list(B),set(B),set2(B),Xs)) = Xs ) ) ) ) ) ).

% folding_insort_key.idem_if_sorted_distinct
tff(fact_7711_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F3: fun(B,A),X: B,A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A3)),S2))
       => ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A3)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F3),X),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),aa(set(B),set(B),minus_minus(set(B),A3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),bot_bot(set(B)))))) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert_remove
tff(fact_7712_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F3: fun(B,A),X: B,A3: set(B)] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A3)),S2))
       => ( pp(aa(set(B),bool,finite_finite(B),A3))
         => ( ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),A3))
           => ( aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A3)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F3),X),aa(set(B),list(B),aa(fun(B,A),fun(set(B),list(B)),sorted8670434370408473282of_set(A,B,Less_eq),F3),A3)) ) ) ) ) ) ).

% folding_insort_key.sorted_key_list_of_set_insert
tff(fact_7713_linorder_Oinsort__key_Ocong,axiom,
    ! [B: $tType,A: $tType,Less_eq: fun(A,fun(A,bool))] : insort_key(A,B,Less_eq) = insort_key(A,B,Less_eq) ).

% linorder.insort_key.cong
tff(fact_7714_folding__insort__key_Oinsort__key__commute,axiom,
    ! [A: $tType,B: $tType,Less_eq: fun(A,fun(A,bool)),Less: fun(A,fun(A,bool)),S2: set(B),F3: fun(B,A),X: B,Y: B] :
      ( folding_insort_key(A,B,Less_eq,Less,S2,F3)
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X),S2))
       => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y),S2))
         => ( aa(fun(list(B),list(B)),fun(list(B),list(B)),comp(list(B),list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F3),Y)),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F3),X)) = aa(fun(list(B),list(B)),fun(list(B),list(B)),comp(list(B),list(B),list(B),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F3),X)),aa(B,fun(list(B),list(B)),aa(fun(B,A),fun(B,fun(list(B),list(B))),insort_key(A,B,Less_eq),F3),Y)) ) ) ) ) ).

% folding_insort_key.insort_key_commute
tff(fact_7715_extract__def,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : extract(A,P,Xs) = aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),case_list(option(product_prod(list(A),product_prod(A,list(A)))),A,none(product_prod(list(A),product_prod(A,list(A)))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_adc(fun(A,bool),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),P),Xs)),dropWhile(A,aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),P),Xs)) ).

% extract_def
tff(fact_7716_sorted__find__Min,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( ? [X2: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
                & pp(aa(A,bool,P,X2)) )
           => ( find(A,P,Xs) = aa(A,option(A),some(A),aa(set(A),A,lattic643756798350308766er_Min(A),collect(A,aa(fun(A,bool),fun(A,bool),aTP_Lamp_add(list(A),fun(fun(A,bool),fun(A,bool)),Xs),P)))) ) ) ) ) ).

% sorted_find_Min
tff(fact_7717_dropWhile__idem,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : dropWhile(A,P,dropWhile(A,P,Xs)) = dropWhile(A,P,Xs) ).

% dropWhile_idem
tff(fact_7718_dropWhile__eq__Nil__conv,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( dropWhile(A,P,Xs) = nil(A) )
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X4)) ) ) ).

% dropWhile_eq_Nil_conv
tff(fact_7719_dropWhile__append2,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,P,X3)) )
     => ( dropWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = dropWhile(A,P,Ys) ) ) ).

% dropWhile_append2
tff(fact_7720_dropWhile__append1,axiom,
    ! [A: $tType,X: A,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( ~ pp(aa(A,bool,P,X))
       => ( dropWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),dropWhile(A,P,Xs)),Ys) ) ) ) ).

% dropWhile_append1
tff(fact_7721_dropWhile__replicate,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,N: nat] :
      ( ( pp(aa(A,bool,P,X))
       => ( dropWhile(A,P,replicate(A,N,X)) = nil(A) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( dropWhile(A,P,replicate(A,N,X)) = replicate(A,N,X) ) ) ) ).

% dropWhile_replicate
tff(fact_7722_takeWhile__dropWhile__id,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),takeWhile(A,P,Xs)),dropWhile(A,P,Xs)) = Xs ).

% takeWhile_dropWhile_id
tff(fact_7723_distinct__dropWhile,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool)] :
      ( distinct(A,Xs)
     => distinct(A,dropWhile(A,P,Xs)) ) ).

% distinct_dropWhile
tff(fact_7724_length__dropWhile__le,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),dropWhile(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs))) ).

% length_dropWhile_le
tff(fact_7725_dropWhile__eq__self__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( dropWhile(A,P,Xs) = Xs )
    <=> ( ( Xs = nil(A) )
        | ~ pp(aa(A,bool,P,aa(list(A),A,hd(A),Xs))) ) ) ).

% dropWhile_eq_self_iff
tff(fact_7726_hd__dropWhile,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( dropWhile(A,P,Xs) != nil(A) )
     => ~ pp(aa(A,bool,P,aa(list(A),A,hd(A),dropWhile(A,P,Xs)))) ) ).

% hd_dropWhile
tff(fact_7727_dropWhile__cong,axiom,
    ! [A: $tType,L: list(A),K: list(A),P: fun(A,bool),Q: fun(A,bool)] :
      ( ( L = K )
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),L)))
           => ( pp(aa(A,bool,P,X3))
            <=> pp(aa(A,bool,Q,X3)) ) )
       => ( dropWhile(A,P,L) = dropWhile(A,Q,K) ) ) ) ).

% dropWhile_cong
tff(fact_7728_set__dropWhileD,axiom,
    ! [A: $tType,X: A,P: fun(A,bool),Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),dropWhile(A,P,Xs))))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs))) ) ).

% set_dropWhileD
tff(fact_7729_find__cong,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),P: fun(A,bool),Q: fun(A,bool)] :
      ( ( Xs = Ys )
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Ys)))
           => ( pp(aa(A,bool,P,X3))
            <=> pp(aa(A,bool,Q,X3)) ) )
       => ( find(A,P,Xs) = find(A,Q,Ys) ) ) ) ).

% find_cong
tff(fact_7730_dropWhile_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
      ( ( pp(aa(A,bool,P,X))
       => ( dropWhile(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = dropWhile(A,P,Xs) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( dropWhile(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs) ) ) ) ).

% dropWhile.simps(2)
tff(fact_7731_dropWhile__append3,axiom,
    ! [A: $tType,P: fun(A,bool),Y: A,Xs: list(A),Ys: list(A)] :
      ( ~ pp(aa(A,bool,P,Y))
     => ( dropWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),dropWhile(A,P,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ) ) ).

% dropWhile_append3
tff(fact_7732_remdups__adj__Cons_H,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),remdups_adj(A,dropWhile(A,aTP_Lamp_ade(A,fun(A,bool),X),Xs))) ).

% remdups_adj_Cons'
tff(fact_7733_dropWhile_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,bool)] : dropWhile(A,P,nil(A)) = nil(A) ).

% dropWhile.simps(1)
tff(fact_7734_sorted__dropWhile,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,bool)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => sorted_wrt(A,ord_less_eq(A),dropWhile(A,P,Xs)) ) ) ).

% sorted_dropWhile
tff(fact_7735_dropWhile__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,bool),F3: fun(B,A),Xs: list(B)] : dropWhile(A,P,aa(list(B),list(A),map(B,A,F3),Xs)) = aa(list(B),list(A),map(B,A,F3),dropWhile(B,aa(fun(B,A),fun(B,bool),comp(A,bool,B,P),F3),Xs)) ).

% dropWhile_map
tff(fact_7736_find__dropWhile,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : find(A,P,Xs) = aa(list(A),option(A),case_list(option(A),A,none(A),aTP_Lamp_adf(A,fun(list(A),option(A)))),dropWhile(A,aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),P),Xs)) ).

% find_dropWhile
tff(fact_7737_find_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] :
      ( ( pp(aa(A,bool,P,X))
       => ( find(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(A,option(A),some(A),X) ) )
      & ( ~ pp(aa(A,bool,P,X))
       => ( find(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = find(A,P,Xs) ) ) ) ).

% find.simps(2)
tff(fact_7738_find_Osimps_I1_J,axiom,
    ! [A: $tType,Uu: fun(A,bool)] : find(A,Uu,nil(A)) = none(A) ).

% find.simps(1)
tff(fact_7739_find__None__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( find(A,P,Xs) = none(A) )
    <=> ~ ? [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
            & pp(aa(A,bool,P,X4)) ) ) ).

% find_None_iff
tff(fact_7740_find__None__iff2,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ( none(A) = find(A,P,Xs) )
    <=> ~ ? [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
            & pp(aa(A,bool,P,X4)) ) ) ).

% find_None_iff2
tff(fact_7741_dropWhile__eq__Cons__conv,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Y: A,Ys: list(A)] :
      ( ( dropWhile(A,P,Xs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys) )
    <=> ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),takeWhile(A,P,Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) )
        & ~ pp(aa(A,bool,P,Y)) ) ) ).

% dropWhile_eq_Cons_conv
tff(fact_7742_takeWhile__eq__filter,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),dropWhile(A,P,Xs))))
         => ~ pp(aa(A,bool,P,X3)) )
     => ( takeWhile(A,P,Xs) = aa(list(A),list(A),filter2(A,P),Xs) ) ) ).

% takeWhile_eq_filter
tff(fact_7743_dropWhile__eq__drop,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : dropWhile(A,P,Xs) = drop(A,aa(list(A),nat,size_size(list(A)),takeWhile(A,P,Xs)),Xs) ).

% dropWhile_eq_drop
tff(fact_7744_dropWhile__append,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,bool),Ys: list(A)] :
      ( ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
           => pp(aa(A,bool,P,X3)) )
       => ( dropWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = dropWhile(A,P,Ys) ) )
      & ( ~ ! [X2: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
             => pp(aa(A,bool,P,X2)) )
       => ( dropWhile(A,P,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),dropWhile(A,P,Xs)),Ys) ) ) ) ).

% dropWhile_append
tff(fact_7745_remdups__adj__append__dropWhile,axiom,
    ! [A: $tType,Xs: list(A),Y: A,Ys: list(A)] : remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),remdups_adj(A,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))),remdups_adj(A,dropWhile(A,aTP_Lamp_ade(A,fun(A,bool),Y),Ys))) ).

% remdups_adj_append_dropWhile
tff(fact_7746_tl__remdups__adj,axiom,
    ! [A: $tType,Ys: list(A)] :
      ( ( Ys != nil(A) )
     => ( aa(list(A),list(A),tl(A),remdups_adj(A,Ys)) = remdups_adj(A,dropWhile(A,aTP_Lamp_adg(list(A),fun(A,bool),Ys),aa(list(A),list(A),tl(A),Ys))) ) ) ).

% tl_remdups_adj
tff(fact_7747_dropWhile__nth,axiom,
    ! [A: $tType,J: nat,P: fun(A,bool),Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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_7748_dropWhile__neq__rev,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( dropWhile(A,aTP_Lamp_acd(A,fun(A,bool),X),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),rev(A),takeWhile(A,aTP_Lamp_acd(A,fun(A,bool),X),Xs))) ) ) ) ).

% dropWhile_neq_rev
tff(fact_7749_takeWhile__neq__rev,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( distinct(A,Xs)
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( takeWhile(A,aTP_Lamp_acd(A,fun(A,bool),X),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),tl(A),dropWhile(A,aTP_Lamp_acd(A,fun(A,bool),X),Xs))) ) ) ) ).

% takeWhile_neq_rev
tff(fact_7750_find__Some__iff,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),X: A] :
      ( ( find(A,P,Xs) = aa(A,option(A),some(A),X) )
    <=> ? [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
          & pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I3)))
          & ( X = aa(nat,A,nth(A,Xs),I3) )
          & ! [J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),I3))
             => ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),J3))) ) ) ) ).

% find_Some_iff
tff(fact_7751_find__Some__iff2,axiom,
    ! [A: $tType,X: A,P: fun(A,bool),Xs: list(A)] :
      ( ( aa(A,option(A),some(A),X) = find(A,P,Xs) )
    <=> ? [I3: nat] :
          ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
          & pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),I3)))
          & ( X = aa(nat,A,nth(A,Xs),I3) )
          & ! [J3: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),J3),I3))
             => ~ pp(aa(A,bool,P,aa(nat,A,nth(A,Xs),J3))) ) ) ) ).

% find_Some_iff2
tff(fact_7752_partition__filter__conv,axiom,
    ! [A: $tType,F3: fun(A,bool),Xs: list(A)] : partition(A,F3,Xs) = aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),filter2(A,F3),Xs)),aa(list(A),list(A),filter2(A,aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),F3)),Xs)) ).

% partition_filter_conv
tff(fact_7753_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_adi(list(product_prod(C,B)),fun(product_prod(A,C),list(product_prod(A,B))),Yzs)),Xys))) ).

% set_relcomp
tff(fact_7754_relcomp__unfold,axiom,
    ! [A: $tType,B: $tType,C: $tType,R3: set(product_prod(A,C)),S: set(product_prod(C,B))] : relcomp(A,C,B,R3,S) = collect(product_prod(A,B),product_case_prod(A,B,bool,aa(set(product_prod(C,B)),fun(A,fun(B,bool)),aTP_Lamp_adj(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,bool))),R3),S))) ).

% relcomp_unfold
tff(fact_7755_partition__filter1,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),partition(A,P,Xs)) = aa(list(A),list(A),filter2(A,P),Xs) ).

% partition_filter1
tff(fact_7756_trancl__Int__subset,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),S: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),S))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),transitive_trancl(A,R3)),S),R3)),S))
       => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R3)),S)) ) ) ).

% trancl_Int_subset
tff(fact_7757_relpow__add,axiom,
    ! [A: $tType,M: nat,N: 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),M),N)),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))),M),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))),N),R)) ).

% relpow_add
tff(fact_7758_relpow_Osimps_I2_J,axiom,
    ! [A: $tType,N: 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,N)),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))),N),R),R) ).

% relpow.simps(2)
tff(fact_7759_relcomp__mono,axiom,
    ! [A: $tType,C: $tType,B: $tType,R4: set(product_prod(A,B)),R3: set(product_prod(A,B)),S10: set(product_prod(B,C)),S: set(product_prod(B,C))] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R4),R3))
     => ( pp(aa(set(product_prod(B,C)),bool,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),bool),ord_less_eq(set(product_prod(B,C))),S10),S))
       => pp(aa(set(product_prod(A,C)),bool,aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),bool),ord_less_eq(set(product_prod(A,C))),relcomp(A,B,C,R4,S10)),relcomp(A,B,C,R3,S))) ) ) ).

% relcomp_mono
tff(fact_7760_partition_Osimps_I1_J,axiom,
    ! [A: $tType,P: fun(A,bool)] : partition(A,P,nil(A)) = aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),nil(A)),nil(A)) ).

% partition.simps(1)
tff(fact_7761_partition__P,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Yes: list(A),No4: list(A)] :
      ( ( partition(A,P,Xs) = aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Yes),No4) )
     => ( ! [X2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Yes)))
           => pp(aa(A,bool,P,X2)) )
        & ! [X2: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),No4)))
           => ~ pp(aa(A,bool,P,X2)) ) ) ) ).

% partition_P
tff(fact_7762_partition_Osimps_I2_J,axiom,
    ! [A: $tType,P: fun(A,bool),X: A,Xs: list(A)] : partition(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(product_prod(list(A),list(A)),product_prod(list(A),list(A)),product_case_prod(list(A),list(A),product_prod(list(A),list(A)),aa(A,fun(list(A),fun(list(A),product_prod(list(A),list(A)))),aTP_Lamp_adk(fun(A,bool),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),P),X)),partition(A,P,Xs)) ).

% partition.simps(2)
tff(fact_7763_partition__filter2,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A)] : aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),partition(A,P,Xs)) = aa(list(A),list(A),filter2(A,aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),P)),Xs) ).

% partition_filter2
tff(fact_7764_partition__set,axiom,
    ! [A: $tType,P: fun(A,bool),Xs: list(A),Yes: list(A),No4: 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_7765_min__ext__compat,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S2)),R))
     => pp(aa(set(product_prod(set(A),set(A))),bool,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),bool),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),min_ext(A,R),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),sup_sup(set(product_prod(set(A),set(A)))),min_ext(A,S2)),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),insert(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),product_Pair(set(A),set(A),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),min_ext(A,R))) ) ).

% min_ext_compat
tff(fact_7766_map__of__map__restrict,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),Ks: list(A)] : map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_adl(fun(A,B),fun(A,product_prod(A,B)),F3)),Ks)) = restrict_map(A,B,aa(fun(A,B),fun(A,option(B)),comp(B,option(B),A,some(B)),F3),aa(list(A),set(A),set2(A),Ks)) ).

% map_of_map_restrict
tff(fact_7767_restrict__out,axiom,
    ! [A: $tType,B: $tType,X: A,A3: set(A),M: fun(A,option(B))] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
     => ( aa(A,option(B),restrict_map(A,B,M,A3),X) = none(B) ) ) ).

% restrict_out
tff(fact_7768_restrict__map__empty,axiom,
    ! [A: $tType,B: $tType,D4: set(A),X2: A] : aa(A,option(B),restrict_map(A,B,aTP_Lamp_zy(A,option(B)),D4),X2) = none(B) ).

% restrict_map_empty
tff(fact_7769_restrict__map__to__empty,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),X2: A] : aa(A,option(B),restrict_map(A,B,M,bot_bot(set(A))),X2) = none(B) ).

% restrict_map_to_empty
tff(fact_7770_restrict__map__def,axiom,
    ! [A: $tType,B: $tType,A3: set(A),M: fun(A,option(B)),X2: A] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
       => ( aa(A,option(B),restrict_map(A,B,M,A3),X2) = aa(A,option(B),M,X2) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
       => ( aa(A,option(B),restrict_map(A,B,M,A3),X2) = none(B) ) ) ) ).

% restrict_map_def
tff(fact_7771_max__ext__compat,axiom,
    ! [A: $tType,R: set(product_prod(A,A)),S2: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,R,S2)),R))
     => pp(aa(set(product_prod(set(A),set(A))),bool,aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),bool),ord_less_eq(set(product_prod(set(A),set(A)))),relcomp(set(A),set(A),set(A),max_ext(A,R),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(set(product_prod(set(A),set(A))),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),sup_sup(set(product_prod(set(A),set(A)))),max_ext(A,S2)),aa(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A))),aa(product_prod(set(A),set(A)),fun(set(product_prod(set(A),set(A))),set(product_prod(set(A),set(A)))),insert(product_prod(set(A),set(A))),aa(set(A),product_prod(set(A),set(A)),product_Pair(set(A),set(A),bot_bot(set(A))),bot_bot(set(A)))),bot_bot(set(product_prod(set(A),set(A)))))))),max_ext(A,R))) ) ).

% max_ext_compat
tff(fact_7772_restrict__map__upds,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),D4: set(A),M: fun(A,option(B))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),D4))
       => ( restrict_map(A,B,map_upds(A,B,M,Xs,Ys),D4) = map_upds(A,B,restrict_map(A,B,M,aa(set(A),set(A),minus_minus(set(A),D4),aa(list(A),set(A),set2(A),Xs))),Xs,Ys) ) ) ) ).

% restrict_map_upds
tff(fact_7773_map__upds__apply__nontin,axiom,
    ! [B: $tType,A: $tType,X: A,Xs: list(A),F3: fun(A,option(B)),Ys: list(B)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(A,option(B),map_upds(A,B,F3,Xs,Ys),X) = aa(A,option(B),F3,X) ) ) ).

% map_upds_apply_nontin
tff(fact_7774_fun__upds__append2__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),M: fun(A,option(B)),Zs: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( map_upds(A,B,M,Xs,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Zs)) = map_upds(A,B,M,Xs,Ys) ) ) ).

% fun_upds_append2_drop
tff(fact_7775_fun__upds__append__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),M: fun(A,option(B)),Zs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( map_upds(A,B,M,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Zs),Ys) = map_upds(A,B,M,Xs,Ys) ) ) ).

% fun_upds_append_drop
tff(fact_7776_map__upds__list__update2__drop,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),I2: nat,M: fun(A,option(B)),Ys: list(B),Y: B] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Xs)),I2))
     => ( map_upds(A,B,M,Xs,list_update(B,Ys,I2,Y)) = map_upds(A,B,M,Xs,Ys) ) ) ).

% map_upds_list_update2_drop
tff(fact_7777_map__upds__append1,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),M: fun(A,option(B)),X: A] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)))
     => ( map_upds(A,B,M,aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))),Ys) = fun_upd(A,option(B),map_upds(A,B,M,Xs,Ys),X,aa(B,option(B),some(B),aa(nat,B,nth(B,Ys),aa(list(A),nat,size_size(list(A)),Xs)))) ) ) ).

% map_upds_append1
tff(fact_7778_map__filter__def,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,option(B)),Xs: list(A)] : map_filter(A,B,F3,Xs) = aa(list(A),list(B),map(A,B,aa(fun(A,option(B)),fun(A,B),comp(option(B),B,A,the2(B)),F3)),aa(list(A),list(A),filter2(A,aTP_Lamp_adm(fun(A,option(B)),fun(A,bool),F3)),Xs)) ).

% map_filter_def
tff(fact_7779_empty__upd__none,axiom,
    ! [A: $tType,B: $tType,X: A,X2: A] : aa(A,option(B),fun_upd(A,option(B),aTP_Lamp_zy(A,option(B)),X,none(B)),X2) = none(B) ).

% empty_upd_none
tff(fact_7780_map__fun__upd,axiom,
    ! [B: $tType,A: $tType,Y: A,Xs: list(A),F3: fun(A,B),V: B] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y),aa(list(A),set(A),set2(A),Xs)))
     => ( aa(list(A),list(B),map(A,B,fun_upd(A,B,F3,Y,V)),Xs) = aa(list(A),list(B),map(A,B,F3),Xs) ) ) ).

% map_fun_upd
tff(fact_7781_option_Ocollapse,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
     => ( aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) = Option ) ) ).

% option.collapse
tff(fact_7782_map__upds__twist,axiom,
    ! [A: $tType,B: $tType,A2: A,As3: list(A),M: fun(A,option(B)),B2: B,Bs: list(B)] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),aa(list(A),set(A),set2(A),As3)))
     => ( map_upds(A,B,fun_upd(A,option(B),M,A2,aa(B,option(B),some(B),B2)),As3,Bs) = fun_upd(A,option(B),map_upds(A,B,M,As3,Bs),A2,aa(B,option(B),some(B),B2)) ) ) ).

% map_upds_twist
tff(fact_7783_ran__map__upd,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A2: B,B2: A] :
      ( ( aa(B,option(A),M,A2) = none(A) )
     => ( ran(B,A,fun_upd(B,option(A),M,A2,aa(A,option(A),some(A),B2))) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),B2),ran(B,A,M)) ) ) ).

% ran_map_upd
tff(fact_7784_fun__upd__None__restrict,axiom,
    ! [B: $tType,A: $tType,X: A,D4: set(A),M: fun(A,option(B))] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),D4))
       => ( fun_upd(A,option(B),restrict_map(A,B,M,D4),X,none(B)) = restrict_map(A,B,M,aa(set(A),set(A),minus_minus(set(A),D4),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),D4))
       => ( fun_upd(A,option(B),restrict_map(A,B,M,D4),X,none(B)) = restrict_map(A,B,M,D4) ) ) ) ).

% fun_upd_None_restrict
tff(fact_7785_map__upd__nonempty,axiom,
    ! [A: $tType,B: $tType,T2: fun(A,option(B)),K: A,X: B] :
      ~ ! [X3: A] : aa(A,option(B),fun_upd(A,option(B),T2,K,aa(B,option(B),some(B),X)),X3) = none(B) ).

% map_upd_nonempty
tff(fact_7786_option_Oexhaust__sel,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
     => ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) ) ) ).

% option.exhaust_sel
tff(fact_7787_option_Oexpand,axiom,
    ! [A: $tType,Option: option(A),Option2: option(A)] :
      ( ( ( Option = none(A) )
      <=> ( Option2 = none(A) ) )
     => ( ( ( Option != none(A) )
         => ( ( Option2 != none(A) )
           => ( aa(option(A),A,the2(A),Option) = aa(option(A),A,the2(A),Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_7788_option_Omap__sel,axiom,
    ! [B: $tType,A: $tType,A2: option(A),F3: fun(A,B)] :
      ( ( A2 != none(A) )
     => ( aa(option(B),B,the2(B),aa(option(A),option(B),map_option(A,B,F3),A2)) = aa(A,B,F3,aa(option(A),A,the2(A),A2)) ) ) ).

% option.map_sel
tff(fact_7789_option_Ocase__eq__if,axiom,
    ! [B: $tType,A: $tType,Option: option(A),F1: B,F2: fun(A,B)] :
      ( ( ( Option = none(A) )
       => ( case_option(B,A,F1,F2,Option) = F1 ) )
      & ( ( Option != none(A) )
       => ( case_option(B,A,F1,F2,Option) = aa(A,B,F2,aa(option(A),A,the2(A),Option)) ) ) ) ).

% option.case_eq_if
tff(fact_7790_Option_Othese__def,axiom,
    ! [A: $tType,A3: set(option(A))] : these(A,A3) = aa(set(option(A)),set(A),image(option(A),A,the2(A)),collect(option(A),aTP_Lamp_zg(set(option(A)),fun(option(A),bool),A3))) ).

% Option.these_def
tff(fact_7791_Max_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),A3))
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Max.infinite
tff(fact_7792_Min_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),A3))
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),A3) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Min.infinite
tff(fact_7793_option_Osplit__sel,axiom,
    ! [B: $tType,A: $tType,P: fun(B,bool),F1: B,F2: fun(A,B),Option: option(A)] :
      ( pp(aa(B,bool,P,case_option(B,A,F1,F2,Option)))
    <=> ( ( ( Option = none(A) )
         => pp(aa(B,bool,P,F1)) )
        & ( ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) )
         => pp(aa(B,bool,P,aa(A,B,F2,aa(option(A),A,the2(A),Option)))) ) ) ) ).

% option.split_sel
tff(fact_7794_option_Osplit__sel__asm,axiom,
    ! [B: $tType,A: $tType,P: fun(B,bool),F1: B,F2: fun(A,B),Option: option(A)] :
      ( pp(aa(B,bool,P,case_option(B,A,F1,F2,Option)))
    <=> ~ ( ( ( Option = none(A) )
            & ~ pp(aa(B,bool,P,F1)) )
          | ( ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) )
            & ~ pp(aa(B,bool,P,aa(A,B,F2,aa(option(A),A,the2(A),Option)))) ) ) ) ).

% option.split_sel_asm
tff(fact_7795_map__of__zip__upd,axiom,
    ! [A: $tType,B: $tType,Ys: list(B),Xs: list(A),Zs: list(B),X: A,Y: B,Z: B] :
      ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(A),nat,size_size(list(A)),Xs) )
     => ( ( aa(list(B),nat,size_size(list(B)),Zs) = aa(list(A),nat,size_size(list(A)),Xs) )
       => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
         => ( ( fun_upd(A,option(B),map_of(A,B,zip(A,B,Xs,Ys)),X,aa(B,option(B),some(B),Y)) = fun_upd(A,option(B),map_of(A,B,zip(A,B,Xs,Zs)),X,aa(B,option(B),some(B),Z)) )
           => ( map_of(A,B,zip(A,B,Xs,Ys)) = map_of(A,B,zip(A,B,Xs,Zs)) ) ) ) ) ) ).

% map_of_zip_upd
tff(fact_7796_restrict__complement__singleton__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B)),X: A] : restrict_map(A,B,F3,aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))))) = fun_upd(A,option(B),F3,X,none(B)) ).

% restrict_complement_singleton_eq
tff(fact_7797_map__upd__upds__conv__if,axiom,
    ! [A: $tType,B: $tType,X: A,Ys: list(B),Xs: list(A),F3: fun(A,option(B)),Y: B] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys),Xs))))
       => ( map_upds(A,B,fun_upd(A,option(B),F3,X,aa(B,option(B),some(B),Y)),Xs,Ys) = map_upds(A,B,F3,Xs,Ys) ) )
      & ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys),Xs))))
       => ( map_upds(A,B,fun_upd(A,option(B),F3,X,aa(B,option(B),some(B),Y)),Xs,Ys) = fun_upd(A,option(B),map_upds(A,B,F3,Xs,Ys),X,aa(B,option(B),some(B),Y)) ) ) ) ).

% map_upd_upds_conv_if
tff(fact_7798_graph__map__upd,axiom,
    ! [A: $tType,B: $tType,M: fun(A,option(B)),K: A,V: B] : graph(A,B,fun_upd(A,option(B),M,K,aa(B,option(B),some(B),V))) = aa(set(product_prod(A,B)),set(product_prod(A,B)),aa(product_prod(A,B),fun(set(product_prod(A,B)),set(product_prod(A,B))),insert(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,K),V)),graph(A,B,fun_upd(A,option(B),M,K,none(B)))) ).

% graph_map_upd
tff(fact_7799_lists__length__Suc__eq,axiom,
    ! [A: $tType,A3: set(A),N: nat] : collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_adn(set(A),fun(nat,fun(list(A),bool)),A3),N)) = aa(set(product_prod(list(A),A)),set(list(A)),image(product_prod(list(A),A),list(A),product_case_prod(list(A),A,list(A),aTP_Lamp_abc(list(A),fun(A,list(A))))),product_Sigma(list(A),A,collect(list(A),aa(nat,fun(list(A),bool),aTP_Lamp_iw(set(A),fun(nat,fun(list(A),bool)),A3),N)),aTP_Lamp_ado(set(A),fun(list(A),set(A)),A3))) ).

% lists_length_Suc_eq
tff(fact_7800_graph__empty,axiom,
    ! [B: $tType,A: $tType] : graph(A,B,aTP_Lamp_zy(A,option(B))) = bot_bot(set(product_prod(A,B))) ).

% graph_empty
tff(fact_7801_set__product,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),product(A,B,Xs,Ys)) = product_Sigma(A,B,aa(list(A),set(A),set2(A),Xs),aTP_Lamp_adp(list(B),fun(A,set(B)),Ys)) ).

% set_product
tff(fact_7802_relcomp__subset__Sigma,axiom,
    ! [B: $tType,A: $tType,C: $tType,R3: set(product_prod(A,B)),A3: set(A),B3: set(B),S: set(product_prod(B,C)),C5: set(C)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),R3),product_Sigma(A,B,A3,aTP_Lamp_adq(set(B),fun(A,set(B)),B3))))
     => ( pp(aa(set(product_prod(B,C)),bool,aa(set(product_prod(B,C)),fun(set(product_prod(B,C)),bool),ord_less_eq(set(product_prod(B,C))),S),product_Sigma(B,C,B3,aTP_Lamp_adr(set(C),fun(B,set(C)),C5))))
       => pp(aa(set(product_prod(A,C)),bool,aa(set(product_prod(A,C)),fun(set(product_prod(A,C)),bool),ord_less_eq(set(product_prod(A,C))),relcomp(A,B,C,R3,S)),product_Sigma(A,C,A3,aTP_Lamp_ads(set(C),fun(A,set(C)),C5)))) ) ) ).

% relcomp_subset_Sigma
tff(fact_7803_trancl__subset__Sigma__aux,axiom,
    ! [A: $tType,A2: A,B2: A,R3: set(product_prod(A,A)),A3: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,A2),B2)),transitive_rtrancl(A,R3)))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A3,aTP_Lamp_adt(set(A),fun(A,set(A)),A3))))
       => ( ( A2 = B2 )
          | pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3)) ) ) ) ).

% trancl_subset_Sigma_aux
tff(fact_7804_Ex__inj__on__UNION__Sigma,axiom,
    ! [A: $tType,B: $tType,A3: fun(B,set(A)),I5: set(B)] :
    ? [F4: fun(A,product_prod(B,A))] :
      ( inj_on(A,product_prod(B,A),F4,aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5)))
      & pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),aa(set(A),set(product_prod(B,A)),image(A,product_prod(B,A),F4),aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(B),set(set(A)),image(B,set(A),A3),I5)))),product_Sigma(B,A,I5,A3))) ) ).

% Ex_inj_on_UNION_Sigma
tff(fact_7805_times__subset__iff,axiom,
    ! [A: $tType,B: $tType,A3: set(A),C5: set(B),B3: set(A),D4: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_adq(set(B),fun(A,set(B)),C5))),product_Sigma(A,B,B3,aTP_Lamp_adq(set(B),fun(A,set(B)),D4))))
    <=> ( ( A3 = bot_bot(set(A)) )
        | ( C5 = bot_bot(set(B)) )
        | ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),C5),D4)) ) ) ) ).

% times_subset_iff
tff(fact_7806_Sigma__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),C5: set(A),B3: fun(A,set(B)),D4: fun(A,set(B))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),C5))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),B3,X3)),aa(A,set(B),D4,X3))) )
       => pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A3,B3)),product_Sigma(A,B,C5,D4))) ) ) ).

% Sigma_mono
tff(fact_7807_Times__subset__cancel2,axiom,
    ! [A: $tType,B: $tType,X: A,C5: set(A),A3: set(B),B3: set(B)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),C5))
     => ( pp(aa(set(product_prod(B,A)),bool,aa(set(product_prod(B,A)),fun(set(product_prod(B,A)),bool),ord_less_eq(set(product_prod(B,A))),product_Sigma(B,A,A3,aTP_Lamp_adu(set(A),fun(B,set(A)),C5))),product_Sigma(B,A,B3,aTP_Lamp_adu(set(A),fun(B,set(A)),C5))))
      <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),B3)) ) ) ).

% Times_subset_cancel2
tff(fact_7808_trancl__subset__Sigma,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A3,aTP_Lamp_adt(set(A),fun(A,set(A)),A3))))
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_trancl(A,R3)),product_Sigma(A,A,A3,aTP_Lamp_adt(set(A),fun(A,set(A)),A3)))) ) ).

% trancl_subset_Sigma
tff(fact_7809_Restr__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A),R3: set(product_prod(A,A))] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,B3,aTP_Lamp_adt(set(A),fun(A,set(A)),B3)))),product_Sigma(A,A,A3,aTP_Lamp_adt(set(A),fun(A,set(A)),A3))) = aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),R3),product_Sigma(A,A,A3,aTP_Lamp_adt(set(A),fun(A,set(A)),A3))) ) ) ).

% Restr_subset
tff(fact_7810_refl__on__def,axiom,
    ! [A: $tType,A3: set(A),R3: set(product_prod(A,A))] :
      ( refl_on(A,A3,R3)
    <=> ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A3,aTP_Lamp_adt(set(A),fun(A,set(A)),A3))))
        & ! [X4: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
           => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X4),X4)),R3)) ) ) ) ).

% refl_on_def
tff(fact_7811_refl__onI,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A3,aTP_Lamp_adt(set(A),fun(A,set(A)),A3))))
     => ( ! [X3: A] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
           => pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X3),X3)),R3)) )
       => refl_on(A,A3,R3) ) ) ).

% refl_onI
tff(fact_7812_open__prod__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S2: set(product_prod(A,B))] :
          ( topolo1002775350975398744n_open(product_prod(A,B),S2)
        <=> ! [X4: product_prod(A,B)] :
              ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),X4),S2))
             => ? [A11: set(A)] :
                  ( topolo1002775350975398744n_open(A,A11)
                  & ? [B9: set(B)] :
                      ( topolo1002775350975398744n_open(B,B9)
                      & pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),X4),product_Sigma(A,B,A11,aTP_Lamp_adv(set(B),fun(A,set(B)),B9))))
                      & pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A11,aTP_Lamp_adv(set(B),fun(A,set(B)),B9))),S2)) ) ) ) ) ) ).

% open_prod_def
tff(fact_7813_open__prod__elim,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S2: set(product_prod(A,B)),X: product_prod(A,B)] :
          ( topolo1002775350975398744n_open(product_prod(A,B),S2)
         => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),X),S2))
           => ~ ! [A7: set(A)] :
                  ( topolo1002775350975398744n_open(A,A7)
                 => ! [B7: set(B)] :
                      ( topolo1002775350975398744n_open(B,B7)
                     => ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),X),product_Sigma(A,B,A7,aTP_Lamp_adv(set(B),fun(A,set(B)),B7))))
                       => ~ pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A7,aTP_Lamp_adv(set(B),fun(A,set(B)),B7))),S2)) ) ) ) ) ) ) ).

% open_prod_elim
tff(fact_7814_open__prod__intro,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [S2: set(product_prod(A,B))] :
          ( ! [X3: product_prod(A,B)] :
              ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),X3),S2))
             => ? [A17: set(A),B8: set(B)] :
                  ( topolo1002775350975398744n_open(A,A17)
                  & topolo1002775350975398744n_open(B,B8)
                  & pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),X3),product_Sigma(A,B,A17,aTP_Lamp_adv(set(B),fun(A,set(B)),B8))))
                  & pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A17,aTP_Lamp_adv(set(B),fun(A,set(B)),B8))),S2)) ) )
         => topolo1002775350975398744n_open(product_prod(A,B),S2) ) ) ).

% open_prod_intro
tff(fact_7815_subset__fst__imageI,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(B),S2: set(product_prod(A,B)),Y: B] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_adq(set(B),fun(A,set(B)),B3))),S2))
     => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Y),B3))
       => pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),S2))) ) ) ).

% subset_fst_imageI
tff(fact_7816_subset__snd__imageI,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B3: set(B),S2: set(product_prod(A,B)),X: A] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),product_Sigma(A,B,A3,aTP_Lamp_adq(set(B),fun(A,set(B)),B3))),S2))
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
       => pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),aa(set(product_prod(A,B)),set(B),image(product_prod(A,B),B,product_snd(A,B)),S2))) ) ) ).

% subset_snd_imageI
tff(fact_7817_graph__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : graph(A,B,M) = collect(product_prod(A,B),aTP_Lamp_adw(fun(A,option(B)),fun(product_prod(A,B),bool),M)) ).

% graph_def
tff(fact_7818_subset__fst__snd,axiom,
    ! [B: $tType,A: $tType,A3: set(product_prod(A,B))] : pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),A3),product_Sigma(A,B,aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),A3),aTP_Lamp_adx(set(product_prod(A,B)),fun(A,set(B)),A3)))) ).

% subset_fst_snd
tff(fact_7819_graph__fun__upd__None,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),K: A] : graph(A,B,fun_upd(A,option(B),M,K,none(B))) = collect(product_prod(A,B),aa(A,fun(product_prod(A,B),bool),aTP_Lamp_ady(fun(A,option(B)),fun(A,fun(product_prod(A,B),bool)),M),K)) ).

% graph_fun_upd_None
tff(fact_7820_Gr__incl,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F3: fun(A,B),B3: set(B)] :
      ( pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),ord_less_eq(set(product_prod(A,B))),bNF_Gr(A,B,A3,F3)),product_Sigma(A,B,A3,aTP_Lamp_adq(set(B),fun(A,set(B)),B3))))
    <=> pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F3),A3)),B3)) ) ).

% Gr_incl
tff(fact_7821_Min_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : aa(set(A),A,lattic643756798350308766er_Min(A),A3) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_adz(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Min.eq_fold'
tff(fact_7822_Gr__def,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F3: fun(A,B)] : bNF_Gr(A,B,A3,F3) = collect(product_prod(A,B),aa(fun(A,B),fun(product_prod(A,B),bool),aTP_Lamp_aea(set(A),fun(fun(A,B),fun(product_prod(A,B),bool)),A3),F3)) ).

% Gr_def
tff(fact_7823_sum_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) = finite_fold(B,A,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_7824_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_7825_Max_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( aa(set(A),A,lattic643756798349783984er_Max(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = finite_fold(A,A,ord_max(A),X,A3) ) ) ) ).

% Max.eq_fold
tff(fact_7826_Min_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( aa(set(A),A,lattic643756798350308766er_Min(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = finite_fold(A,A,ord_min(A),X,A3) ) ) ) ).

% Min.eq_fold
tff(fact_7827_comp__fun__idem__on_Ofold__insert__idem2,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),X: A,A3: set(A),Z: B] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)),S2))
       => ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = finite_fold(A,B,F3,aa(B,B,aa(A,fun(B,B),F3,X),Z),A3) ) ) ) ) ).

% comp_fun_idem_on.fold_insert_idem2
tff(fact_7828_comp__fun__idem__on_Ofold__insert__idem,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),X: A,A3: set(A),Z: B] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)),S2))
       => ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(B,B,aa(A,fun(B,B),F3,X),finite_fold(A,B,F3,Z,A3)) ) ) ) ) ).

% comp_fun_idem_on.fold_insert_idem
tff(fact_7829_pairs__le__eq__Sigma,axiom,
    ! [M: nat] : collect(product_prod(nat,nat),product_case_prod(nat,nat,bool,aTP_Lamp_hi(nat,fun(nat,fun(nat,bool)),M))) = product_Sigma(nat,nat,aa(nat,set(nat),set_ord_atMost(nat),M),aTP_Lamp_aeb(nat,fun(nat,set(nat)),M)) ).

% pairs_le_eq_Sigma
tff(fact_7830_Max_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : aa(set(A),A,lattic643756798349783984er_Max(A),A3) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_aec(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Max.eq_fold'
tff(fact_7831_Sup__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A)] : aa(set(A),A,lattic5882676163264333800up_fin(A),A3) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_aed(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Sup_fin.eq_fold'
tff(fact_7832_Inf__fin_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A)] : aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_aee(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Inf_fin.eq_fold'
tff(fact_7833_inf__Sup__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),A2),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) = A2 ) ) ) ) ).

% inf_Sup_absorb
tff(fact_7834_Inf__fin_Osingleton,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A] : aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% Inf_fin.singleton
tff(fact_7835_Sup__fin_Osingleton,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A] : aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = X ) ).

% Sup_fin.singleton
tff(fact_7836_sup__Inf__absorb,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)),A2) = A2 ) ) ) ) ).

% sup_Inf_absorb
tff(fact_7837_Inf__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)) ) ) ) ) ).

% Inf_fin.insert
tff(fact_7838_Sup__fin_Oinsert,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) ) ) ) ) ).

% Sup_fin.insert
tff(fact_7839_Inf__fin__le__Sup__fin,axiom,
    ! [A: $tType] :
      ( lattice(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)),aa(set(A),A,lattic5882676163264333800up_fin(A),A3))) ) ) ) ).

% Inf_fin_le_Sup_fin
tff(fact_7840_Sup__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A3) ) ) ) ) ).

% Sup_fin.in_idem
tff(fact_7841_Inf__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)),A2)) ) ) ) ).

% Inf_fin.coboundedI
tff(fact_7842_Sup__fin_OcoboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),A2: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A2),aa(set(A),A,lattic5882676163264333800up_fin(A),A3))) ) ) ) ).

% Sup_fin.coboundedI
tff(fact_7843_Inf__fin_Oin__idem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) ) ) ) ) ).

% Inf_fin.in_idem
tff(fact_7844_Sup__fin__Max,axiom,
    ! [A: $tType] :
      ( ( semilattice_sup(A)
        & linorder(A) )
     => ( lattic5882676163264333800up_fin(A) = lattic643756798349783984er_Max(A) ) ) ).

% Sup_fin_Max
tff(fact_7845_Inf__fin__Min,axiom,
    ! [A: $tType] :
      ( ( semilattice_inf(A)
        & linorder(A) )
     => ( lattic7752659483105999362nf_fin(A) = lattic643756798350308766er_Min(A) ) ) ).

% Inf_fin_Min
tff(fact_7846_Sup__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)),X))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),X)) ) ) ) ) ) ).

% Sup_fin.bounded_iff
tff(fact_7847_Inf__fin_Obounded__iff,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)))
            <=> ! [X4: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),X4)) ) ) ) ) ) ).

% Inf_fin.bounded_iff
tff(fact_7848_Sup__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),X)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)),X)) ) ) ) ) ).

% Sup_fin.boundedI
tff(fact_7849_Sup__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)),X))
             => ! [A16: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A16),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A16),X)) ) ) ) ) ) ).

% Sup_fin.boundedE
tff(fact_7850_Inf__fin_OboundedI,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A5)) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3))) ) ) ) ) ).

% Inf_fin.boundedI
tff(fact_7851_Inf__fin_OboundedE,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)))
             => ! [A16: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A16),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),A16)) ) ) ) ) ) ).

% Inf_fin.boundedE
tff(fact_7852_card_Oeq__fold,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),nat,finite_card(A),A3) = finite_fold(A,nat,aTP_Lamp_zh(A,fun(nat,nat)),zero_zero(nat),A3) ).

% card.eq_fold
tff(fact_7853_Sup__fin__Sup,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A3) = aa(set(A),A,complete_Sup_Sup(A),A3) ) ) ) ) ).

% Sup_fin_Sup
tff(fact_7854_Inf__fin__Inf,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) = aa(set(A),A,complete_Inf_Inf(A),A3) ) ) ) ) ).

% Inf_fin_Inf
tff(fact_7855_Inf__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),A3))
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Inf_fin.infinite
tff(fact_7856_Sup__fin_Oinfinite,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A)] :
          ( ~ pp(aa(set(A),bool,finite_finite(A),A3))
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A3) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Sup_fin.infinite
tff(fact_7857_sorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : aa(set(A),list(A),linord4507533701916653071of_set(A),A3) = finite_fold(A,list(A),linorder_insort_key(A,A,aTP_Lamp_ma(A,A)),nil(A),A3) ) ).

% sorted_list_of_set.fold_insort_key.eq_fold
tff(fact_7858_Sup__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite(A),B3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)),aa(set(A),A,lattic5882676163264333800up_fin(A),B3))) ) ) ) ) ).

% Sup_fin.subset_imp
tff(fact_7859_Inf__fin_Osubset__imp,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite(A),B3))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B3)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3))) ) ) ) ) ).

% Inf_fin.subset_imp
tff(fact_7860_Inf__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [H: fun(A,A),N3: set(A)] :
          ( ! [X3: A,Y4: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y4)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(A,A,H,X3)),aa(A,A,H,Y4))
         => ( pp(aa(set(A),bool,finite_finite(A),N3))
           => ( ( N3 != bot_bot(set(A)) )
             => ( aa(A,A,H,aa(set(A),A,lattic7752659483105999362nf_fin(A),N3)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),image(A,A,H),N3)) ) ) ) ) ) ).

% Inf_fin.hom_commute
tff(fact_7861_Sup__fin_Ohom__commute,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [H: fun(A,A),N3: set(A)] :
          ( ! [X3: A,Y4: A] : aa(A,A,H,aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y4)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(A,A,H,X3)),aa(A,A,H,Y4))
         => ( pp(aa(set(A),bool,finite_finite(A),N3))
           => ( ( N3 != bot_bot(set(A)) )
             => ( aa(A,A,H,aa(set(A),A,lattic5882676163264333800up_fin(A),N3)) = aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),image(A,A,H),N3)) ) ) ) ) ) ).

% Sup_fin.hom_commute
tff(fact_7862_Inf__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( B3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
             => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),B3)),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) ) ) ) ) ) ).

% Inf_fin.subset
tff(fact_7863_Sup__fin_Osubset,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( B3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
             => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),B3)),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) = aa(set(A),A,lattic5882676163264333800up_fin(A),A3) ) ) ) ) ) ).

% Sup_fin.subset
tff(fact_7864_Inf__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( ( A3 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
tff(fact_7865_Inf__fin_Oclosed,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X3: A,Y4: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),inf_inf(A),X3),Y4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y4),bot_bot(set(A))))))
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)),A3)) ) ) ) ) ).

% Inf_fin.closed
tff(fact_7866_Sup__fin_Oinsert__not__elem,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( ( A3 != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
tff(fact_7867_Sup__fin_Oclosed,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X3: A,Y4: A] : pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),X3),Y4)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),Y4),bot_bot(set(A))))))
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)),A3)) ) ) ) ) ).

% Sup_fin.closed
tff(fact_7868_Inf__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite(A),B3))
             => ( ( B3 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B3)) ) ) ) ) ) ) ).

% Inf_fin.union
tff(fact_7869_Sup__fin_Ounion,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite(A),B3))
             => ( ( B3 != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)),aa(set(A),A,lattic5882676163264333800up_fin(A),B3)) ) ) ) ) ) ) ).

% Sup_fin.union
tff(fact_7870_Inf__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = finite_fold(A,A,inf_inf(A),X,A3) ) ) ) ).

% Inf_fin.eq_fold
tff(fact_7871_Sup__fin_Oeq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = finite_fold(A,A,sup_sup(A),X,A3) ) ) ) ).

% Sup_fin.eq_fold
tff(fact_7872_inf__Sup2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite(A),B3))
             => ( ( B3 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)),aa(set(A),A,lattic5882676163264333800up_fin(A),B3)) = aa(set(A),A,lattic5882676163264333800up_fin(A),collect(A,aa(set(A),fun(A,bool),aTP_Lamp_aef(set(A),fun(set(A),fun(A,bool)),A3),B3))) ) ) ) ) ) ) ).

% inf_Sup2_distrib
tff(fact_7873_inf__Sup1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),A3)) = aa(set(A),A,lattic5882676163264333800up_fin(A),collect(A,aa(A,fun(A,bool),aTP_Lamp_aeg(set(A),fun(A,fun(A,bool)),A3),X))) ) ) ) ) ).

% inf_Sup1_distrib
tff(fact_7874_sup__Inf1__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),collect(A,aa(A,fun(A,bool),aTP_Lamp_aeh(set(A),fun(A,fun(A,bool)),A3),X))) ) ) ) ) ).

% sup_Inf1_distrib
tff(fact_7875_sup__Inf2__distrib,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( A3 != bot_bot(set(A)) )
           => ( pp(aa(set(A),bool,finite_finite(A),B3))
             => ( ( B3 != bot_bot(set(A)) )
               => ( aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,lattic7752659483105999362nf_fin(A),A3)),aa(set(A),A,lattic7752659483105999362nf_fin(A),B3)) = aa(set(A),A,lattic7752659483105999362nf_fin(A),collect(A,aa(set(A),fun(A,bool),aTP_Lamp_aei(set(A),fun(set(A),fun(A,bool)),A3),B3))) ) ) ) ) ) ) ).

% sup_Inf2_distrib
tff(fact_7876_Inf__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( ( aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = X ) )
            & ( ( aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Inf_fin.insert_remove
tff(fact_7877_Inf__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( ( ( aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) = X ) )
              & ( ( aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic7752659483105999362nf_fin(A),A3) = aa(A,A,aa(A,fun(A,A),inf_inf(A),X),aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Inf_fin.remove
tff(fact_7878_Sup__fin_Oremove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( ( ( aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
               => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A3) = X ) )
              & ( ( aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
               => ( aa(set(A),A,lattic5882676163264333800up_fin(A),A3) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ) ).

% Sup_fin.remove
tff(fact_7879_Sup__fin_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [A3: set(A),X: A] :
          ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ( ( aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) = bot_bot(set(A)) )
             => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = X ) )
            & ( ( aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) != bot_bot(set(A)) )
             => ( aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(A,A,aa(A,fun(A,A),sup_sup(A),X),aa(set(A),A,lattic5882676163264333800up_fin(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% Sup_fin.insert_remove
tff(fact_7880_comp__fun__commute__on_Ofold__set__union__disj,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),A3: set(A),B3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),S2))
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),S2))
         => ( pp(aa(set(A),bool,finite_finite(A),A3))
           => ( pp(aa(set(A),bool,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)) )
               => ( finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),A3),B3)) = finite_fold(A,B,F3,finite_fold(A,B,F3,Z,A3),B3) ) ) ) ) ) ) ) ).

% comp_fun_commute_on.fold_set_union_disj
tff(fact_7881_comp__fun__commute__on_Ofold__insert__remove,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),X: A,A3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)),S2))
       => ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(B,B,aa(A,fun(B,B),F3,X),finite_fold(A,B,F3,Z,aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ).

% comp_fun_commute_on.fold_insert_remove
tff(fact_7882_Finite__Set_Ofold__cong,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),G: fun(A,fun(B,B)),A3: set(A),S: B,T2: B,B3: set(A)] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( finite4664212375090638736ute_on(A,B,S2,G)
       => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),S2))
         => ( pp(aa(set(A),bool,finite_finite(A),A3))
           => ( ! [X3: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
                 => ( aa(A,fun(B,B),F3,X3) = aa(A,fun(B,B),G,X3) ) )
             => ( ( S = T2 )
               => ( ( A3 = B3 )
                 => ( finite_fold(A,B,F3,S,A3) = finite_fold(A,B,G,T2,B3) ) ) ) ) ) ) ) ) ).

% Finite_Set.fold_cong
tff(fact_7883_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
    ! [B: $tType,A: $tType,C: $tType,S2: set(A),F3: fun(A,fun(B,B)),G: fun(C,A),R: set(C)] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,G),top_top(set(C)))),S2))
       => finite4664212375090638736ute_on(C,B,R,aa(fun(C,A),fun(C,fun(B,B)),comp(A,fun(B,B),C,F3),G)) ) ) ).

% comp_fun_commute_on.comp_comp_fun_commute_on
tff(fact_7884_comp__fun__commute__on_Ofold__insert,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),X: A,A3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)),S2))
       => ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = aa(B,B,aa(A,fun(B,B),F3,X),finite_fold(A,B,F3,Z,A3)) ) ) ) ) ) ).

% comp_fun_commute_on.fold_insert
tff(fact_7885_comp__fun__commute__on_Ofold__insert2,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),X: A,A3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)),S2))
       => ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( finite_fold(A,B,F3,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)) = finite_fold(A,B,F3,aa(B,B,aa(A,fun(B,B),F3,X),Z),A3) ) ) ) ) ) ).

% comp_fun_commute_on.fold_insert2
tff(fact_7886_comp__fun__commute__on_Ofold__fun__left__comm,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),X: A,A3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)),S2))
       => ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( aa(B,B,aa(A,fun(B,B),F3,X),finite_fold(A,B,F3,Z,A3)) = finite_fold(A,B,F3,aa(B,B,aa(A,fun(B,B),F3,X),Z),A3) ) ) ) ) ).

% comp_fun_commute_on.fold_fun_left_comm
tff(fact_7887_comp__fun__commute__on_Ofold__rec,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),A3: set(A),X: A,Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),S2))
       => ( pp(aa(set(A),bool,finite_finite(A),A3))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ( finite_fold(A,B,F3,Z,A3) = aa(B,B,aa(A,fun(B,B),F3,X),finite_fold(A,B,F3,Z,aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))))) ) ) ) ) ) ).

% comp_fun_commute_on.fold_rec
tff(fact_7888_ntrancl__Suc,axiom,
    ! [A: $tType,N: nat,R: set(product_prod(A,A))] : transitive_ntrancl(A,aa(nat,nat,suc,N),R) = relcomp(A,A,A,transitive_ntrancl(A,N,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_7889_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_abf(A,product_prod(A,A))),Xs)) ).

% Id_on_set
tff(fact_7890_Id__def,axiom,
    ! [A: $tType] : id2(A) = collect(product_prod(A,A),aTP_Lamp_aej(product_prod(A,A),bool)) ).

% Id_def
tff(fact_7891_Id__on__subset__Times,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),id_on(A,A3)),product_Sigma(A,A,A3,aTP_Lamp_adt(set(A),fun(A,set(A)),A3)))) ).

% Id_on_subset_Times
tff(fact_7892_rtrancl__Int__subset,axiom,
    ! [A: $tType,S: set(product_prod(A,A)),R3: set(product_prod(A,A))] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),id2(A)),S))
     => ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),relcomp(A,A,A,aa(set(product_prod(A,A)),set(product_prod(A,A)),aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),set(product_prod(A,A))),inf_inf(set(product_prod(A,A))),transitive_rtrancl(A,R3)),S),R3)),S))
       => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),transitive_rtrancl(A,R3)),S)) ) ) ).

% rtrancl_Int_subset
tff(fact_7893_relInvImage__Id__on,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,B),A3: set(A),B3: set(B)] :
      ( ! [A12: A,A23: A] :
          ( ( aa(A,B,F3,A12) = aa(A,B,F3,A23) )
        <=> ( A12 = A23 ) )
     => pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),bNF_Gr7122648621184425601vImage(A,B,A3,id_on(B,B3),F3)),id2(A))) ) ).

% relInvImage_Id_on
tff(fact_7894_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)) = groups5270119922927024881d_list(A,aa(list(B),list(A),map(B,A,G),remdups(B,Xs))) ) ).

% prod.set_conv_list
tff(fact_7895_prod__list_ONil,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( groups5270119922927024881d_list(A,nil(A)) = one_one(A) ) ) ).

% prod_list.Nil
tff(fact_7896_prod__list__zero__iff,axiom,
    ! [A: $tType] :
      ( ( semiring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [Xs: list(A)] :
          ( ( groups5270119922927024881d_list(A,Xs) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),zero_zero(A)),aa(list(A),set(A),set2(A),Xs))) ) ) ).

% prod_list_zero_iff
tff(fact_7897_relInvImage__mono,axiom,
    ! [A: $tType,B: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(A,A)),A3: set(B),F3: fun(B,A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R1),R22))
     => pp(aa(set(product_prod(B,B)),bool,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),bool),ord_less_eq(set(product_prod(B,B))),bNF_Gr7122648621184425601vImage(B,A,A3,R1,F3)),bNF_Gr7122648621184425601vImage(B,A,A3,R22,F3))) ) ).

% relInvImage_mono
tff(fact_7898_prod__list_Oeq__foldr,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xs: list(A)] : groups5270119922927024881d_list(A,Xs) = aa(A,A,foldr(A,A,times_times(A),Xs),one_one(A)) ) ).

% prod_list.eq_foldr
tff(fact_7899_relInvImage__def,axiom,
    ! [B: $tType,A: $tType,A3: set(A),R: set(product_prod(B,B)),F3: fun(A,B)] : bNF_Gr7122648621184425601vImage(A,B,A3,R,F3) = collect(product_prod(A,A),aa(fun(A,B),fun(product_prod(A,A),bool),aa(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool)),aTP_Lamp_aek(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool))),A3),R),F3)) ).

% relInvImage_def
tff(fact_7900_prod_Odistinct__set__conv__list,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Xs: list(B),G: fun(B,A)] :
          ( distinct(B,Xs)
         => ( 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)) = groups5270119922927024881d_list(A,aa(list(B),list(A),map(B,A,G),Xs)) ) ) ) ).

% prod.distinct_set_conv_list
tff(fact_7901_relImage__relInvImage,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),F3: fun(B,A),A3: set(B)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),product_Sigma(A,A,aa(set(B),set(A),image(B,A,F3),A3),aa(set(B),fun(A,set(A)),aTP_Lamp_ael(fun(B,A),fun(set(B),fun(A,set(A))),F3),A3))))
     => ( bNF_Gr4221423524335903396lImage(B,A,bNF_Gr7122648621184425601vImage(B,A,A3,R,F3),F3) = R ) ) ).

% relImage_relInvImage
tff(fact_7902_relInvImage__UNIV__relImage,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,A)),F3: fun(A,B)] : pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R),bNF_Gr7122648621184425601vImage(A,B,top_top(set(A)),bNF_Gr4221423524335903396lImage(A,B,R,F3),F3))) ).

% relInvImage_UNIV_relImage
tff(fact_7903_relImage__mono,axiom,
    ! [B: $tType,A: $tType,R1: set(product_prod(A,A)),R22: set(product_prod(A,A)),F3: fun(A,B)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R1),R22))
     => pp(aa(set(product_prod(B,B)),bool,aa(set(product_prod(B,B)),fun(set(product_prod(B,B)),bool),ord_less_eq(set(product_prod(B,B))),bNF_Gr4221423524335903396lImage(A,B,R1,F3)),bNF_Gr4221423524335903396lImage(A,B,R22,F3))) ) ).

% relImage_mono
tff(fact_7904_relImage__def,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,B)),F3: fun(B,A)] : bNF_Gr4221423524335903396lImage(B,A,R,F3) = collect(product_prod(A,A),aa(fun(B,A),fun(product_prod(A,A),bool),aTP_Lamp_aem(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),bool)),R),F3)) ).

% relImage_def
tff(fact_7905_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M: nat] : aa(product_prod(nat,nat),nat,nat_prod_encode,nat_prod_decode_aux(K,M)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(K)),M) ).

% prod_encode_prod_decode_aux
tff(fact_7906_lists__empty,axiom,
    ! [A: $tType] : lists(A,bot_bot(set(A))) = aa(set(list(A)),set(list(A)),aa(list(A),fun(set(list(A)),set(list(A))),insert(list(A)),nil(A)),bot_bot(set(list(A)))) ).

% lists_empty
tff(fact_7907_Cons__in__lists__iff,axiom,
    ! [A: $tType,X: A,Xs: list(A),A3: set(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),lists(A,A3)))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
        & pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Xs),lists(A,A3))) ) ) ).

% Cons_in_lists_iff
tff(fact_7908_in__listsI,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A)] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3)) )
     => pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Xs),lists(A,A3))) ) ).

% in_listsI
tff(fact_7909_lists__Int__eq,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] : lists(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)) = aa(set(list(A)),set(list(A)),aa(set(list(A)),fun(set(list(A)),set(list(A))),inf_inf(set(list(A))),lists(A,A3)),lists(A,B3)) ).

% lists_Int_eq
tff(fact_7910_append__in__lists__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),A3: set(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),lists(A,A3)))
    <=> ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Xs),lists(A,A3)))
        & pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Ys),lists(A,A3))) ) ) ).

% append_in_lists_conv
tff(fact_7911_lists__UNIV,axiom,
    ! [A: $tType] : lists(A,top_top(set(A))) = top_top(set(list(A))) ).

% lists_UNIV
tff(fact_7912_in__listsD,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Xs),lists(A,A3)))
     => ! [X2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3)) ) ) ).

% in_listsD
tff(fact_7913_in__lists__conv__set,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Xs),lists(A,A3)))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3)) ) ) ).

% in_lists_conv_set
tff(fact_7914_listrel__refl__on,axiom,
    ! [A: $tType,A3: set(A),R3: set(product_prod(A,A))] :
      ( refl_on(A,A3,R3)
     => refl_on(list(A),lists(A,A3),listrel(A,A,R3)) ) ).

% listrel_refl_on
tff(fact_7915_listsE,axiom,
    ! [A: $tType,X: A,L: list(A),A3: set(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),L)),lists(A,A3)))
     => ~ ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
         => ~ pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),L),lists(A,A3))) ) ) ).

% listsE
tff(fact_7916_lists_OCons,axiom,
    ! [A: $tType,A2: A,A3: set(A),L: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
     => ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),L),lists(A,A3)))
       => pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),L)),lists(A,A3))) ) ) ).

% lists.Cons
tff(fact_7917_lists_Ocases,axiom,
    ! [A: $tType,A2: list(A),A3: set(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),A2),lists(A,A3)))
     => ( ( A2 != nil(A) )
       => ~ ! [A5: A,L4: list(A)] :
              ( ( A2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A5),L4) )
             => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),A3))
               => ~ pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),L4),lists(A,A3))) ) ) ) ) ).

% lists.cases
tff(fact_7918_lists_Osimps,axiom,
    ! [A: $tType,A2: list(A),A3: set(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),A2),lists(A,A3)))
    <=> ( ( A2 = nil(A) )
        | ? [A6: A,L2: list(A)] :
            ( ( A2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),L2) )
            & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A6),A3))
            & pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),L2),lists(A,A3))) ) ) ) ).

% lists.simps
tff(fact_7919_lists_ONil,axiom,
    ! [A: $tType,A3: set(A)] : pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),nil(A)),lists(A,A3))) ).

% lists.Nil
tff(fact_7920_lists__IntI,axiom,
    ! [A: $tType,L: list(A),A3: set(A),B3: set(A)] :
      ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),L),lists(A,A3)))
     => ( pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),L),lists(A,B3)))
       => pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),L),lists(A,aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)))) ) ) ).

% lists_IntI
tff(fact_7921_lists__eq__set,axiom,
    ! [A: $tType,A3: set(A)] : lists(A,A3) = collect(list(A),aTP_Lamp_aen(set(A),fun(list(A),bool),A3)) ).

% lists_eq_set
tff(fact_7922_lists__mono,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => pp(aa(set(list(A)),bool,aa(set(list(A)),fun(set(list(A)),bool),ord_less_eq(set(list(A))),lists(A,A3)),lists(A,B3))) ) ).

% lists_mono
tff(fact_7923_le__prod__encode__1,axiom,
    ! [A2: nat,B2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),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_7924_le__prod__encode__2,axiom,
    ! [B2: nat,A2: nat] : pp(aa(nat,bool,aa(nat,fun(nat,bool),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_7925_lists__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),A3: set(B)] : lists(A,aa(set(B),set(A),image(B,A,F3),A3)) = aa(set(list(B)),set(list(A)),image(list(B),list(A),map(B,A,F3)),lists(B,A3)) ).

% lists_image
tff(fact_7926_prod__encode__def,axiom,
    nat_prod_encode = product_case_prod(nat,nat,nat,aTP_Lamp_aeo(nat,fun(nat,nat))) ).

% prod_encode_def
tff(fact_7927_Collect__finite__eq__lists,axiom,
    ! [A: $tType] : collect(set(A),finite_finite(A)) = aa(set(list(A)),set(set(A)),image(list(A),set(A),set2(A)),lists(A,top_top(set(A)))) ).

% Collect_finite_eq_lists
tff(fact_7928_Collect__finite__subset__eq__lists,axiom,
    ! [A: $tType,T6: set(A)] : collect(set(A),aTP_Lamp_aai(set(A),fun(set(A),bool),T6)) = aa(set(list(A)),set(set(A)),image(list(A),set(A),set2(A)),lists(A,T6)) ).

% Collect_finite_subset_eq_lists
tff(fact_7929_listrel__subset,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),A3: set(A)] :
      ( pp(aa(set(product_prod(A,A)),bool,aa(set(product_prod(A,A)),fun(set(product_prod(A,A)),bool),ord_less_eq(set(product_prod(A,A))),R3),product_Sigma(A,A,A3,aTP_Lamp_adt(set(A),fun(A,set(A)),A3))))
     => pp(aa(set(product_prod(list(A),list(A))),bool,aa(set(product_prod(list(A),list(A))),fun(set(product_prod(list(A),list(A))),bool),ord_less_eq(set(product_prod(list(A),list(A)))),listrel(A,A,R3)),product_Sigma(list(A),list(A),lists(A,A3),aTP_Lamp_aep(set(A),fun(list(A),set(list(A))),A3)))) ) ).

% listrel_subset
tff(fact_7930_list__encode_Oelims,axiom,
    ! [X: list(nat),Y: nat] :
      ( ( nat_list_encode(X) = Y )
     => ( ( ( X = nil(nat) )
         => ( Y != zero_zero(nat) ) )
       => ~ ! [X3: nat,Xs2: list(nat)] :
              ( ( X = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X3),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,X3),nat_list_encode(Xs2)))) ) ) ) ) ).

% list_encode.elims
tff(fact_7931_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list(nat)] : nat_list_encode(aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X),Xs)) = aa(nat,nat,suc,aa(product_prod(nat,nat),nat,nat_prod_encode,aa(nat,product_prod(nat,nat),product_Pair(nat,nat,X),nat_list_encode(Xs)))) ).

% list_encode.simps(2)
tff(fact_7932_list__encode_Opelims,axiom,
    ! [X: list(nat),Y: nat] :
      ( ( nat_list_encode(X) = Y )
     => ( pp(aa(list(nat),bool,accp(list(nat),nat_list_encode_rel),X))
       => ( ( ( X = nil(nat) )
           => ( ( Y = zero_zero(nat) )
             => ~ pp(aa(list(nat),bool,accp(list(nat),nat_list_encode_rel),nil(nat))) ) )
         => ~ ! [X3: nat,Xs2: list(nat)] :
                ( ( X = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X3),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,X3),nat_list_encode(Xs2)))) )
                 => ~ pp(aa(list(nat),bool,accp(list(nat),nat_list_encode_rel),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),X3),Xs2))) ) ) ) ) ) ).

% list_encode.pelims
tff(fact_7933_INF__set__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),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)),F3),Xs),top_top(A)) ) ).

% INF_set_fold
tff(fact_7934_fold__append,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),Xs: list(B),Ys: list(B)] : fold(B,A,F3,aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Xs),Ys)) = aa(fun(A,A),fun(A,A),comp(A,A,A,fold(B,A,F3,Ys)),fold(B,A,F3,Xs)) ).

% fold_append
tff(fact_7935_fold__replicate,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),N: nat,X: B] : fold(B,A,F3,replicate(B,N,X)) = aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),aa(B,fun(A,A),F3,X)) ).

% fold_replicate
tff(fact_7936_fold__remove1__split,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F3: fun(A,fun(B,B)),X: A] :
      ( ! [X3: A,Y4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),aa(list(A),set(A),set2(A),Xs)))
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X3)),aa(A,fun(B,B),F3,Y4)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Y4)),aa(A,fun(B,B),F3,X3)) ) ) )
     => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs)))
       => ( fold(A,B,F3,Xs) = aa(fun(B,B),fun(B,B),comp(B,B,B,fold(A,B,F3,remove1(A,X,Xs))),aa(A,fun(B,B),F3,X)) ) ) ) ).

% fold_remove1_split
tff(fact_7937_foldr__fold,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F3: fun(A,fun(B,B))] :
      ( ! [X3: A,Y4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),aa(list(A),set(A),set2(A),Xs)))
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Y4)),aa(A,fun(B,B),F3,X3)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X3)),aa(A,fun(B,B),F3,Y4)) ) ) )
     => ( foldr(A,B,F3,Xs) = fold(A,B,F3,Xs) ) ) ).

% foldr_fold
tff(fact_7938_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),groups8242544230860333062m_list(A,aa(list(A),list(A),rev(A),Xs))) ) ).

% fold_plus_sum_list_rev
tff(fact_7939_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),insert(A),Xs),A3) ).

% union_set_fold
tff(fact_7940_List_Ofold__cong,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: A,Xs: list(B),Ys: list(B),F3: fun(B,fun(A,A)),G: fun(B,fun(A,A))] :
      ( ( A2 = B2 )
     => ( ( Xs = Ys )
       => ( ! [X3: B] :
              ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),aa(list(B),set(B),set2(B),Xs)))
             => ( aa(B,fun(A,A),F3,X3) = aa(B,fun(A,A),G,X3) ) )
         => ( aa(A,A,fold(B,A,F3,Xs),A2) = aa(A,A,fold(B,A,G,Ys),B2) ) ) ) ) ).

% List.fold_cong
tff(fact_7941_fold__invariant,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Q: fun(A,bool),P: fun(B,bool),S: B,F3: fun(A,fun(B,B))] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,Q,X3)) )
     => ( pp(aa(B,bool,P,S))
       => ( ! [X3: A,S3: B] :
              ( pp(aa(A,bool,Q,X3))
             => ( pp(aa(B,bool,P,S3))
               => pp(aa(B,bool,P,aa(B,B,aa(A,fun(B,B),F3,X3),S3))) ) )
         => pp(aa(B,bool,P,aa(B,B,fold(A,B,F3,Xs),S))) ) ) ) ).

% fold_invariant
tff(fact_7942_fold__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,fun(A,A)),X: B,Xs: list(B),S: A] : aa(A,A,fold(B,A,F3,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)),S) = aa(A,A,fold(B,A,F3,Xs),aa(A,A,aa(B,fun(A,A),F3,X),S)) ).

% fold_simps(2)
tff(fact_7943_fold__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,F3: fun(B,fun(A,A)),S: A] : aa(A,A,fold(B,A,F3,nil(B)),S) = S ).

% fold_simps(1)
tff(fact_7944_fold__Cons__rev,axiom,
    ! [A: $tType,Xs: list(A)] : fold(A,list(A),cons(A),Xs) = aa(list(A),fun(list(A),list(A)),append(A),aa(list(A),list(A),rev(A),Xs)) ).

% fold_Cons_rev
tff(fact_7945_rev__conv__fold,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),rev(A),Xs) = aa(list(A),list(A),fold(A,list(A),cons(A),Xs),nil(A)) ).

% rev_conv_fold
tff(fact_7946_foldr__conv__fold,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),Xs: list(B)] : foldr(B,A,F3,Xs) = fold(B,A,F3,aa(list(B),list(B),rev(B),Xs)) ).

% foldr_conv_fold
tff(fact_7947_fold__rev,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F3: fun(A,fun(B,B))] :
      ( ! [X3: A,Y4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Y4),aa(list(A),set(A),set2(A),Xs)))
           => ( aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,Y4)),aa(A,fun(B,B),F3,X3)) = aa(fun(B,B),fun(B,B),comp(B,B,B,aa(A,fun(B,B),F3,X3)),aa(A,fun(B,B),F3,Y4)) ) ) )
     => ( fold(A,B,F3,aa(list(A),list(A),rev(A),Xs)) = fold(A,B,F3,Xs) ) ) ).

% fold_rev
tff(fact_7948_fold__map,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: fun(B,fun(A,A)),F3: fun(C,B),Xs: list(C)] : fold(B,A,G,aa(list(C),list(B),map(C,B,F3),Xs)) = fold(C,A,aa(fun(C,B),fun(C,fun(A,A)),comp(B,fun(A,A),C,G),F3),Xs) ).

% fold_map
tff(fact_7949_fold__commute,axiom,
    ! [A: $tType,C: $tType,B: $tType,Xs: list(A),H: fun(B,C),G: fun(A,fun(B,B)),F3: fun(A,fun(C,C))] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),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,X3)) = aa(fun(B,C),fun(B,C),comp(C,C,B,aa(A,fun(C,C),F3,X3)),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,F3,Xs)),H) ) ) ).

% fold_commute
tff(fact_7950_fold__commute__apply,axiom,
    ! [A: $tType,C: $tType,B: $tType,Xs: list(A),H: fun(B,C),G: fun(A,fun(B,B)),F3: fun(A,fun(C,C)),S: B] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),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,X3)) = aa(fun(B,C),fun(B,C),comp(C,C,B,aa(A,fun(C,C),F3,X3)),H) ) )
     => ( aa(B,C,H,aa(B,B,fold(A,B,G,Xs),S)) = aa(C,C,fold(A,C,F3,Xs),aa(B,C,H,S)) ) ) ).

% fold_commute_apply
tff(fact_7951_fold__Cons,axiom,
    ! [B: $tType,A: $tType,F3: fun(A,fun(B,B)),X: A,Xs: list(A)] : fold(A,B,F3,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(fun(B,B),fun(B,B),comp(B,B,B,fold(A,B,F3,Xs)),aa(A,fun(B,B),F3,X)) ).

% fold_Cons
tff(fact_7952_fold__append__concat__rev,axiom,
    ! [A: $tType,Xss: list(list(A))] : fold(list(A),list(A),append(A),Xss) = aa(list(A),fun(list(A),list(A)),append(A),concat(A,aa(list(list(A)),list(list(A)),rev(list(A)),Xss))) ).

% fold_append_concat_rev
tff(fact_7953_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_7954_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_7955_Inf__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [X: A,Xs: list(A)] : aa(set(A),A,lattic7752659483105999362nf_fin(A),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) = aa(A,A,fold(A,A,inf_inf(A),Xs),X) ) ).

% Inf_fin.set_eq_fold
tff(fact_7956_Sup__fin_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [X: A,Xs: list(A)] : aa(set(A),A,lattic5882676163264333800up_fin(A),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) = aa(A,A,fold(A,A,sup_sup(A),Xs),X) ) ).

% Sup_fin.set_eq_fold
tff(fact_7957_Max_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] : aa(set(A),A,lattic643756798349783984er_Max(A),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) = aa(A,A,fold(A,A,ord_max(A),Xs),X) ) ).

% Max.set_eq_fold
tff(fact_7958_Min_Oset__eq__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X: A,Xs: list(A)] : aa(set(A),A,lattic643756798350308766er_Min(A),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs))) = aa(A,A,fold(A,A,ord_min(A),Xs),X) ) ).

% Min.set_eq_fold
tff(fact_7959_comp__fun__idem__on_Ofold__set__fold,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xs: list(A),Y: B] :
      ( finite673082921795544331dem_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),S2))
       => ( finite_fold(A,B,F3,Y,aa(list(A),set(A),set2(A),Xs)) = aa(B,B,fold(A,B,F3,Xs),Y) ) ) ) ).

% comp_fun_idem_on.fold_set_fold
tff(fact_7960_comp__fun__commute__on_Ofold__set__fold__remdups,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(A,fun(B,B)),Xs: list(A),Y: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),S2))
       => ( finite_fold(A,B,F3,Y,aa(list(A),set(A),set2(A),Xs)) = aa(B,B,fold(A,B,F3,remdups(A,Xs)),Y) ) ) ) ).

% comp_fun_commute_on.fold_set_fold_remdups
tff(fact_7961_SUP__set__fold,axiom,
    ! [B: $tType,A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [F3: fun(B,A),Xs: list(B)] : aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),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)),F3),Xs),bot_bot(A)) ) ).

% SUP_set_fold
tff(fact_7962_Gcd__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( ( gcd_Gcd(A,A3) = zero_zero(A) )
        <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),zero_zero(A)),bot_bot(set(A))))) ) ) ).

% Gcd_0_iff
tff(fact_7963_finite__sequence__to__countable__set,axiom,
    ! [A: $tType,X6: set(A)] :
      ( countable_countable(A,X6)
     => ~ ! [F7: fun(nat,set(A))] :
            ( ! [I: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),F7,I)),X6))
           => ( ! [I: nat] : pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),F7,I)),aa(nat,set(A),F7,aa(nat,nat,suc,I))))
             => ( ! [I: nat] : pp(aa(set(A),bool,finite_finite(A),aa(nat,set(A),F7,I)))
               => ( aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),F7),top_top(set(nat)))) != X6 ) ) ) ) ) ).

% finite_sequence_to_countable_set
tff(fact_7964_Gcd__UNIV,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ( gcd_Gcd(A,top_top(set(A))) = one_one(A) ) ) ).

% Gcd_UNIV
tff(fact_7965_countable__image__eq__inj,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),S2: set(B)] :
      ( countable_countable(A,aa(set(B),set(A),image(B,A,F3),S2))
    <=> ? [T9: set(B)] :
          ( countable_countable(B,T9)
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T9),S2))
          & ( aa(set(B),set(A),image(B,A,F3),S2) = aa(set(B),set(A),image(B,A,F3),T9) )
          & inj_on(B,A,F3,T9) ) ) ).

% countable_image_eq_inj
tff(fact_7966_ex__countable__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),S2: set(B),P: fun(set(A),bool)] :
      ( ? [T9: set(A)] :
          ( countable_countable(A,T9)
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T9),aa(set(B),set(A),image(B,A,F3),S2)))
          & pp(aa(set(A),bool,P,T9)) )
    <=> ? [T9: set(B)] :
          ( countable_countable(B,T9)
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T9),S2))
          & inj_on(B,A,F3,T9)
          & pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F3),T9))) ) ) ).

% ex_countable_subset_image_inj
tff(fact_7967_all__countable__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),S2: set(B),P: fun(set(A),bool)] :
      ( ! [T9: set(A)] :
          ( ( countable_countable(A,T9)
            & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T9),aa(set(B),set(A),image(B,A,F3),S2))) )
         => pp(aa(set(A),bool,P,T9)) )
    <=> ! [T9: set(B)] :
          ( ( countable_countable(B,T9)
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T9),S2))
            & inj_on(B,A,F3,T9) )
         => pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F3),T9))) ) ) ).

% all_countable_subset_image_inj
tff(fact_7968_countable__image__eq,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),S2: set(B)] :
      ( countable_countable(A,aa(set(B),set(A),image(B,A,F3),S2))
    <=> ? [T9: set(B)] :
          ( countable_countable(B,T9)
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T9),S2))
          & ( aa(set(B),set(A),image(B,A,F3),S2) = aa(set(B),set(A),image(B,A,F3),T9) ) ) ) ).

% countable_image_eq
tff(fact_7969_countable__subset__image,axiom,
    ! [A: $tType,B: $tType,B3: set(A),F3: fun(B,A),A3: set(B)] :
      ( ( countable_countable(A,B3)
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),aa(set(B),set(A),image(B,A,F3),A3))) )
    <=> ? [A18: set(B)] :
          ( countable_countable(B,A18)
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A18),A3))
          & ( B3 = aa(set(B),set(A),image(B,A,F3),A18) ) ) ) ).

% countable_subset_image
tff(fact_7970_ex__countable__subset__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),S2: set(B),P: fun(set(A),bool)] :
      ( ? [T9: set(A)] :
          ( countable_countable(A,T9)
          & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T9),aa(set(B),set(A),image(B,A,F3),S2)))
          & pp(aa(set(A),bool,P,T9)) )
    <=> ? [T9: set(B)] :
          ( countable_countable(B,T9)
          & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T9),S2))
          & pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F3),T9))) ) ) ).

% ex_countable_subset_image
tff(fact_7971_all__countable__subset__image,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),S2: set(B),P: fun(set(A),bool)] :
      ( ! [T9: set(A)] :
          ( ( countable_countable(A,T9)
            & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),T9),aa(set(B),set(A),image(B,A,F3),S2))) )
         => pp(aa(set(A),bool,P,T9)) )
    <=> ! [T9: set(B)] :
          ( ( countable_countable(B,T9)
            & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),T9),S2)) )
         => pp(aa(set(A),bool,P,aa(set(B),set(A),image(B,A,F3),T9))) ) ) ).

% all_countable_subset_image
tff(fact_7972_ccINF__greatest,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),U: A,F3: fun(B,A)] :
          ( countable_countable(B,A3)
         => ( ! [I4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,I4))) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3)))) ) ) ) ).

% ccINF_greatest
tff(fact_7973_le__ccINF__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),U: A,F3: fun(B,A)] :
          ( countable_countable(B,A3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3))))
          <=> ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,X4))) ) ) ) ) ).

% le_ccINF_iff
tff(fact_7974_ccINF__lower2,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),I2: B,F3: fun(B,A),U: A] :
          ( countable_countable(B,A3)
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I2)),U))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3))),U)) ) ) ) ) ).

% ccINF_lower2
tff(fact_7975_ccINF__lower,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),I2: B,F3: fun(B,A)] :
          ( countable_countable(B,A3)
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3))),aa(B,A,F3,I2))) ) ) ) ).

% ccINF_lower
tff(fact_7976_ccINF__mono,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),B3: set(C),F3: fun(B,A),G: fun(C,A)] :
          ( countable_countable(B,A3)
         => ( countable_countable(C,B3)
           => ( ! [M5: C] :
                  ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),M5),B3))
                 => ? [X2: B] :
                      ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X2),A3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X2)),aa(C,A,G,M5))) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image(C,A,G),B3)))) ) ) ) ) ).

% ccINF_mono
tff(fact_7977_ccSUP__mono,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),B3: set(C),F3: fun(B,A),G: fun(C,A)] :
          ( countable_countable(B,A3)
         => ( countable_countable(C,B3)
           => ( ! [N2: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),N2),A3))
                 => ? [X2: C] :
                      ( pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),X2),B3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,N2)),aa(C,A,G,X2))) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,G),B3)))) ) ) ) ) ).

% ccSUP_mono
tff(fact_7978_ccSUP__least,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),F3: fun(B,A),U: A] :
          ( countable_countable(B,A3)
         => ( ! [I4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I4)),U)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3))),U)) ) ) ) ).

% ccSUP_least
tff(fact_7979_ccSUP__upper,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),I2: B,F3: fun(B,A)] :
          ( countable_countable(B,A3)
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,I2)),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3)))) ) ) ) ).

% ccSUP_upper
tff(fact_7980_ccSUP__le__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),F3: fun(B,A),U: A] :
          ( countable_countable(B,A3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3))),U))
          <=> ! [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X4)),U)) ) ) ) ) ).

% ccSUP_le_iff
tff(fact_7981_ccSUP__upper2,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),I2: B,U: A,F3: fun(B,A)] :
          ( countable_countable(B,A3)
         => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),I2),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(B,A,F3,I2)))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3)))) ) ) ) ) ).

% ccSUP_upper2
tff(fact_7982_less__ccSUP__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( counta3822494911875563373attice(A)
        & linorder(A) )
     => ! [A3: set(B),A2: A,F3: fun(B,A)] :
          ( countable_countable(B,A3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3))))
          <=> ? [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(B,A,F3,X4))) ) ) ) ) ).

% less_ccSUP_iff
tff(fact_7983_ccINF__less__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta3822494911875563373attice(A)
        & linorder(A) )
     => ! [A3: set(B),F3: fun(B,A),A2: A] :
          ( countable_countable(B,A3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3))),A2))
          <=> ? [X4: B] :
                ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,F3,X4)),A2)) ) ) ) ) ).

% ccINF_less_iff
tff(fact_7984_ccInf__superset__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( countable_countable(A,A3)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B3))) ) ) ) ).

% ccInf_superset_mono
tff(fact_7985_ccInf__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B3: set(A),A3: set(A)] :
          ( countable_countable(A,B3)
         => ( countable_countable(A,A3)
           => ( ! [B5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B5),B3))
                 => ? [X2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),A3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X2),B5)) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B3))) ) ) ) ) ).

% ccInf_mono
tff(fact_7986_ccInf__lower,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),X: A] :
          ( countable_countable(A,A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),X)) ) ) ) ).

% ccInf_lower
tff(fact_7987_ccInf__lower2,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),U: A,V: A] :
          ( countable_countable(A,A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),U),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),U),V))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),A3)),V)) ) ) ) ) ).

% ccInf_lower2
tff(fact_7988_le__ccInf__iff,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),B2: A] :
          ( countable_countable(A,A3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),aa(set(A),A,complete_Inf_Inf(A),A3)))
          <=> ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),B2),X4)) ) ) ) ) ).

% le_ccInf_iff
tff(fact_7989_ccInf__greatest,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),Z: A] :
          ( countable_countable(A,A3)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),X3)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Z),aa(set(A),A,complete_Inf_Inf(A),A3))) ) ) ) ).

% ccInf_greatest
tff(fact_7990_countable__Collect__finite__subset,axiom,
    ! [A: $tType,T6: set(A)] :
      ( countable_countable(A,T6)
     => countable_countable(set(A),collect(set(A),aTP_Lamp_aai(set(A),fun(set(A),bool),T6))) ) ).

% countable_Collect_finite_subset
tff(fact_7991_infinite__countable__subset_H,axiom,
    ! [A: $tType,X6: set(A)] :
      ( ~ pp(aa(set(A),bool,finite_finite(A),X6))
     => ? [C6: set(A)] :
          ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),C6),X6))
          & countable_countable(A,C6)
          & ~ pp(aa(set(A),bool,finite_finite(A),C6)) ) ) ).

% infinite_countable_subset'
tff(fact_7992_ccSup__upper2,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),U: A,V: A] :
          ( countable_countable(A,A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),U),A3))
           => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V),U))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),V),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ) ) ).

% ccSup_upper2
tff(fact_7993_ccSup__le__iff,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),B2: A] :
          ( countable_countable(A,A3)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),B2))
          <=> ! [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),B2)) ) ) ) ) ).

% ccSup_le_iff
tff(fact_7994_ccSup__upper,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),X: A] :
          ( countable_countable(A,A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ) ).

% ccSup_upper
tff(fact_7995_ccSup__least,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),Z: A] :
          ( countable_countable(A,A3)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),Z)) ) ) ) ).

% ccSup_least
tff(fact_7996_ccSup__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B3: set(A),A3: set(A)] :
          ( countable_countable(A,B3)
         => ( countable_countable(A,A3)
           => ( ! [A5: A] :
                  ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A5),A3))
                 => ? [X2: A] :
                      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X2),B3))
                      & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),A5),X2)) ) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3))) ) ) ) ) ).

% ccSup_mono
tff(fact_7997_Gcd__int__greater__eq__0,axiom,
    ! [K5: set(int)] : pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),gcd_Gcd(int,K5))) ).

% Gcd_int_greater_eq_0
tff(fact_7998_countable__subset,axiom,
    ! [A: $tType,A3: set(A),B3: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
     => ( countable_countable(A,B3)
       => countable_countable(A,A3) ) ) ).

% countable_subset
tff(fact_7999_ccSup__subset__mono,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B3: set(A),A3: set(A)] :
          ( countable_countable(A,B3)
         => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),B3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3))) ) ) ) ).

% ccSup_subset_mono
tff(fact_8000_Gcd__1,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A3: set(A)] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),one_one(A)),A3))
         => ( gcd_Gcd(A,A3) = one_one(A) ) ) ) ).

% Gcd_1
tff(fact_8001_Gcd__nat__eq__one,axiom,
    ! [N3: set(nat)] :
      ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),one_one(nat)),N3))
     => ( gcd_Gcd(nat,N3) = one_one(nat) ) ) ).

% Gcd_nat_eq_one
tff(fact_8002_less__ccSup__iff,axiom,
    ! [A: $tType] :
      ( ( counta3822494911875563373attice(A)
        & linorder(A) )
     => ! [S2: set(A),A2: A] :
          ( countable_countable(A,S2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),aa(set(A),A,complete_Sup_Sup(A),S2)))
          <=> ? [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),A2),X4)) ) ) ) ) ).

% less_ccSup_iff
tff(fact_8003_Gcd__eq__1__I,axiom,
    ! [A: $tType] :
      ( semiring_Gcd(A)
     => ! [A2: A,A3: set(A)] :
          ( pp(dvd_dvd(A,A2,one_one(A)))
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => ( gcd_Gcd(A,A3) = one_one(A) ) ) ) ) ).

% Gcd_eq_1_I
tff(fact_8004_ccInf__less__iff,axiom,
    ! [A: $tType] :
      ( ( counta3822494911875563373attice(A)
        & linorder(A) )
     => ! [S2: set(A),A2: A] :
          ( countable_countable(A,S2)
         => ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(set(A),A,complete_Inf_Inf(A),S2)),A2))
          <=> ? [X4: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),S2))
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),A2)) ) ) ) ) ).

% ccInf_less_iff
tff(fact_8005_ccSup__inter__less__eq,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( countable_countable(A,A3)
         => ( countable_countable(A,B3)
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3))),aa(A,A,aa(A,fun(A,A),inf_inf(A),aa(set(A),A,complete_Sup_Sup(A),A3)),aa(set(A),A,complete_Sup_Sup(A),B3)))) ) ) ) ).

% ccSup_inter_less_eq
tff(fact_8006_less__eq__ccInf__inter,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(A),B3: set(A)] :
          ( countable_countable(A,A3)
         => ( countable_countable(A,B3)
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),sup_sup(A),aa(set(A),A,complete_Inf_Inf(A),A3)),aa(set(A),A,complete_Inf_Inf(A),B3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),inf_inf(set(A)),A3),B3)))) ) ) ) ).

% less_eq_ccInf_inter
tff(fact_8007_ccSUP__subset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [B3: set(B),A3: set(B),F3: fun(B,A),G: fun(B,A)] :
          ( countable_countable(B,B3)
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),B3))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),A3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G,X3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,F3),A3))),aa(set(A),A,complete_Sup_Sup(A),aa(set(B),set(A),image(B,A,G),B3)))) ) ) ) ) ).

% ccSUP_subset_mono
tff(fact_8008_ccINF__superset__mono,axiom,
    ! [A: $tType,B: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [A3: set(B),B3: set(B),F3: fun(B,A),G: fun(B,A)] :
          ( countable_countable(B,A3)
         => ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),B3),A3))
           => ( ! [X3: B] :
                  ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X3),B3))
                 => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,F3,X3)),aa(B,A,G,X3))) )
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,F3),A3))),aa(set(A),A,complete_Inf_Inf(A),aa(set(B),set(A),image(B,A,G),B3)))) ) ) ) ) ).

% ccINF_superset_mono
tff(fact_8009_mono__ccSUP,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F3: fun(A,B),I5: set(C),A3: fun(C,A)] :
          ( order_mono(A,B,F3)
         => ( countable_countable(C,I5)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(C),set(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aeq(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A3)),I5))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),aa(set(C),set(A),image(C,A,A3),I5))))) ) ) ) ).

% mono_ccSUP
tff(fact_8010_mono__ccSup,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F3: fun(A,B),A3: set(A)] :
          ( order_mono(A,B,F3)
         => ( countable_countable(A,A3)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(set(B),B,complete_Sup_Sup(B),aa(set(A),set(B),image(A,B,F3),A3))),aa(A,B,F3,aa(set(A),A,complete_Sup_Sup(A),A3)))) ) ) ) ).

% mono_ccSup
tff(fact_8011_mono__ccINF,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( counta3822494911875563373attice(B)
        & counta4013691401010221786attice(A) )
     => ! [F3: fun(A,B),I5: set(C),A3: fun(C,A)] :
          ( order_mono(A,B,F3)
         => ( countable_countable(C,I5)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),aa(set(C),set(A),image(C,A,A3),I5)))),aa(set(B),B,complete_Inf_Inf(B),aa(set(C),set(B),image(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aeq(fun(A,B),fun(fun(C,A),fun(C,B)),F3),A3)),I5)))) ) ) ) ).

% mono_ccINF
tff(fact_8012_mono__ccInf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [F3: fun(A,B),A3: set(A)] :
          ( order_mono(A,B,F3)
         => ( countable_countable(A,A3)
           => pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,F3,aa(set(A),A,complete_Inf_Inf(A),A3))),aa(set(B),B,complete_Inf_Inf(B),aa(set(A),set(B),image(A,B,F3),A3)))) ) ) ) ).

% mono_ccInf
tff(fact_8013_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_8014_sort__key__conv__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] :
          ( inj_on(B,A,F3,aa(list(B),set(B),set2(B),Xs))
         => ( linorder_sort_key(B,A,F3,Xs) = aa(list(B),list(B),fold(B,list(B),linorder_insort_key(B,A,F3),Xs),nil(B)) ) ) ) ).

% sort_key_conv_fold
tff(fact_8015_minus__set__fold,axiom,
    ! [A: $tType,A3: set(A),Xs: list(A)] : aa(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_8016_sort__upt,axiom,
    ! [M: nat,N: nat] : linorder_sort_key(nat,nat,aTP_Lamp_fu(nat,nat),upt(M,N)) = upt(M,N) ).

% sort_upt
tff(fact_8017_sort__upto,axiom,
    ! [I2: int,J: int] : linorder_sort_key(int,int,aTP_Lamp_hd(int,int),upto(I2,J)) = upto(I2,J) ).

% sort_upto
tff(fact_8018_sort__key__simps_I1_J,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A)] : linorder_sort_key(B,A,F3,nil(B)) = nil(B) ) ).

% sort_key_simps(1)
tff(fact_8019_set__sort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] : aa(list(B),set(B),set2(B),linorder_sort_key(B,A,F3,Xs)) = aa(list(B),set(B),set2(B),Xs) ) ).

% set_sort
tff(fact_8020_length__sort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] : aa(list(B),nat,size_size(list(B)),linorder_sort_key(B,A,F3,Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ) ).

% length_sort
tff(fact_8021_distinct__sort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] :
          ( distinct(B,linorder_sort_key(B,A,F3,Xs))
        <=> distinct(B,Xs) ) ) ).

% distinct_sort
tff(fact_8022_sort__key__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),X: B,Xs: list(B)] : linorder_sort_key(B,A,F3,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),X),Xs)) = aa(list(B),list(B),aa(B,fun(list(B),list(B)),linorder_insort_key(B,A,F3),X),linorder_sort_key(B,A,F3,Xs)) ) ).

% sort_key_simps(2)
tff(fact_8023_sort__key__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [C2: B,Xs: list(A)] : linorder_sort_key(A,B,aTP_Lamp_aer(B,fun(A,B),C2),Xs) = Xs ) ).

% sort_key_const
tff(fact_8024_filter__sort,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [P: fun(B,bool),F3: fun(B,A),Xs: list(B)] : aa(list(B),list(B),filter2(B,P),linorder_sort_key(B,A,F3,Xs)) = linorder_sort_key(B,A,F3,aa(list(B),list(B),filter2(B,P),Xs)) ) ).

% filter_sort
tff(fact_8025_sort__key__stable,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F3: fun(A,B),K: B,Xs: list(A)] : aa(list(A),list(A),filter2(A,aa(B,fun(A,bool),aTP_Lamp_aes(fun(A,B),fun(B,fun(A,bool)),F3),K)),linorder_sort_key(A,B,F3,Xs)) = aa(list(A),list(A),filter2(A,aa(B,fun(A,bool),aTP_Lamp_aes(fun(A,B),fun(B,fun(A,bool)),F3),K)),Xs) ) ).

% sort_key_stable
tff(fact_8026_sorted__sort__id,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( sorted_wrt(A,ord_less_eq(A),Xs)
         => ( linorder_sort_key(A,A,aTP_Lamp_ma(A,A),Xs) = Xs ) ) ) ).

% sorted_sort_id
tff(fact_8027_sorted__sort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : sorted_wrt(A,ord_less_eq(A),linorder_sort_key(A,A,aTP_Lamp_ma(A,A),Xs)) ) ).

% sorted_sort
tff(fact_8028_sorted__sort__key,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] : sorted_wrt(A,ord_less_eq(A),aa(list(B),list(A),map(B,A,F3),linorder_sort_key(B,A,F3,Xs))) ) ).

% sorted_sort_key
tff(fact_8029_remove__code_I1_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),remove(A),X),aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),removeAll(A,X),Xs)) ).

% remove_code(1)
tff(fact_8030_sorted__list__of__set__sort__remdups,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(set(A),list(A),linord4507533701916653071of_set(A),aa(list(A),set(A),set2(A),Xs)) = linorder_sort_key(A,A,aTP_Lamp_ma(A,A),remdups(A,Xs)) ) ).

% sorted_list_of_set_sort_remdups
tff(fact_8031_sort__conv__fold,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : linorder_sort_key(A,A,aTP_Lamp_ma(A,A),Xs) = aa(list(A),list(A),fold(A,list(A),linorder_insort_key(A,A,aTP_Lamp_ma(A,A)),Xs),nil(A)) ) ).

% sort_conv_fold
tff(fact_8032_sort__key__def,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [F3: fun(B,A),Xs: list(B)] : linorder_sort_key(B,A,F3,Xs) = aa(list(B),list(B),foldr(B,list(B),linorder_insort_key(B,A,F3),Xs),nil(B)) ) ).

% sort_key_def
tff(fact_8033_Bleast__code,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,bool)] : bleast(A,aa(list(A),set(A),set2(A),Xs),P) = aa(list(A),A,case_list(A,A,abort_Bleast(A,aa(list(A),set(A),set2(A),Xs),P),aTP_Lamp_aet(A,fun(list(A),A))),aa(list(A),list(A),filter2(A,P),linorder_sort_key(A,A,aTP_Lamp_ma(A,A),Xs))) ) ).

% Bleast_code
tff(fact_8034_inter__coset__fold,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),coset(A,Xs)) = aa(set(A),set(A),fold(A,set(A),remove(A),Xs),A3) ).

% inter_coset_fold
tff(fact_8035_UNIV__coset,axiom,
    ! [A: $tType] : top_top(set(A)) = coset(A,nil(A)) ).

% UNIV_coset
tff(fact_8036_subset__code_I2_J,axiom,
    ! [B: $tType,A3: set(B),Ys: list(B)] :
      ( pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),A3),coset(B,Ys)))
    <=> ! [X4: B] :
          ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),aa(list(B),set(B),set2(B),Ys)))
         => ~ pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),A3)) ) ) ).

% subset_code(2)
tff(fact_8037_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_8038_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_8039_insert__code_I2_J,axiom,
    ! [A: $tType,X: A,Xs: list(A)] : aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),coset(A,Xs)) = coset(A,aa(list(A),list(A),removeAll(A,X),Xs)) ).

% insert_code(2)
tff(fact_8040_union__coset__filter,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)),coset(A,Xs)),A3) = coset(A,aa(list(A),list(A),filter2(A,aTP_Lamp_aeu(set(A),fun(A,bool),A3)),Xs)) ).

% union_coset_filter
tff(fact_8041_subset__code_I3_J,axiom,
    ! [C: $tType] : ~ pp(aa(set(C),bool,aa(set(C),fun(set(C),bool),ord_less_eq(set(C)),coset(C,nil(C))),aa(list(C),set(C),set2(C),nil(C)))) ).

% subset_code(3)
tff(fact_8042_minus__coset__filter,axiom,
    ! [A: $tType,A3: set(A),Xs: list(A)] : aa(set(A),set(A),minus_minus(set(A),A3),coset(A,Xs)) = aa(list(A),set(A),set2(A),aa(list(A),list(A),filter2(A,aTP_Lamp_a(set(A),fun(A,bool),A3)),Xs)) ).

% minus_coset_filter
tff(fact_8043_sum__mult__sum__if__inj,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [F3: fun(A,B),G: fun(C,B),A3: set(A),B3: set(C)] :
          ( inj_on(product_prod(A,C),B,product_case_prod(A,C,B,aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_aev(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),F3),G)),product_Sigma(A,C,A3,aTP_Lamp_ads(set(C),fun(A,set(C)),B3)))
         => ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F3),A3)),aa(set(C),B,groups7311177749621191930dd_sum(C,B,G),B3)) = aa(set(B),B,groups7311177749621191930dd_sum(B,B,id(B)),collect(B,aa(set(C),fun(B,bool),aa(set(A),fun(set(C),fun(B,bool)),aa(fun(C,B),fun(set(A),fun(set(C),fun(B,bool))),aTP_Lamp_aew(fun(A,B),fun(fun(C,B),fun(set(A),fun(set(C),fun(B,bool)))),F3),G),A3),B3))) ) ) ) ).

% sum_mult_sum_if_inj
tff(fact_8044_butlast__power,axiom,
    ! [A: $tType,N: 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))),N),butlast(A)),Xs) = take(A,aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),N),Xs) ).

% butlast_power
tff(fact_8045_of__nat__eq__id,axiom,
    semiring_1_of_nat(nat) = id(nat) ).

% of_nat_eq_id
tff(fact_8046_list_Omap__id0,axiom,
    ! [A: $tType] : map(A,A,id(A)) = id(list(A)) ).

% list.map_id0
tff(fact_8047_id__funpow,axiom,
    ! [A: $tType,N: nat] : aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N),id(A)) = id(A) ).

% id_funpow
tff(fact_8048_rotate0,axiom,
    ! [A: $tType] : rotate(A,zero_zero(nat)) = id(list(A)) ).

% rotate0
tff(fact_8049_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_8050_drop__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se4197421643247451524op_bit(A,zero_zero(nat)) = id(A) ) ) ).

% drop_bit_0
tff(fact_8051_butlast__rev,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),butlast(A),aa(list(A),list(A),rev(A),Xs)) = aa(list(A),list(A),rev(A),aa(list(A),list(A),tl(A),Xs)) ).

% butlast_rev
tff(fact_8052_butlast__snoc,axiom,
    ! [A: $tType,Xs: list(A),X: A] : aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A)))) = Xs ).

% butlast_snoc
tff(fact_8053_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,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ).

% length_butlast
tff(fact_8054_foldr__Nil,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,B))] : foldr(A,B,F3,nil(A)) = id(B) ).

% foldr_Nil
tff(fact_8055_foldr__filter,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),P: fun(B,bool),Xs: list(B)] : foldr(B,A,F3,aa(list(B),list(B),filter2(B,P),Xs)) = foldr(B,A,aa(fun(B,bool),fun(B,fun(A,A)),aTP_Lamp_aex(fun(B,fun(A,A)),fun(fun(B,bool),fun(B,fun(A,A))),F3),P),Xs) ).

% foldr_filter
tff(fact_8056_fold__id,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),F3: fun(A,fun(B,B))] :
      ( ! [X3: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),aa(list(A),set(A),set2(A),Xs)))
         => ( aa(A,fun(B,B),F3,X3) = id(B) ) )
     => ( fold(A,B,F3,Xs) = id(B) ) ) ).

% fold_id
tff(fact_8057_fold__Nil,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,fun(B,B))] : fold(A,B,F3,nil(A)) = id(B) ).

% fold_Nil
tff(fact_8058_fold__filter,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,fun(A,A)),P: fun(B,bool),Xs: list(B)] : fold(B,A,F3,aa(list(B),list(B),filter2(B,P),Xs)) = fold(B,A,aa(fun(B,bool),fun(B,fun(A,A)),aTP_Lamp_aex(fun(B,fun(A,A)),fun(fun(B,bool),fun(B,fun(A,A))),F3),P),Xs) ).

% fold_filter
tff(fact_8059_List_Omap_Oidentity,axiom,
    ! [A: $tType] : map(A,A,aTP_Lamp_aam(A,A)) = id(list(A)) ).

% List.map.identity
tff(fact_8060_map__butlast,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,A),Xs: list(B)] : aa(list(B),list(A),map(B,A,F3),aa(list(B),list(B),butlast(B),Xs)) = aa(list(A),list(A),butlast(A),aa(list(B),list(A),map(B,A,F3),Xs)) ).

% map_butlast
tff(fact_8061_list_Omap__id,axiom,
    ! [A: $tType,T2: list(A)] : aa(list(A),list(A),map(A,A,id(A)),T2) = T2 ).

% list.map_id
tff(fact_8062_distinct__butlast,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( distinct(A,Xs)
     => distinct(A,aa(list(A),list(A),butlast(A),Xs)) ) ).

% distinct_butlast
tff(fact_8063_butlast__tl,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),butlast(A),aa(list(A),list(A),tl(A),Xs)) = aa(list(A),list(A),tl(A),aa(list(A),list(A),butlast(A),Xs)) ).

% butlast_tl
tff(fact_8064_drop__butlast,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : drop(A,N,aa(list(A),list(A),butlast(A),Xs)) = aa(list(A),list(A),butlast(A),drop(A,N,Xs)) ).

% drop_butlast
tff(fact_8065_in__set__butlast__appendI,axiom,
    ! [A: $tType,X: A,Xs: list(A),Ys: list(A)] :
      ( ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),Xs))))
        | pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),Ys)))) )
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys))))) ) ).

% in_set_butlast_appendI
tff(fact_8066_in__set__butlastD,axiom,
    ! [A: $tType,X: A,Xs: list(A)] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),aa(list(A),list(A),butlast(A),Xs))))
     => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),aa(list(A),set(A),set2(A),Xs))) ) ).

% in_set_butlastD
tff(fact_8067_butlast__append,axiom,
    ! [A: $tType,Ys: list(A),Xs: list(A)] :
      ( ( ( Ys = nil(A) )
       => ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),butlast(A),Xs) ) )
      & ( ( Ys != nil(A) )
       => ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),aa(list(A),list(A),butlast(A),Ys)) ) ) ) ).

% butlast_append
tff(fact_8068_butlast_Osimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A),X: A] :
      ( ( ( Xs = nil(A) )
       => ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = nil(A) ) )
      & ( ( Xs != nil(A) )
       => ( aa(list(A),list(A),butlast(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),aa(list(A),list(A),butlast(A),Xs)) ) ) ) ).

% butlast.simps(2)
tff(fact_8069_butlast_Osimps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),list(A),butlast(A),nil(A)) = nil(A) ).

% butlast.simps(1)
tff(fact_8070_funpow__simps__right_I1_J,axiom,
    ! [A: $tType,F3: 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)),F3) = id(A) ).

% funpow_simps_right(1)
tff(fact_8071_sorted__butlast,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] :
          ( ( Xs != nil(A) )
         => ( sorted_wrt(A,ord_less_eq(A),Xs)
           => sorted_wrt(A,ord_less_eq(A),aa(list(A),list(A),butlast(A),Xs)) ) ) ) ).

% sorted_butlast
tff(fact_8072_nth__butlast,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),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)),N) = aa(nat,A,nth(A,Xs),N) ) ) ).

% nth_butlast
tff(fact_8073_take__butlast,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( take(A,N,aa(list(A),list(A),butlast(A),Xs)) = take(A,N,Xs) ) ) ).

% take_butlast
tff(fact_8074_butlast__conv__take,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),list(A),butlast(A),Xs) = take(A,aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)),Xs) ).

% butlast_conv_take
tff(fact_8075_butlast__list__update,axiom,
    ! [A: $tType,K: nat,Xs: list(A),X: A] :
      ( ( ( K = aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) )
       => ( aa(list(A),list(A),butlast(A),list_update(A,Xs,K,X)) = aa(list(A),list(A),butlast(A),Xs) ) )
      & ( ( K != aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) )
       => ( aa(list(A),list(A),butlast(A),list_update(A,Xs,K,X)) = list_update(A,aa(list(A),list(A),butlast(A),Xs),K,X) ) ) ) ).

% butlast_list_update
tff(fact_8076_butlast__take,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),N),aa(list(A),nat,size_size(list(A)),Xs)))
     => ( aa(list(A),list(A),butlast(A),take(A,N,Xs)) = take(A,aa(nat,nat,minus_minus(nat,N),one_one(nat)),Xs) ) ) ).

% butlast_take
tff(fact_8077_Rat_Opositive__def,axiom,
    positive = map_fun(rat,product_prod(int,int),bool,bool,rep_Rat,id(bool),aTP_Lamp_aey(product_prod(int,int),bool)) ).

% Rat.positive_def
tff(fact_8078_rcis__inverse,axiom,
    ! [R3: real,A2: real] : aa(complex,complex,inverse_inverse(complex),rcis(R3,A2)) = rcis(divide_divide(real,one_one(real),R3),aa(real,real,uminus_uminus(real),A2)) ).

% rcis_inverse
tff(fact_8079_cis__rcis__eq,axiom,
    ! [A2: real] : cis(A2) = rcis(one_one(real),A2) ).

% cis_rcis_eq
tff(fact_8080_DeMoivre2,axiom,
    ! [R3: real,A2: real,N: nat] : aa(nat,complex,power_power(complex,rcis(R3,A2)),N) = rcis(aa(nat,real,power_power(real,R3),N),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),A2)) ).

% DeMoivre2
tff(fact_8081_of__rat__def,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ( field_char_0_of_rat(A) = map_fun(rat,product_prod(int,int),A,A,rep_Rat,id(A),aTP_Lamp_aez(product_prod(int,int),A)) ) ) ).

% of_rat_def
tff(fact_8082_power_Opower__eq__if,axiom,
    ! [A: $tType,M: nat,One: A,Times: fun(A,fun(A,A)),P2: A] :
      ( ( ( M = zero_zero(nat) )
       => ( aa(nat,A,aa(A,fun(nat,A),power2(A,One,Times),P2),M) = One ) )
      & ( ( M != zero_zero(nat) )
       => ( aa(nat,A,aa(A,fun(nat,A),power2(A,One,Times),P2),M) = aa(A,A,aa(A,fun(A,A),Times,P2),aa(nat,A,aa(A,fun(nat,A),power2(A,One,Times),P2),aa(nat,nat,minus_minus(nat,M),one_one(nat)))) ) ) ) ).

% power.power_eq_if
tff(fact_8083_of__rat__numeral__eq,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [W: num] : aa(rat,A,field_char_0_of_rat(A),aa(num,rat,numeral_numeral(rat),W)) = aa(num,A,numeral_numeral(A),W) ) ).

% of_rat_numeral_eq
tff(fact_8084_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_8085_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_8086_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_8087_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_8088_of__rat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R3)),zero_zero(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R3),zero_zero(rat))) ) ) ).

% of_rat_less_0_iff
tff(fact_8089_zero__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),zero_zero(A)),aa(rat,A,field_char_0_of_rat(A),R3)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),zero_zero(rat)),R3)) ) ) ).

% zero_less_of_rat_iff
tff(fact_8090_one__less__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),one_one(A)),aa(rat,A,field_char_0_of_rat(A),R3)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),one_one(rat)),R3)) ) ) ).

% one_less_of_rat_iff
tff(fact_8091_of__rat__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R3)),one_one(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R3),one_one(rat))) ) ) ).

% of_rat_less_1_iff
tff(fact_8092_of__rat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R3)),zero_zero(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R3),zero_zero(rat))) ) ) ).

% of_rat_le_0_iff
tff(fact_8093_zero__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),aa(rat,A,field_char_0_of_rat(A),R3)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),zero_zero(rat)),R3)) ) ) ).

% zero_le_of_rat_iff
tff(fact_8094_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_8095_of__rat__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R3)),one_one(A)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R3),one_one(rat))) ) ) ).

% of_rat_le_1_iff
tff(fact_8096_one__le__of__rat__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),one_one(A)),aa(rat,A,field_char_0_of_rat(A),R3)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),one_one(rat)),R3)) ) ) ).

% one_le_of_rat_iff
tff(fact_8097_power_Opower_Opower__Suc,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),A2: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power2(A,One,Times),A2),aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),Times,A2),aa(nat,A,aa(A,fun(nat,A),power2(A,One,Times),A2),N)) ).

% power.power.power_Suc
tff(fact_8098_power_Opower_Opower__0,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A)),A2: A] : aa(nat,A,aa(A,fun(nat,A),power2(A,One,Times),A2),zero_zero(nat)) = One ).

% power.power.power_0
tff(fact_8099_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_8100_of__rat__dense,axiom,
    ! [X: real,Y: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),Y))
     => ? [Q2: rat] :
          ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),X),aa(rat,real,field_char_0_of_rat(real),Q2)))
          & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(rat,real,field_char_0_of_rat(real),Q2)),Y)) ) ) ).

% of_rat_dense
tff(fact_8101_power_Opower_Ocong,axiom,
    ! [A: $tType,One: A,Times: fun(A,fun(A,A))] : power2(A,One,Times) = power2(A,One,Times) ).

% power.power.cong
tff(fact_8102_of__rat__power,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat,N: nat] : aa(rat,A,field_char_0_of_rat(A),aa(nat,rat,power_power(rat,A2),N)) = aa(nat,A,power_power(A,aa(rat,A,field_char_0_of_rat(A),A2)),N) ) ).

% of_rat_power
tff(fact_8103_of__rat__divide,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: rat,B2: rat] : aa(rat,A,field_char_0_of_rat(A),divide_divide(rat,A2,B2)) = divide_divide(A,aa(rat,A,field_char_0_of_rat(A),A2),aa(rat,A,field_char_0_of_rat(A),B2)) ) ).

% of_rat_divide
tff(fact_8104_nonzero__of__rat__divide,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [B2: rat,A2: rat] :
          ( ( B2 != zero_zero(rat) )
         => ( aa(rat,A,field_char_0_of_rat(A),divide_divide(rat,A2,B2)) = divide_divide(A,aa(rat,A,field_char_0_of_rat(A),A2),aa(rat,A,field_char_0_of_rat(A),B2)) ) ) ) ).

% nonzero_of_rat_divide
tff(fact_8105_of__rat__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat,S: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(rat,A,field_char_0_of_rat(A),R3)),aa(rat,A,field_char_0_of_rat(A),S)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),R3),S)) ) ) ).

% of_rat_less
tff(fact_8106_of__rat__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [R3: rat,S: rat] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(rat,A,field_char_0_of_rat(A),R3)),aa(rat,A,field_char_0_of_rat(A),S)))
        <=> pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),R3),S)) ) ) ).

% of_rat_less_eq
tff(fact_8107_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),fract(A2,B2)) = divide_divide(A,ring_1_of_int(A,A2),ring_1_of_int(A,B2)) ) ) ) ).

% of_rat_rat
tff(fact_8108_of__rat_Orep__eq,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [X: rat] : aa(rat,A,field_char_0_of_rat(A),X) = divide_divide(A,ring_1_of_int(A,aa(product_prod(int,int),int,product_fst(int,int),aa(rat,product_prod(int,int),rep_Rat,X))),ring_1_of_int(A,aa(product_prod(int,int),int,product_snd(int,int),aa(rat,product_prod(int,int),rep_Rat,X)))) ) ).

% of_rat.rep_eq
tff(fact_8109_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(A,fun(B,B)),A3: set(A),Z: B,Y: B,A2: A] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),S2))
       => ( finite_fold_graph(A,B,F3,Z,A3,Y)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),A3))
           => ? [Y8: B] :
                ( ( Y = aa(B,B,aa(A,fun(B,B),F3,A2),Y8) )
                & finite_fold_graph(A,B,F3,Z,aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A2),bot_bot(set(A)))),Y8) ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_insertE_aux
tff(fact_8110_list_Oin__rel,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),A2: list(A),B2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,R),A2),B2))
    <=> ? [Z5: list(product_prod(A,B))] :
          ( pp(aa(set(list(product_prod(A,B))),bool,aa(list(product_prod(A,B)),fun(set(list(product_prod(A,B))),bool),member(list(product_prod(A,B))),Z5),collect(list(product_prod(A,B)),aTP_Lamp_afa(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),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_8111_list__all2__Nil,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),nil(A)),Ys))
    <=> ( Ys = nil(B) ) ) ).

% list_all2_Nil
tff(fact_8112_list__all2__Nil2,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),Xs: list(A)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),nil(B)))
    <=> ( Xs = nil(A) ) ) ).

% list_all2_Nil2
tff(fact_8113_list__all2__rev,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),aa(list(A),list(A),rev(A),Xs)),aa(list(B),list(B),rev(B),Ys)))
    <=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys)) ) ).

% list_all2_rev
tff(fact_8114_comp__fun__commute__on_Ofold__graph__determ,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(A,fun(B,B)),A3: set(A),Z: B,X: B,Y: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),S2))
       => ( finite_fold_graph(A,B,F3,Z,A3,X)
         => ( finite_fold_graph(A,B,F3,Z,A3,Y)
           => ( Y = X ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_determ
tff(fact_8115_list_Orel__mono,axiom,
    ! [B: $tType,A: $tType,R: fun(A,fun(B,bool)),Ra: fun(A,fun(B,bool))] :
      ( pp(aa(fun(A,fun(B,bool)),bool,aa(fun(A,fun(B,bool)),fun(fun(A,fun(B,bool)),bool),ord_less_eq(fun(A,fun(B,bool))),R),Ra))
     => pp(aa(fun(list(A),fun(list(B),bool)),bool,aa(fun(list(A),fun(list(B),bool)),fun(fun(list(A),fun(list(B),bool)),bool),ord_less_eq(fun(list(A),fun(list(B),bool))),list_all2(A,B,R)),list_all2(A,B,Ra))) ) ).

% list.rel_mono
tff(fact_8116_list__all2__rev1,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),aa(list(A),list(A),rev(A),Xs)),Ys))
    <=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),aa(list(B),list(B),rev(B),Ys))) ) ).

% list_all2_rev1
tff(fact_8117_list__all2__nthD,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),P2: nat] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),P2),aa(list(A),nat,size_size(list(A)),Xs)))
       => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,Xs),P2)),aa(nat,B,nth(B,Ys),P2))) ) ) ).

% list_all2_nthD
tff(fact_8118_list__all2__nthD2,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),P2: nat] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys))
     => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),P2),aa(list(B),nat,size_size(list(B)),Ys)))
       => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,Xs),P2)),aa(nat,B,nth(B,Ys),P2))) ) ) ).

% list_all2_nthD2
tff(fact_8119_list__all2__all__nthI,axiom,
    ! [A: $tType,B: $tType,A2: list(A),B2: list(B),P: fun(A,fun(B,bool))] :
      ( ( aa(list(A),nat,size_size(list(A)),A2) = aa(list(B),nat,size_size(list(B)),B2) )
     => ( ! [N2: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),N2),aa(list(A),nat,size_size(list(A)),A2)))
           => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,A2),N2)),aa(nat,B,nth(B,B2),N2))) )
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),A2),B2)) ) ) ).

% list_all2_all_nthI
tff(fact_8120_list__all2__conv__all__nth,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Xs)))
           => pp(aa(B,bool,aa(A,fun(B,bool),P,aa(nat,A,nth(A,Xs),I3)),aa(nat,B,nth(B,Ys),I3))) ) ) ) ).

% list_all2_conv_all_nth
tff(fact_8121_list_Octr__transfer_I1_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool))] : pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,R),nil(A)),nil(B))) ).

% list.ctr_transfer(1)
tff(fact_8122_list_Orel__eq,axiom,
    ! [A: $tType] : list_all2(A,A,fequal(A)) = fequal(list(A)) ).

% list.rel_eq
tff(fact_8123_list_Orel__refl,axiom,
    ! [B: $tType,Ra: fun(B,fun(B,bool)),X: list(B)] :
      ( ! [X3: B] : pp(aa(B,bool,aa(B,fun(B,bool),Ra,X3),X3))
     => pp(aa(list(B),bool,aa(list(B),fun(list(B),bool),list_all2(B,B,Ra),X),X)) ) ).

% list.rel_refl
tff(fact_8124_list__all2__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),list_all2(A,A,fequal(A)),Xs),Ys)) ) ).

% list_all2_eq
tff(fact_8125_list__all2__mono,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),Q: fun(A,fun(B,bool))] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys))
     => ( ! [Xs2: A,Ys3: B] :
            ( pp(aa(B,bool,aa(A,fun(B,bool),P,Xs2),Ys3))
           => pp(aa(B,bool,aa(A,fun(B,bool),Q,Xs2),Ys3)) )
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,Q),Xs),Ys)) ) ) ).

% list_all2_mono
tff(fact_8126_list__all2__refl,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
      ( ! [X3: A] : pp(aa(A,bool,aa(A,fun(A,bool),P,X3),X3))
     => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),list_all2(A,A,P),Xs),Xs)) ) ).

% list_all2_refl
tff(fact_8127_list__all2__trans,axiom,
    ! [B: $tType,A: $tType,C: $tType,P1: fun(A,fun(B,bool)),P22: fun(B,fun(C,bool)),P32: fun(A,fun(C,bool)),As3: list(A),Bs: list(B),Cs: list(C)] :
      ( ! [A5: A,B5: B,C4: C] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),P1,A5),B5))
         => ( pp(aa(C,bool,aa(B,fun(C,bool),P22,B5),C4))
           => pp(aa(C,bool,aa(A,fun(C,bool),P32,A5),C4)) ) )
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P1),As3),Bs))
       => ( pp(aa(list(C),bool,aa(list(B),fun(list(C),bool),list_all2(B,C,P22),Bs),Cs))
         => pp(aa(list(C),bool,aa(list(A),fun(list(C),bool),list_all2(A,C,P32),As3),Cs)) ) ) ) ).

% list_all2_trans
tff(fact_8128_list__all2__antisym,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),Q: fun(A,fun(A,bool)),Xs: list(A),Ys: list(A)] :
      ( ! [X3: A,Y4: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),P,X3),Y4))
         => ( pp(aa(A,bool,aa(A,fun(A,bool),Q,Y4),X3))
           => ( X3 = Y4 ) ) )
     => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),list_all2(A,A,P),Xs),Ys))
       => ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),list_all2(A,A,Q),Ys),Xs))
         => ( Xs = Ys ) ) ) ) ).

% list_all2_antisym
tff(fact_8129_list__all2__appendI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),A2: list(A),B2: list(B),C2: list(A),D2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),A2),B2))
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),C2),D2))
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),A2),C2)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),B2),D2))) ) ) ).

% list_all2_appendI
tff(fact_8130_list_Orel__inject_I2_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),X21: A,X222: list(A),Y21: B,Y22: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,R),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y21),Y22)))
    <=> ( pp(aa(B,bool,aa(A,fun(B,bool),R,X21),Y21))
        & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,R),X222),Y22)) ) ) ).

% list.rel_inject(2)
tff(fact_8131_list_Orel__intros_I2_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),X21: A,Y21: B,X222: list(A),Y22: list(B)] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),R,X21),Y21))
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,R),X222),Y22))
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,R),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X222)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y21),Y22))) ) ) ).

% list.rel_intros(2)
tff(fact_8132_list__all2__Cons,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),X: A,Xs: list(A),Y: B,Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)))
    <=> ( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y))
        & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys)) ) ) ).

% list_all2_Cons
tff(fact_8133_list__all2__Cons1,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),X: A,Xs: list(A),Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs)),Ys))
    <=> ? [Z5: B,Zs3: list(B)] :
          ( ( Ys = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Z5),Zs3) )
          & pp(aa(B,bool,aa(A,fun(B,bool),P,X),Z5))
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Zs3)) ) ) ).

% list_all2_Cons1
tff(fact_8134_list__all2__Cons2,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Y: B,Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y),Ys)))
    <=> ? [Z5: A,Zs3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z5),Zs3) )
          & pp(aa(B,bool,aa(A,fun(B,bool),P,Z5),Y))
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Zs3),Ys)) ) ) ).

% list_all2_Cons2
tff(fact_8135_list__all2__update__cong,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),X: A,Y: B,I2: nat] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys))
     => ( pp(aa(B,bool,aa(A,fun(B,bool),P,X),Y))
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),list_update(A,Xs,I2,X)),list_update(B,Ys,I2,Y))) ) ) ).

% list_all2_update_cong
tff(fact_8136_list_Orel__distinct_I2_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),Y21: A,Y22: list(A)] : ~ pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,R),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y21),Y22)),nil(B))) ).

% list.rel_distinct(2)
tff(fact_8137_list_Orel__distinct_I1_J,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),Y21: B,Y22: list(B)] : ~ pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,R),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y21),Y22))) ).

% list.rel_distinct(1)
tff(fact_8138_list_Orel__cases,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),A2: list(A),B2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,R),A2),B2))
     => ( ( ( A2 = nil(A) )
         => ( B2 != nil(B) ) )
       => ~ ! [X1: A,X23: list(A)] :
              ( ( A2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),X23) )
             => ! [Y1: B,Y23: list(B)] :
                  ( ( B2 = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y1),Y23) )
                 => ( pp(aa(B,bool,aa(A,fun(B,bool),R,X1),Y1))
                   => ~ pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,R),X23),Y23)) ) ) ) ) ) ).

% list.rel_cases
tff(fact_8139_list_Orel__induct,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),X: list(A),Y: list(B),Q: fun(list(A),fun(list(B),bool))] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,R),X),Y))
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,nil(A)),nil(B)))
       => ( ! [A21: A,A222: list(A),B21: B,B222: list(B)] :
              ( pp(aa(B,bool,aa(A,fun(B,bool),R,A21),B21))
             => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,A222),B222))
               => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A21),A222)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B21),B222))) ) )
         => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),Q,X),Y)) ) ) ) ).

% list.rel_induct
tff(fact_8140_list__all2__induct,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),R: fun(list(A),fun(list(B),bool))] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys))
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),R,nil(A)),nil(B)))
       => ( ! [X3: A,Xs2: list(A),Y4: B,Ys3: list(B)] :
              ( pp(aa(B,bool,aa(A,fun(B,bool),P,X3),Y4))
             => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs2),Ys3))
               => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),R,Xs2),Ys3))
                 => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),R,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X3),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y4),Ys3))) ) ) )
         => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),R,Xs),Ys)) ) ) ) ).

% list_all2_induct
tff(fact_8141_list_Orel__sel,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),A2: list(A),B2: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,R),A2),B2))
    <=> ( ( ( A2 = nil(A) )
        <=> ( B2 = nil(B) ) )
        & ( ( A2 != nil(A) )
         => ( ( B2 != nil(B) )
           => ( pp(aa(B,bool,aa(A,fun(B,bool),R,aa(list(A),A,hd(A),A2)),aa(list(B),B,hd(B),B2)))
              & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,R),aa(list(A),list(A),tl(A),A2)),aa(list(B),list(B),tl(B),B2))) ) ) ) ) ) ).

% list.rel_sel
tff(fact_8142_list__all2__same,axiom,
    ! [A: $tType,P: fun(A,fun(A,bool)),Xs: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),list_all2(A,A,P),Xs),Xs))
    <=> ! [X4: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),aa(list(A),set(A),set2(A),Xs)))
         => pp(aa(A,bool,aa(A,fun(A,bool),P,X4),X4)) ) ) ).

% list_all2_same
tff(fact_8143_list_Orel__refl__strong,axiom,
    ! [A: $tType,X: list(A),Ra: fun(A,fun(A,bool))] :
      ( ! [Z2: A] :
          ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z2),aa(list(A),set(A),set2(A),X)))
         => pp(aa(A,bool,aa(A,fun(A,bool),Ra,Z2),Z2)) )
     => pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),list_all2(A,A,Ra),X),X)) ) ).

% list.rel_refl_strong
tff(fact_8144_list_Orel__mono__strong,axiom,
    ! [A: $tType,B: $tType,R: fun(A,fun(B,bool)),X: list(A),Y: list(B),Ra: fun(A,fun(B,bool))] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,R),X),Y))
     => ( ! [Z2: A,Yb: B] :
            ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z2),aa(list(A),set(A),set2(A),X)))
           => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Yb),aa(list(B),set(B),set2(B),Y)))
             => ( pp(aa(B,bool,aa(A,fun(B,bool),R,Z2),Yb))
               => pp(aa(B,bool,aa(A,fun(B,bool),Ra,Z2),Yb)) ) ) )
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,Ra),X),Y)) ) ) ).

% list.rel_mono_strong
tff(fact_8145_list_Orel__cong,axiom,
    ! [A: $tType,B: $tType,X: list(A),Ya: list(A),Y: list(B),Xa: list(B),R: fun(A,fun(B,bool)),Ra: fun(A,fun(B,bool))] :
      ( ( X = Ya )
     => ( ( Y = Xa )
       => ( ! [Z2: A,Yb: B] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Z2),aa(list(A),set(A),set2(A),Ya)))
             => ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Yb),aa(list(B),set(B),set2(B),Xa)))
               => ( pp(aa(B,bool,aa(A,fun(B,bool),R,Z2),Yb))
                <=> pp(aa(B,bool,aa(A,fun(B,bool),Ra,Z2),Yb)) ) ) )
         => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,R),X),Y))
          <=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,Ra),Ya),Xa)) ) ) ) ) ).

% list.rel_cong
tff(fact_8146_list__all2__append2,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),Zs: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Zs)))
    <=> ? [Us2: list(A),Vs3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),Vs3) )
          & ( aa(list(A),nat,size_size(list(A)),Us2) = aa(list(B),nat,size_size(list(B)),Ys) )
          & ( aa(list(A),nat,size_size(list(A)),Vs3) = aa(list(B),nat,size_size(list(B)),Zs) )
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Us2),Ys))
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Vs3),Zs)) ) ) ).

% list_all2_append2
tff(fact_8147_list__all2__append1,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(A),Zs: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Ys)),Zs))
    <=> ? [Us2: list(B),Vs3: list(B)] :
          ( ( Zs = aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Us2),Vs3) )
          & ( aa(list(B),nat,size_size(list(B)),Us2) = aa(list(A),nat,size_size(list(A)),Xs) )
          & ( aa(list(B),nat,size_size(list(B)),Vs3) = aa(list(A),nat,size_size(list(A)),Ys) )
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Us2))
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Ys),Vs3)) ) ) ).

% list_all2_append1
tff(fact_8148_list__all2__append,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(A,fun(B,bool)),Us: list(A),Vs: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xs),Us)),aa(list(B),list(B),aa(list(B),fun(list(B),list(B)),append(B),Ys),Vs)))
      <=> ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys))
          & pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Us),Vs)) ) ) ) ).

% list_all2_append
tff(fact_8149_list__all2__dropI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),N: nat] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys))
     => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),drop(A,N,Xs)),drop(B,N,Ys))) ) ).

% list_all2_dropI
tff(fact_8150_list__all2__takeI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B),N: nat] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys))
     => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),take(A,N,Xs)),take(B,N,Ys))) ) ).

% list_all2_takeI
tff(fact_8151_list__all2__lengthD,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) ) ) ).

% list_all2_lengthD
tff(fact_8152_list_Orel__map_I2_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sa: fun(A,fun(C,bool)),X: list(A),G: fun(B,C),Y: list(B)] :
      ( pp(aa(list(C),bool,aa(list(A),fun(list(C),bool),list_all2(A,C,Sa),X),aa(list(B),list(C),map(B,C,G),Y)))
    <=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_afb(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Sa),G)),X),Y)) ) ).

% list.rel_map(2)
tff(fact_8153_list_Orel__map_I1_J,axiom,
    ! [A: $tType,C: $tType,B: $tType,Sb: fun(C,fun(B,bool)),I2: fun(A,C),X: list(A),Y: list(B)] :
      ( pp(aa(list(B),bool,aa(list(C),fun(list(B),bool),list_all2(C,B,Sb),aa(list(A),list(C),map(A,C,I2),X)),Y))
    <=> pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_afc(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Sb),I2)),X),Y)) ) ).

% list.rel_map(1)
tff(fact_8154_list__all2__map1,axiom,
    ! [C: $tType,A: $tType,B: $tType,P: fun(A,fun(B,bool)),F3: fun(C,A),As3: list(C),Bs: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),aa(list(C),list(A),map(C,A,F3),As3)),Bs))
    <=> pp(aa(list(B),bool,aa(list(C),fun(list(B),bool),list_all2(C,B,aa(fun(C,A),fun(C,fun(B,bool)),aTP_Lamp_afd(fun(A,fun(B,bool)),fun(fun(C,A),fun(C,fun(B,bool))),P),F3)),As3),Bs)) ) ).

% list_all2_map1
tff(fact_8155_list__all2__map2,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: fun(A,fun(B,bool)),As3: list(A),F3: fun(C,B),Bs: list(C)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),As3),aa(list(C),list(B),map(C,B,F3),Bs)))
    <=> pp(aa(list(C),bool,aa(list(A),fun(list(C),bool),list_all2(A,C,aa(fun(C,B),fun(A,fun(C,bool)),aTP_Lamp_afe(fun(A,fun(B,bool)),fun(fun(C,B),fun(A,fun(C,bool))),P),F3)),As3),Bs)) ) ).

% list_all2_map2
tff(fact_8156_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_afg(list(list(A)),fun(list(A),bool),Xss)) ).

% product_lists_set
tff(fact_8157_comp__fun__commute__on_Ofold__graph__insertE,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(A,fun(B,B)),X: A,A3: set(A),Z: B,V: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3)),S2))
       => ( finite_fold_graph(A,B,F3,Z,aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),A3),V)
         => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => ~ ! [Y4: B] :
                  ( ( V = aa(B,B,aa(A,fun(B,B),F3,X),Y4) )
                 => ~ finite_fold_graph(A,B,F3,Z,A3,Y4) ) ) ) ) ) ).

% comp_fun_commute_on.fold_graph_insertE
tff(fact_8158_list__all2I,axiom,
    ! [A: $tType,B: $tType,A2: list(A),B2: list(B),P: fun(A,fun(B,bool))] :
      ( ! [X3: product_prod(A,B)] :
          ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),X3),aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),zip(A,B,A2,B2))))
         => pp(aa(product_prod(A,B),bool,product_case_prod(A,B,bool,P),X3)) )
     => ( ( aa(list(A),nat,size_size(list(A)),A2) = aa(list(B),nat,size_size(list(B)),B2) )
       => pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),A2),B2)) ) ) ).

% list_all2I
tff(fact_8159_comp__fun__commute__on_Ofold__equality,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F3: fun(A,fun(B,B)),A3: set(A),Z: B,Y: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),S2))
       => ( finite_fold_graph(A,B,F3,Z,A3,Y)
         => ( finite_fold(A,B,F3,Z,A3) = Y ) ) ) ) ).

% comp_fun_commute_on.fold_equality
tff(fact_8160_comp__fun__commute__on_Ofold__graph__fold,axiom,
    ! [B: $tType,A: $tType,S2: set(A),F3: fun(A,fun(B,B)),A3: set(A),Z: B] :
      ( finite4664212375090638736ute_on(A,B,S2,F3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),A3),S2))
       => ( pp(aa(set(A),bool,finite_finite(A),A3))
         => finite_fold_graph(A,B,F3,Z,A3,finite_fold(A,B,F3,Z,A3)) ) ) ) ).

% comp_fun_commute_on.fold_graph_fold
tff(fact_8161_list__all2__iff,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,bool)),Xs: list(A),Ys: list(B)] :
      ( pp(aa(list(B),bool,aa(list(A),fun(list(B),bool),list_all2(A,B,P),Xs),Ys))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
        & ! [X4: product_prod(A,B)] :
            ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),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,Ys))))
           => pp(aa(product_prod(A,B),bool,product_case_prod(A,B,bool,P),X4)) ) ) ) ).

% list_all2_iff
tff(fact_8162_iteratesp_Omono,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [F3: fun(A,A)] : order_mono(fun(A,bool),fun(A,bool),aTP_Lamp_afh(fun(A,A),fun(fun(A,bool),fun(A,bool)),F3)) ) ).

% iteratesp.mono
tff(fact_8163_dom__map__upds,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),Xs: list(A),Ys: list(B)] : dom(A,B,map_upds(A,B,M,Xs,Ys)) = aa(set(A),set(A),aa(set(A),fun(set(A),set(A)),sup_sup(set(A)),aa(list(A),set(A),set2(A),take(A,aa(list(B),nat,size_size(list(B)),Ys),Xs))),dom(A,B,M)) ).

% dom_map_upds
tff(fact_8164_dom__eq__empty__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B))] :
      ( ( dom(A,B,F3) = bot_bot(set(A)) )
    <=> ! [X4: A] : aa(A,option(B),F3,X4) = none(B) ) ).

% dom_eq_empty_conv
tff(fact_8165_fun__upd__None__if__notin__dom,axiom,
    ! [B: $tType,A: $tType,K: A,M: fun(A,option(B))] :
      ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),K),dom(A,B,M)))
     => ( fun_upd(A,option(B),M,K,none(B)) = M ) ) ).

% fun_upd_None_if_notin_dom
tff(fact_8166_dom__empty,axiom,
    ! [B: $tType,A: $tType] : dom(A,B,aTP_Lamp_zy(A,option(B))) = bot_bot(set(A)) ).

% dom_empty
tff(fact_8167_dom__map__of__zip,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( dom(A,B,map_of(A,B,zip(A,B,Xs,Ys))) = aa(list(A),set(A),set2(A),Xs) ) ) ).

% dom_map_of_zip
tff(fact_8168_dom__fun__upd,axiom,
    ! [B: $tType,A: $tType,Y: option(B),F3: fun(A,option(B)),X: A] :
      ( ( ( Y = none(B) )
       => ( dom(A,B,fun_upd(A,option(B),F3,X,Y)) = aa(set(A),set(A),minus_minus(set(A),dom(A,B,F3)),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ) )
      & ( ( Y != none(B) )
       => ( dom(A,B,fun_upd(A,option(B),F3,X,Y)) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),dom(A,B,F3)) ) ) ) ).

% dom_fun_upd
tff(fact_8169_dom__minus,axiom,
    ! [A: $tType,B: $tType,F3: fun(B,option(A)),X: B,A3: set(B)] :
      ( ( aa(B,option(A),F3,X) = none(A) )
     => ( aa(set(B),set(B),minus_minus(set(B),dom(B,A,F3)),aa(set(B),set(B),aa(B,fun(set(B),set(B)),insert(B),X),A3)) = aa(set(B),set(B),minus_minus(set(B),dom(B,A,F3)),A3) ) ) ).

% dom_minus
tff(fact_8170_chain__singleton,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [X: A] : comple1602240252501008431_chain(A,ord_less_eq(A),aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A)))) ) ).

% chain_singleton
tff(fact_8171_finite__map__freshness,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B))] :
      ( pp(aa(set(A),bool,finite_finite(A),dom(A,B,F3)))
     => ( ~ pp(aa(set(A),bool,finite_finite(A),top_top(set(A))))
       => ? [X3: A] : aa(A,option(B),F3,X3) = none(B) ) ) ).

% finite_map_freshness
tff(fact_8172_domIff,axiom,
    ! [A: $tType,B: $tType,A2: A,M: fun(A,option(B))] :
      ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A2),dom(A,B,M)))
    <=> ( aa(A,option(B),M,A2) != none(B) ) ) ).

% domIff
tff(fact_8173_dom__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B))] : dom(A,B,M) = collect(A,aTP_Lamp_adm(fun(A,option(B)),fun(A,bool),M)) ).

% dom_def
tff(fact_8174_chain__subset,axiom,
    ! [A: $tType,Ord: fun(A,fun(A,bool)),A3: set(A),B3: set(A)] :
      ( comple1602240252501008431_chain(A,Ord,A3)
     => ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),B3),A3))
       => comple1602240252501008431_chain(A,Ord,B3) ) ) ).

% chain_subset
tff(fact_8175_ccpo__Sup__least,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A3: set(A),Z: A] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A3)
         => ( ! [X3: A] :
                ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X3),A3))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X3),Z)) )
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(set(A),A,complete_Sup_Sup(A),A3)),Z)) ) ) ) ).

% ccpo_Sup_least
tff(fact_8176_ccpo__Sup__upper,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A3: set(A),X: A] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A3)
         => ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X),A3))
           => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X),aa(set(A),A,complete_Sup_Sup(A),A3))) ) ) ) ).

% ccpo_Sup_upper
tff(fact_8177_finite__set__of__finite__maps,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B3: set(B)] :
      ( pp(aa(set(A),bool,finite_finite(A),A3))
     => ( pp(aa(set(B),bool,finite_finite(B),B3))
       => pp(aa(set(fun(A,option(B))),bool,finite_finite(fun(A,option(B))),collect(fun(A,option(B)),aa(set(B),fun(fun(A,option(B)),bool),aTP_Lamp_afi(set(A),fun(set(B),fun(fun(A,option(B)),bool)),A3),B3)))) ) ) ).

% finite_set_of_finite_maps
tff(fact_8178_finite__Map__induct,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),P: fun(fun(A,option(B)),bool)] :
      ( pp(aa(set(A),bool,finite_finite(A),dom(A,B,M)))
     => ( pp(aa(fun(A,option(B)),bool,P,aTP_Lamp_zy(A,option(B))))
       => ( ! [K2: A,V3: B,M5: fun(A,option(B))] :
              ( pp(aa(set(A),bool,finite_finite(A),dom(A,B,M5)))
             => ( ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),K2),dom(A,B,M5)))
               => ( pp(aa(fun(A,option(B)),bool,P,M5))
                 => pp(aa(fun(A,option(B)),bool,P,fun_upd(A,option(B),M5,K2,aa(B,option(B),some(B),V3)))) ) ) )
         => pp(aa(fun(A,option(B)),bool,P,M)) ) ) ) ).

% finite_Map_induct
tff(fact_8179_dom__eq__singleton__conv,axiom,
    ! [A: $tType,B: $tType,F3: fun(A,option(B)),X: A] :
      ( ( dom(A,B,F3) = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),X),bot_bot(set(A))) )
    <=> ? [V6: B] : F3 = fun_upd(A,option(B),aTP_Lamp_zy(A,option(B)),X,aa(B,option(B),some(B),V6)) ) ).

% dom_eq_singleton_conv
tff(fact_8180_map__of__map__keys,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),M: fun(A,option(B))] :
      ( ( aa(list(A),set(A),set2(A),Xs) = dom(A,B,M) )
     => ( map_of(A,B,aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aTP_Lamp_afj(fun(A,option(B)),fun(A,product_prod(A,B)),M)),Xs)) = M ) ) ).

% map_of_map_keys
tff(fact_8181_in__chain__finite,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [A3: set(A)] :
          ( comple1602240252501008431_chain(A,ord_less_eq(A),A3)
         => ( pp(aa(set(A),bool,finite_finite(A),A3))
           => ( ( A3 != bot_bot(set(A)) )
             => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,complete_Sup_Sup(A),A3)),A3)) ) ) ) ) ).

% in_chain_finite
tff(fact_8182_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_an(nat,real),Uu) = divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uu)),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,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),aa(num,num,bit0,one2)))),one_one(nat)))) ).

% ATP.lambda_1
tff(fact_8183_ATP_Olambda__2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_rr(A,A),Uu) = divide_divide(A,aa(A,A,minus_minus(A,exp(A,Uu)),one_one(A)),Uu) ) ).

% ATP.lambda_2
tff(fact_8184_ATP_Olambda__3,axiom,
    ! [A: $tType,Uu: set(set(A))] : aa(set(set(A)),int,aTP_Lamp_kp(set(set(A)),int),Uu) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),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_3
tff(fact_8185_ATP_Olambda__4,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A] :
          ( pp(aa(A,bool,aTP_Lamp_zf(A,bool),Uu))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uu),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),zero_zero(A)),Uu)) ) ) ) ).

% ATP.lambda_4
tff(fact_8186_ATP_Olambda__5,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_bn(nat,real),Uu) = aa(nat,real,power_power(real,divide_divide(real,one_one(real),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))),aa(nat,nat,suc,Uu)) ).

% ATP.lambda_5
tff(fact_8187_ATP_Olambda__6,axiom,
    ! [Uu: real] :
      ( pp(aa(real,bool,aTP_Lamp_hc(real,bool),Uu))
    <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Uu))
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Uu),aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))
        & ( cos(real,Uu) = zero_zero(real) ) ) ) ).

% ATP.lambda_6
tff(fact_8188_ATP_Olambda__7,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: product_prod(int,int)] : aa(product_prod(int,int),A,aTP_Lamp_aez(product_prod(int,int),A),Uu) = divide_divide(A,ring_1_of_int(A,aa(product_prod(int,int),int,product_fst(int,int),Uu)),ring_1_of_int(A,aa(product_prod(int,int),int,product_snd(int,int),Uu))) ) ).

% ATP.lambda_7
tff(fact_8189_ATP_Olambda__8,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_so(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_8
tff(fact_8190_ATP_Olambda__9,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_ei(nat,real),Uu) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uu)),aa(nat,real,power_power(real,zero_zero(real)),Uu)) ).

% ATP.lambda_9
tff(fact_8191_ATP_Olambda__10,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_sn(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_10
tff(fact_8192_ATP_Olambda__11,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_ru(real,real),Uu) = divide_divide(real,cos(real,Uu),sin(real,Uu)) ).

% ATP.lambda_11
tff(fact_8193_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_sb(nat,real),Uu) = aa(real,real,root(Uu),aa(nat,real,semiring_1_of_nat(real),Uu)) ).

% ATP.lambda_12
tff(fact_8194_ATP_Olambda__13,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_ym(nat,nat),Uu) = aa(nat,nat,minus_minus(nat,Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_13
tff(fact_8195_ATP_Olambda__14,axiom,
    ! [A: $tType,Uu: A] : aa(A,product_prod(A,A),aTP_Lamp_abf(A,product_prod(A,A)),Uu) = aa(A,product_prod(A,A),product_Pair(A,A,Uu),Uu) ).

% ATP.lambda_14
tff(fact_8196_ATP_Olambda__15,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ci(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).

% ATP.lambda_15
tff(fact_8197_ATP_Olambda__16,axiom,
    ! [A: $tType,Uu: A] : aa(A,list(A),aTP_Lamp_qj(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ).

% ATP.lambda_16
tff(fact_8198_ATP_Olambda__17,axiom,
    ! [Uu: product_prod(int,int)] :
      ( pp(aa(product_prod(int,int),bool,aTP_Lamp_aey(product_prod(int,int),bool),Uu))
    <=> pp(aa(int,bool,aa(int,fun(int,bool),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_17
tff(fact_8199_ATP_Olambda__18,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_si(nat,real),Uu) = divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),Uu)) ).

% ATP.lambda_18
tff(fact_8200_ATP_Olambda__19,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_sm(nat,A),Uu) = divide_divide(A,one_one(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_19
tff(fact_8201_ATP_Olambda__20,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_acs(list(B),fun(nat,nat)),Uu) = aa(nat,fun(nat,nat),ord_max(nat),aa(nat,nat,minus_minus(nat,aa(list(B),nat,size_size(list(B)),Uu)),aa(nat,nat,suc,zero_zero(nat)))) ).

% ATP.lambda_20
tff(fact_8202_ATP_Olambda__21,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( pp(aa(list(B),bool,aTP_Lamp_act(list(B),bool),Uu))
    <=> ( Uu != nil(B) ) ) ).

% ATP.lambda_21
tff(fact_8203_ATP_Olambda__22,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_abw(list(A),bool),Uu))
    <=> ( Uu != nil(A) ) ) ).

% ATP.lambda_22
tff(fact_8204_ATP_Olambda__23,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: B] : aa(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C))),aTP_Lamp_abk(B,fun(product_prod(A,C),product_prod(A,product_prod(B,C)))),Uu) = product_case_prod(A,C,product_prod(A,product_prod(B,C)),aTP_Lamp_abj(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu)) ).

% ATP.lambda_23
tff(fact_8205_ATP_Olambda__24,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_pf(real,real),Uu) = suminf(real,aTP_Lamp_ao(real,fun(nat,real),Uu)) ).

% ATP.lambda_24
tff(fact_8206_ATP_Olambda__25,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real] : aa(real,filter(A),aTP_Lamp_yv(real,filter(A)),Uu) = principal(A,collect(A,aTP_Lamp_yu(real,fun(A,bool),Uu))) ) ).

% ATP.lambda_25
tff(fact_8207_ATP_Olambda__26,axiom,
    ! [Uu: real] : aa(real,filter(product_prod(complex,complex)),aTP_Lamp_zd(real,filter(product_prod(complex,complex))),Uu) = principal(product_prod(complex,complex),collect(product_prod(complex,complex),product_case_prod(complex,complex,bool,aTP_Lamp_zc(real,fun(complex,fun(complex,bool)),Uu)))) ).

% ATP.lambda_26
tff(fact_8208_ATP_Olambda__27,axiom,
    ! [Uu: real] : aa(real,filter(product_prod(real,real)),aTP_Lamp_zb(real,filter(product_prod(real,real))),Uu) = principal(product_prod(real,real),collect(product_prod(real,real),product_case_prod(real,real,bool,aTP_Lamp_za(real,fun(real,fun(real,bool)),Uu)))) ).

% ATP.lambda_27
tff(fact_8209_ATP_Olambda__28,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [Uu: real] : aa(real,filter(product_prod(A,A)),aTP_Lamp_yz(real,filter(product_prod(A,A))),Uu) = principal(product_prod(A,A),collect(product_prod(A,A),product_case_prod(A,A,bool,aTP_Lamp_yy(real,fun(A,fun(A,bool)),Uu)))) ) ).

% ATP.lambda_28
tff(fact_8210_ATP_Olambda__29,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_sk(nat,real),Uu) = aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uu))) ).

% ATP.lambda_29
tff(fact_8211_ATP_Olambda__30,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_acr(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_30
tff(fact_8212_ATP_Olambda__31,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_acj(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_31
tff(fact_8213_ATP_Olambda__32,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_lt(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit1,Uu)) ).

% ATP.lambda_32
tff(fact_8214_ATP_Olambda__33,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_lp(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit0,Uu)) ).

% ATP.lambda_33
tff(fact_8215_ATP_Olambda__34,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_hg(nat,fun(nat,product_prod(nat,nat))),Uu) = product_Pair(nat,nat,aa(nat,nat,suc,Uu)) ).

% ATP.lambda_34
tff(fact_8216_ATP_Olambda__35,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: A] :
          ( pp(aa(A,bool,aTP_Lamp_ze(A,bool),Uu))
        <=> ? [N5: int] :
              ( ( Uu = ring_1_of_int(A,N5) )
              & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),zero_zero(int)),N5)) ) ) ) ).

% ATP.lambda_35
tff(fact_8217_ATP_Olambda__36,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_xv(product_prod(A,A),bool),Uu))
        <=> ? [X4: A] : Uu = aa(A,product_prod(A,A),product_Pair(A,A,X4),X4) ) ) ).

% ATP.lambda_36
tff(fact_8218_ATP_Olambda__37,axiom,
    ! [A: $tType,Uu: product_prod(A,A)] :
      ( pp(aa(product_prod(A,A),bool,aTP_Lamp_aej(product_prod(A,A),bool),Uu))
    <=> ? [X4: A] : Uu = aa(A,product_prod(A,A),product_Pair(A,A,X4),X4) ) ).

% ATP.lambda_37
tff(fact_8219_ATP_Olambda__38,axiom,
    ! [Uu: real] :
      ( pp(aa(real,bool,aTP_Lamp_mf(real,bool),Uu))
    <=> ? [I3: int,N5: nat] :
          ( ( Uu = divide_divide(real,ring_1_of_int(real,I3),aa(nat,real,semiring_1_of_nat(real),N5)) )
          & ( N5 != zero_zero(nat) ) ) ) ).

% ATP.lambda_38
tff(fact_8220_ATP_Olambda__39,axiom,
    ! [Uu: real] :
      ( pp(aa(real,bool,aTP_Lamp_mq(real,bool),Uu))
    <=> ? [I3: int,J3: int] :
          ( ( Uu = divide_divide(real,ring_1_of_int(real,I3),ring_1_of_int(real,J3)) )
          & ( J3 != zero_zero(int) ) ) ) ).

% ATP.lambda_39
tff(fact_8221_ATP_Olambda__40,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_xx(product_prod(A,A),bool),Uu))
        <=> ? [X4: A,Y5: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X4),Y5) )
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Y5)) ) ) ) ).

% ATP.lambda_40
tff(fact_8222_ATP_Olambda__41,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_xw(product_prod(A,A),bool),Uu))
        <=> ? [X4: A,Y5: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X4),Y5) )
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y5),X4)) ) ) ) ).

% ATP.lambda_41
tff(fact_8223_ATP_Olambda__42,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_up(product_prod(A,A),bool),Uu))
        <=> ? [X4: A,Y5: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X4),Y5) )
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y5)) ) ) ) ).

% ATP.lambda_42
tff(fact_8224_ATP_Olambda__43,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_uq(product_prod(A,A),bool),Uu))
        <=> ? [X4: A,Y5: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X4),Y5) )
              & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X4)) ) ) ) ).

% ATP.lambda_43
tff(fact_8225_ATP_Olambda__44,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: product_prod(A,A)] :
          ( pp(aa(product_prod(A,A),bool,aTP_Lamp_un(product_prod(A,A),bool),Uu))
        <=> ? [X4: A,Y5: A] :
              ( ( Uu = aa(A,product_prod(A,A),product_Pair(A,A,X4),Y5) )
              & ( X4 != Y5 ) ) ) ) ).

% ATP.lambda_44
tff(fact_8226_ATP_Olambda__45,axiom,
    ! [Uu: nat] : aa(nat,option(num),aTP_Lamp_lr(nat,option(num)),Uu) = aa(num,option(num),some(num),one2) ).

% ATP.lambda_45
tff(fact_8227_ATP_Olambda__46,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_ly(num,fun(nat,option(num)),Uu),Uua) = case_num(option(num),aa(num,option(num),some(num),one2),aTP_Lamp_lw(nat,fun(num,option(num)),Uua),aTP_Lamp_lx(nat,fun(num,option(num)),Uua),Uu) ).

% ATP.lambda_46
tff(fact_8228_ATP_Olambda__47,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_cg(A,fun(nat,A),Uu),Uua) = if(A,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uua),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,power_power(A,Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_47
tff(fact_8229_ATP_Olambda__48,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_dy(nat,fun(nat,A),Uu),Uua) = if(A,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_48
tff(fact_8230_ATP_Olambda__49,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ch(A,fun(nat,A),Uu),Uua) = if(A,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uua),zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,power_power(A,Uu),Uua))) ) ).

% ATP.lambda_49
tff(fact_8231_ATP_Olambda__50,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_bo(fun(nat,real),fun(nat,real),Uu),Uua) = if(real,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uua),zero_zero(real),aa(nat,real,Uu,divide_divide(nat,aa(nat,nat,minus_minus(nat,Uua),one_one(nat)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% ATP.lambda_50
tff(fact_8232_ATP_Olambda__51,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_adh(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uu),Uua) = if(list(product_prod(A,B)),aa(C,bool,aa(C,fun(C,bool),fequal(C),aa(product_prod(A,C),C,product_snd(A,C),Uu)),aa(product_prod(C,B),C,product_fst(C,B),Uua)),aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),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_51
tff(fact_8233_ATP_Olambda__52,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_ku(code_integer,fun(code_integer,int)),Uu),Uua) = if(int,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),code_int_of_integer(Uu)),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),aa(num,num,bit0,one2))),code_int_of_integer(Uu))),one_one(int))) ).

% ATP.lambda_52
tff(fact_8234_ATP_Olambda__53,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hp(nat,fun(nat,A)),Uu),Uua) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uua),zero_zero(nat)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),aa(nat,A,semiring_1_of_nat(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),aa(num,num,bit0,one2))),aa(nat,A,semiring_1_of_nat(A),Uu))),one_one(A))) ) ).

% ATP.lambda_53
tff(fact_8235_ATP_Olambda__54,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_aac(code_integer,fun(code_integer,num)),Uu),Uua) = if(num,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(num,num,aa(num,fun(num,num),plus_plus(num),code_num_of_integer(Uu)),code_num_of_integer(Uu)),aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),code_num_of_integer(Uu)),code_num_of_integer(Uu))),one2)) ).

% ATP.lambda_54
tff(fact_8236_ATP_Olambda__55,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_kr(code_integer,fun(code_integer,nat)),Uu),Uua) = if(nat,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uua),zero_zero(code_integer)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),code_nat_of_integer(Uu)),code_nat_of_integer(Uu)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),code_nat_of_integer(Uu)),code_nat_of_integer(Uu))),one_one(nat))) ).

% ATP.lambda_55
tff(fact_8237_ATP_Olambda__56,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_dx(nat,fun(nat,A),Uu),Uua) = if(A,aa(bool,bool,fNot,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_56
tff(fact_8238_ATP_Olambda__57,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_top(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_xq(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_57
tff(fact_8239_ATP_Olambda__58,axiom,
    ! [A: $tType] :
      ( ( lattice(A)
        & order_bot(A) )
     => ! [Uu: fun(A,A),Uua: nat] : aa(nat,A,aTP_Lamp_xo(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_58
tff(fact_8240_ATP_Olambda__59,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_afg(list(list(A)),fun(list(A),bool),Uu),Uua))
    <=> pp(aa(list(list(A)),bool,aa(list(A),fun(list(list(A)),bool),list_all2(A,list(A),aTP_Lamp_aff(A,fun(list(A),bool))),Uua),Uu)) ) ).

% ATP.lambda_59
tff(fact_8241_ATP_Olambda__60,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_lz(nat,fun(num,option(num))),Uu),Uua) = case_nat(option(num),none(num),aTP_Lamp_ly(num,fun(nat,option(num)),Uua),Uu) ).

% ATP.lambda_60
tff(fact_8242_ATP_Olambda__61,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_lw(nat,fun(num,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_lp(num,option(num)),bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_61
tff(fact_8243_ATP_Olambda__62,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_lq(num,fun(nat,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_lp(num,option(num)),bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_62
tff(fact_8244_ATP_Olambda__63,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_aay(list(list(A)),fun(nat,list(A)),Uu),Uua) = aa(list(nat),list(A),map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aax(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_63
tff(fact_8245_ATP_Olambda__64,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_hk(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_hj(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_64
tff(fact_8246_ATP_Olambda__65,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_hn(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hm(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_65
tff(fact_8247_ATP_Olambda__66,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_al(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua)),divide_divide(real,one_one(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))))),aa(nat,real,power_power(real,aa(real,real,minus_minus(real,Uu),one_one(real))),aa(nat,nat,suc,Uua))) ).

% ATP.lambda_66
tff(fact_8248_ATP_Olambda__67,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_bh(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),one_one(nat))))),aa(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),aa(num,num,bit0,one2))),Uua)),one_one(nat)))) ).

% ATP.lambda_67
tff(fact_8249_ATP_Olambda__68,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_bq(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)))),aa(nat,real,power_power(real,Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua))) ).

% ATP.lambda_68
tff(fact_8250_ATP_Olambda__69,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ds(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),Uua))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_69
tff(fact_8251_ATP_Olambda__70,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_dt(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,power_power(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)) ) ).

% ATP.lambda_70
tff(fact_8252_ATP_Olambda__71,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_hf(real,fun(real,bool),Uu),Uua))
    <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Uua))
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Uua),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
        & ( sin(real,Uua) = Uu ) ) ) ).

% ATP.lambda_71
tff(fact_8253_ATP_Olambda__72,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_gc(real,fun(real,bool),Uu),Uua))
    <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2))))),Uua))
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Uua),divide_divide(real,pi,aa(num,real,numeral_numeral(real),aa(num,num,bit0,one2)))))
        & ( aa(real,real,tan(real),Uua) = Uu ) ) ) ).

% ATP.lambda_72
tff(fact_8254_ATP_Olambda__73,axiom,
    ! [Uu: complex,Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_cj(complex,fun(real,bool),Uu),Uua))
    <=> ( ( sgn_sgn(complex,Uu) = cis(Uua) )
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(real,real,uminus_uminus(real),pi)),Uua))
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Uua),pi)) ) ) ).

% ATP.lambda_73
tff(fact_8255_ATP_Olambda__74,axiom,
    ! [Uu: real,Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_ij(real,fun(int,bool),Uu),Uua))
    <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),ring_1_of_int(real,Uua)),Uu))
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),Uu),ring_1_of_int(real,aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int))))) ) ) ).

% ATP.lambda_74
tff(fact_8256_ATP_Olambda__75,axiom,
    ! [Uu: rat,Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_im(rat,fun(int,bool),Uu),Uua))
    <=> ( pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less_eq(rat),ring_1_of_int(rat,Uua)),Uu))
        & pp(aa(rat,bool,aa(rat,fun(rat,bool),ord_less(rat),Uu),ring_1_of_int(rat,aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int))))) ) ) ).

% ATP.lambda_75
tff(fact_8257_ATP_Olambda__76,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ao(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(real,real,aa(real,fun(real,real),times_times(real),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),aa(num,num,bit0,one2)))),one_one(nat))))),aa(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),aa(num,num,bit0,one2)))),one_one(nat))))) ).

% ATP.lambda_76
tff(fact_8258_ATP_Olambda__77,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_pg(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,power_power(real,Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% ATP.lambda_77
tff(fact_8259_ATP_Olambda__78,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_rv(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,power_power(real,aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_78
tff(fact_8260_ATP_Olambda__79,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_dw(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_79
tff(fact_8261_ATP_Olambda__80,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_cw(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_80
tff(fact_8262_ATP_Olambda__81,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_dz(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(A,A,minus_minus(A,divide_divide(A,Uu,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_81
tff(fact_8263_ATP_Olambda__82,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_de(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),one_one(A))),Uua)) ) ).

% ATP.lambda_82
tff(fact_8264_ATP_Olambda__83,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(set(A))] :
      ( pp(aa(set(set(A)),bool,aTP_Lamp_kq(set(set(A)),fun(set(set(A)),bool),Uu),Uua))
    <=> ( pp(aa(set(set(A)),bool,aa(set(set(A)),fun(set(set(A)),bool),ord_less_eq(set(set(A))),Uua),Uu))
        & ( Uua != bot_bot(set(set(A))) ) ) ) ).

% ATP.lambda_83
tff(fact_8265_ATP_Olambda__84,axiom,
    ! [A: $tType,Uu: set(option(A)),Uua: option(A)] :
      ( pp(aa(option(A),bool,aTP_Lamp_zg(set(option(A)),fun(option(A),bool),Uu),Uua))
    <=> ( pp(aa(set(option(A)),bool,aa(option(A),fun(set(option(A)),bool),member(option(A)),Uua),Uu))
        & ( Uua != none(A) ) ) ) ).

% ATP.lambda_84
tff(fact_8266_ATP_Olambda__85,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_dj(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,power_power(nat,aa(nat,nat,binomial(Uu),Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))) ).

% ATP.lambda_85
tff(fact_8267_ATP_Olambda__86,axiom,
    ! [Uu: set(int),Uua: int] :
      ( pp(aa(int,bool,aTP_Lamp_ll(set(int),fun(int,bool),Uu),Uua))
    <=> ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),Uua),Uu))
        & ! [X4: int] :
            ( pp(aa(set(int),bool,aa(int,fun(set(int),bool),member(int),X4),Uu))
           => pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),X4),Uua)) ) ) ) ).

% ATP.lambda_86
tff(fact_8268_ATP_Olambda__87,axiom,
    ! [A: $tType,Uu: set(set(A)),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_lm(set(set(A)),fun(set(A),bool),Uu),Uua))
    <=> ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),Uua),Uu))
        & ! [X4: set(A)] :
            ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),Uu))
           => ~ pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uua),X4)) ) ) ) ).

% ATP.lambda_87
tff(fact_8269_ATP_Olambda__88,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_fd(real,fun(nat,real),Uu),Uua) = divide_divide(real,aa(nat,real,power_power(real,Uu),Uua),semiring_char_0_fact(real,Uua)) ).

% ATP.lambda_88
tff(fact_8270_ATP_Olambda__89,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_vb(nat,fun(real,real),Uu),Uua) = divide_divide(real,aa(nat,real,power_power(real,Uua),Uu),exp(real,Uua)) ).

% ATP.lambda_89
tff(fact_8271_ATP_Olambda__90,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_ny(set(A),fun(set(A),bool),Uu),Uua))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu))
        & pp(aa(set(A),bool,finite_finite(A),Uua)) ) ) ).

% ATP.lambda_90
tff(fact_8272_ATP_Olambda__91,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_jz(set(A),fun(set(A),bool)),Uu),Uua))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less(set(A)),Uu),Uua))
        & pp(aa(set(A),bool,finite_finite(A),Uua)) ) ) ).

% ATP.lambda_91
tff(fact_8273_ATP_Olambda__92,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_xr(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,minus_minus(nat,aa(nat,nat,power_power(nat,Uu),Uua)),Uua) ).

% ATP.lambda_92
tff(fact_8274_ATP_Olambda__93,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_cu(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_93
tff(fact_8275_ATP_Olambda__94,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ct(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_94
tff(fact_8276_ATP_Olambda__95,axiom,
    ! [Uu: nat,Uua: complex] :
      ( pp(aa(complex,bool,aTP_Lamp_eq(nat,fun(complex,bool),Uu),Uua))
    <=> ( aa(nat,complex,power_power(complex,Uua),Uu) = one_one(complex) ) ) ).

% ATP.lambda_95
tff(fact_8277_ATP_Olambda__96,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: nat,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ji(nat,fun(A,bool),Uu),Uua))
        <=> ( aa(nat,A,power_power(A,Uua),Uu) = one_one(A) ) ) ) ).

% ATP.lambda_96
tff(fact_8278_ATP_Olambda__97,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_jh(A,fun(A,bool),Uu),Uua))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(A,A,abs_abs(A),Uua)),Uu)) ) ) ) ).

% ATP.lambda_97
tff(fact_8279_ATP_Olambda__98,axiom,
    ! [A: $tType,Uu: fun(set(A),bool),Uua: set(A)] :
      ( pp(aa(set(A),bool,aa(fun(set(A),bool),fun(set(A),bool),aTP_Lamp_xy(fun(set(A),bool),fun(set(A),bool)),Uu),Uua))
    <=> ( ( Uua = bot_bot(set(A)) )
        | ? [A11: set(A),A6: A] :
            ( ( Uua = aa(set(A),set(A),aa(A,fun(set(A),set(A)),insert(A),A6),A11) )
            & pp(aa(set(A),bool,Uu,A11)) ) ) ) ).

% ATP.lambda_98
tff(fact_8280_ATP_Olambda__99,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_st(real,fun(nat,real),Uu),Uua) = aa(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_99
tff(fact_8281_ATP_Olambda__100,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_vx(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_100
tff(fact_8282_ATP_Olambda__101,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_vc(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_101
tff(fact_8283_ATP_Olambda__102,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ak(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),aa(num,num,bit0,one2)))),one_one(nat))))),aa(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),aa(num,num,bit0,one2)))),one_one(nat)))) ).

% ATP.lambda_102
tff(fact_8284_ATP_Olambda__103,axiom,
    ! [Uu: real,Uua: real] :
      ( pp(aa(real,bool,aTP_Lamp_gt(real,fun(real,bool),Uu),Uua))
    <=> ( pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),Uua))
        & pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Uua),pi))
        & ( cos(real,Uua) = Uu ) ) ) ).

% ATP.lambda_103
tff(fact_8285_ATP_Olambda__104,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_we(set(product_prod(A,A)),fun(nat,bool),Uu),Uua))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Uua))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu))) ) ) ).

% ATP.lambda_104
tff(fact_8286_ATP_Olambda__105,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_wg(nat,fun(nat,bool),Uu),Uua))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),zero_zero(nat)),Uua))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(nat,nat,suc,Uu))) ) ) ).

% ATP.lambda_105
tff(fact_8287_ATP_Olambda__106,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gw(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),aa(num,num,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),aa(num,num,bit0,one2))),Uua)))) ) ).

% ATP.lambda_106
tff(fact_8288_ATP_Olambda__107,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_da(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),aa(num,num,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),aa(num,num,bit0,one2))),Uua)))) ) ).

% ATP.lambda_107
tff(fact_8289_ATP_Olambda__108,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_se(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_108
tff(fact_8290_ATP_Olambda__109,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_en(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_109
tff(fact_8291_ATP_Olambda__110,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),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_110
tff(fact_8292_ATP_Olambda__111,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,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_111
tff(fact_8293_ATP_Olambda__112,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(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,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_112
tff(fact_8294_ATP_Olambda__113,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_am(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_113
tff(fact_8295_ATP_Olambda__114,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_it(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_114
tff(fact_8296_ATP_Olambda__115,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_aw(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,power_power(A,zero_zero(A)),Uua)) ) ).

% ATP.lambda_115
tff(fact_8297_ATP_Olambda__116,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fs(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,Uua),one_one(nat)))) ) ).

% ATP.lambda_116
tff(fact_8298_ATP_Olambda__117,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_fr(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_117
tff(fact_8299_ATP_Olambda__118,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_sf(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_118
tff(fact_8300_ATP_Olambda__119,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cq(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_119
tff(fact_8301_ATP_Olambda__120,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_lo(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ( pp(aa(A,bool,Uu,Uua))
            & ! [Y5: A] :
                ( pp(aa(A,bool,Uu,Y5))
               => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Y5),Uua)) ) ) ) ) ).

% ATP.lambda_120
tff(fact_8302_ATP_Olambda__121,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_go(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)),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_121
tff(fact_8303_ATP_Olambda__122,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_fn(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),Uu,Uua)),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_122
tff(fact_8304_ATP_Olambda__123,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_vv(fun(A,real),fun(A,bool),Uu),Uua))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(A,real,Uu,Uua)),zero_zero(real))) ) ).

% ATP.lambda_123
tff(fact_8305_ATP_Olambda__124,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,list(A),aTP_Lamp_aan(fun(B,A),fun(B,list(A)),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,Uu,Uua)),nil(A)) ).

% ATP.lambda_124
tff(fact_8306_ATP_Olambda__125,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: B] :
          ( pp(aa(B,bool,aTP_Lamp_jf(fun(B,A),fun(B,bool),Uu),Uua))
        <=> ( aa(B,A,Uu,Uua) = one_one(A) ) ) ) ).

% ATP.lambda_125
tff(fact_8307_ATP_Olambda__126,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_td(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_rv(fun(nat,real),fun(nat,real),Uu)),aa(nat,set(nat),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),aa(num,num,bit0,one2))),Uua)),one_one(nat)))) ).

% ATP.lambda_126
tff(fact_8308_ATP_Olambda__127,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_tc(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_rv(fun(nat,real),fun(nat,real),Uu)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua))) ).

% ATP.lambda_127
tff(fact_8309_ATP_Olambda__128,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_abd(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_abc(list(A),fun(A,list(A))),Uua)),Uu) ).

% ATP.lambda_128
tff(fact_8310_ATP_Olambda__129,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_aeo(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_129
tff(fact_8311_ATP_Olambda__130,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_ui(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_130
tff(fact_8312_ATP_Olambda__131,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_bw(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))),aa(nat,A,power_power(A,Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% ATP.lambda_131
tff(fact_8313_ATP_Olambda__132,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_cf(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,suc,Uua)))),aa(nat,A,power_power(A,Uu),aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_132
tff(fact_8314_ATP_Olambda__133,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_bx(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_133
tff(fact_8315_ATP_Olambda__134,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(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),times_times(A),aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,Uua))),aa(nat,A,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_134
tff(fact_8316_ATP_Olambda__135,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_bi(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uua))),aa(nat,A,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_135
tff(fact_8317_ATP_Olambda__136,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aTP_Lamp_ln(num,fun(num,int),Uu),Uua) = aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),Uu)),aa(int,int,minus_minus(int,aa(nat,int,power_power(int,aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),aa(num,nat,numeral_numeral(nat),Uu))),aa(num,int,numeral_numeral(int),Uua))) ).

% ATP.lambda_136
tff(fact_8318_ATP_Olambda__137,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ce(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),Uu)),Uua)) ) ).

% ATP.lambda_137
tff(fact_8319_ATP_Olambda__138,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_nz(nat,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sgn_sgn(real,Uua)),aa(nat,real,power_power(real,aa(real,real,abs_abs(real),Uua)),Uu)) ).

% ATP.lambda_138
tff(fact_8320_ATP_Olambda__139,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_by(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_139
tff(fact_8321_ATP_Olambda__140,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_bz(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_140
tff(fact_8322_ATP_Olambda__141,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_sy(A,fun(nat,A),Uu),Uua) = divide_divide(A,aa(nat,A,semiring_1_of_nat(A),Uua),aa(nat,A,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_141
tff(fact_8323_ATP_Olambda__142,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_sx(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(nat,A,power_power(A,Uu),Uua)) ) ).

% ATP.lambda_142
tff(fact_8324_ATP_Olambda__143,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_fh(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sin_coeff(Uua)),aa(nat,real,power_power(real,Uu),Uua)) ).

% ATP.lambda_143
tff(fact_8325_ATP_Olambda__144,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_eh(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uua)),aa(nat,real,power_power(real,Uu),Uua)) ).

% ATP.lambda_144
tff(fact_8326_ATP_Olambda__145,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_aai(set(A),fun(set(A),bool),Uu),Uua))
    <=> ( pp(aa(set(A),bool,finite_finite(A),Uua))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu)) ) ) ).

% ATP.lambda_145
tff(fact_8327_ATP_Olambda__146,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: list(product_prod(A,B))] :
      ( pp(aa(list(product_prod(A,B)),bool,aTP_Lamp_afa(fun(A,fun(B,bool)),fun(list(product_prod(A,B)),bool),Uu),Uua))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),bool),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,bool,Uu)))) ) ).

% ATP.lambda_146
tff(fact_8328_ATP_Olambda__147,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: A] : aa(A,list(list(A)),aTP_Lamp_abe(list(list(A)),fun(A,list(list(A))),Uu),Uua) = aa(list(list(A)),list(list(A)),map(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua)),product_lists(A,Uu)) ).

% ATP.lambda_147
tff(fact_8329_ATP_Olambda__148,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_zo(list(A),fun(list(A),bool)),Uu),Uua))
    <=> ( aa(list(A),nat,size_size(list(A)),Uu) = aa(list(A),nat,size_size(list(A)),Uua) ) ) ).

% ATP.lambda_148
tff(fact_8330_ATP_Olambda__149,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_aen(set(A),fun(list(A),bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uua)),Uu)) ) ).

% ATP.lambda_149
tff(fact_8331_ATP_Olambda__150,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_ft(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).

% ATP.lambda_150
tff(fact_8332_ATP_Olambda__151,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_abb(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_151
tff(fact_8333_ATP_Olambda__152,axiom,
    ! [A: $tType,Uu: set(nat),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_acp(set(nat),fun(product_prod(A,nat),bool),Uu),Uua))
    <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(product_prod(A,nat),nat,product_snd(A,nat),Uua)),Uu)) ) ).

% ATP.lambda_152
tff(fact_8334_ATP_Olambda__153,axiom,
    ! [Uu: set(nat),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_ql(set(nat),fun(nat,bool),Uu),Uua))
    <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,suc,Uua)),Uu)) ) ).

% ATP.lambda_153
tff(fact_8335_ATP_Olambda__154,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_gb(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),aa(num,num,bit0,one2))),Uua)),one_one(A))) ) ).

% ATP.lambda_154
tff(fact_8336_ATP_Olambda__155,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_ge(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),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)) ) ).

% ATP.lambda_155
tff(fact_8337_ATP_Olambda__156,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ga(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),aa(num,num,bit0,one2)))) ) ).

% ATP.lambda_156
tff(fact_8338_ATP_Olambda__157,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_dv(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_157
tff(fact_8339_ATP_Olambda__158,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_sv(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_158
tff(fact_8340_ATP_Olambda__159,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_aao(fun(nat,A),fun(nat,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),product_Pair(nat,A,Uua),aa(nat,A,Uu,Uua)) ).

% ATP.lambda_159
tff(fact_8341_ATP_Olambda__160,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_adl(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_160
tff(fact_8342_ATP_Olambda__161,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_hx(A,fun(nat,A),Uu),Uua) = aa(A,A,bit_se4730199178511100633sh_bit(A,Uua),aa(bool,A,zero_neq_one_of_bool(A),aa(nat,bool,bit_se5641148757651400278ts_bit(A,Uu),Uua))) ) ).

% ATP.lambda_161
tff(fact_8343_ATP_Olambda__162,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_afj(fun(A,option(B)),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),product_Pair(A,B,Uua),aa(option(B),B,the2(B),aa(A,option(B),Uu,Uua))) ).

% ATP.lambda_162
tff(fact_8344_ATP_Olambda__163,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_sr(fun(nat,A),fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua))) ) ).

% ATP.lambda_163
tff(fact_8345_ATP_Olambda__164,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_abq(int,fun(int,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),aa(bool,int,zero_neq_one_of_bool(int),aa(bool,bool,fNot,aa(int,bool,aa(int,fun(int,bool),fequal(int),Uua),zero_zero(int))))) ).

% ATP.lambda_164
tff(fact_8346_ATP_Olambda__165,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_ss(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_165
tff(fact_8347_ATP_Olambda__166,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_sl(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_166
tff(fact_8348_ATP_Olambda__167,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_nq(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,Uu),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ).

% ATP.lambda_167
tff(fact_8349_ATP_Olambda__168,axiom,
    ! [A: $tType] :
      ( ( comple5582772986160207858norder(A)
        & canoni5634975068530333245id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_nr(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),aa(nat,set(nat),set_ord_lessThan(nat),Uua)) ) ).

% ATP.lambda_168
tff(fact_8350_ATP_Olambda__169,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_yu(real,fun(A,bool),Uu),Uua))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Uu),real_V7770717601297561774m_norm(A,Uua))) ) ) ).

% ATP.lambda_169
tff(fact_8351_ATP_Olambda__170,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_wf(set(product_prod(A,A)),fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),aa(set(product_prod(A,A)),nat,finite_card(product_prod(A,A)),Uu))) ) ).

% ATP.lambda_170
tff(fact_8352_ATP_Olambda__171,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_sc(A,fun(nat,A),Uu),Uua) = divide_divide(A,Uu,aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_171
tff(fact_8353_ATP_Olambda__172,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( pp(aa(list(A),bool,aTP_Lamp_aci(nat,fun(list(A),bool),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uu),aa(list(A),nat,size_size(list(A)),Uua))) ) ).

% ATP.lambda_172
tff(fact_8354_ATP_Olambda__173,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gz(A,fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_173
tff(fact_8355_ATP_Olambda__174,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_gu(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_174
tff(fact_8356_ATP_Olambda__175,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_aff(A,fun(list(A),bool)),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uu),aa(list(A),set(A),set2(A),Uua))) ) ).

% ATP.lambda_175
tff(fact_8357_ATP_Olambda__176,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_acf(list(A),fun(A,bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),aa(list(A),set(A),set2(A),Uu))) ) ).

% ATP.lambda_176
tff(fact_8358_ATP_Olambda__177,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_adg(list(A),fun(A,bool),Uu),Uua))
    <=> ( Uua = aa(list(A),A,hd(A),Uu) ) ) ).

% ATP.lambda_177
tff(fact_8359_ATP_Olambda__178,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_lf(nat,fun(nat,bool)),Uu),Uua))
    <=> ( Uua = aa(nat,nat,suc,Uu) ) ) ).

% ATP.lambda_178
tff(fact_8360_ATP_Olambda__179,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( pp(aa(set(A),bool,aTP_Lamp_jp(set(A),fun(set(A),bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uu)) ) ).

% ATP.lambda_179
tff(fact_8361_ATP_Olambda__180,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_jo(nat,fun(nat,bool)),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uua),Uu)) ) ).

% ATP.lambda_180
tff(fact_8362_ATP_Olambda__181,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_qo(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_181
tff(fact_8363_ATP_Olambda__182,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_vr(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_182
tff(fact_8364_ATP_Olambda__183,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_cs(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Uu)) ) ) ).

% ATP.lambda_183
tff(fact_8365_ATP_Olambda__184,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_nx(nat,fun(nat,nat),Uu),Uua) = modulo_modulo(nat,Uua,Uu) ).

% ATP.lambda_184
tff(fact_8366_ATP_Olambda__185,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_tj(A,fun(A,A),Uu),Uua) = divide_divide(A,Uua,Uu) ) ).

% ATP.lambda_185
tff(fact_8367_ATP_Olambda__186,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mj(A,fun(A,A),Uu),Uua) = divide_divide(A,Uua,Uu) ) ).

% ATP.lambda_186
tff(fact_8368_ATP_Olambda__187,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_yf(A,fun(A,A),Uu),Uua) = divide_divide(A,Uua,Uu) ) ).

% ATP.lambda_187
tff(fact_8369_ATP_Olambda__188,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ea(nat,fun(nat,bool)),Uu),Uua))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uua),Uu)) ) ).

% ATP.lambda_188
tff(fact_8370_ATP_Olambda__189,axiom,
    ! [A: $tType] :
      ( bounde4967611905675639751up_bot(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_qp(A,fun(A,bool)),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_189
tff(fact_8371_ATP_Olambda__190,axiom,
    ! [A: $tType] :
      ( unboun7993243217541854897norder(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_vs(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_190
tff(fact_8372_ATP_Olambda__191,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ek(A,fun(A,bool),Uu),Uua))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),Uu)) ) ) ).

% ATP.lambda_191
tff(fact_8373_ATP_Olambda__192,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_sa(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uu) ).

% ATP.lambda_192
tff(fact_8374_ATP_Olambda__193,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mv(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_193
tff(fact_8375_ATP_Olambda__194,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_mg(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,minus_minus(nat,Uua),Uu) ).

% ATP.lambda_194
tff(fact_8376_ATP_Olambda__195,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mi(A,fun(A,A),Uu),Uua) = aa(A,A,minus_minus(A,Uua),Uu) ) ).

% ATP.lambda_195
tff(fact_8377_ATP_Olambda__196,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_ol(nat,fun(real,real),Uu),Uua) = aa(nat,real,power_power(real,Uua),Uu) ).

% ATP.lambda_196
tff(fact_8378_ATP_Olambda__197,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_abi(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).

% ATP.lambda_197
tff(fact_8379_ATP_Olambda__198,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_ni(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).

% ATP.lambda_198
tff(fact_8380_ATP_Olambda__199,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_yg(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_199
tff(fact_8381_ATP_Olambda__200,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_mh(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_200
tff(fact_8382_ATP_Olambda__201,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_on(real,fun(real,real),Uu),Uua) = powr(real,Uua,Uu) ).

% ATP.lambda_201
tff(fact_8383_ATP_Olambda__202,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_jg(nat,fun(nat,bool),Uu),Uua))
    <=> pp(dvd_dvd(nat,Uua,Uu)) ) ).

% ATP.lambda_202
tff(fact_8384_ATP_Olambda__203,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_ae(A,fun(A,bool),Uu),Uua))
        <=> pp(dvd_dvd(A,Uua,Uu)) ) ) ).

% ATP.lambda_203
tff(fact_8385_ATP_Olambda__204,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_abl(B,fun(A,product_prod(A,B))),Uu),Uua) = aa(B,product_prod(A,B),product_Pair(A,B,Uua),Uu) ).

% ATP.lambda_204
tff(fact_8386_ATP_Olambda__205,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_abr(A,fun(nat,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),product_Pair(nat,A,Uua),Uu) ).

% ATP.lambda_205
tff(fact_8387_ATP_Olambda__206,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_cn(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(Uua),Uu) ).

% ATP.lambda_206
tff(fact_8388_ATP_Olambda__207,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_abc(list(A),fun(A,list(A))),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uu) ).

% ATP.lambda_207
tff(fact_8389_ATP_Olambda__208,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_sj(real,fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),Uu) ).

% ATP.lambda_208
tff(fact_8390_ATP_Olambda__209,axiom,
    ! [B: $tType,Uu: set(B),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_yn(set(B),fun(B,bool),Uu),Uua))
    <=> pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uua),Uu)) ) ).

% ATP.lambda_209
tff(fact_8391_ATP_Olambda__210,axiom,
    ! [A: $tType] :
      ( topological_t3_space(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_yb(set(A),fun(A,bool),Uu),Uua))
        <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ) ).

% ATP.lambda_210
tff(fact_8392_ATP_Olambda__211,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_a(set(A),fun(A,bool),Uu),Uua))
    <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ).

% ATP.lambda_211
tff(fact_8393_ATP_Olambda__212,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat] : aa(nat,set(product_prod(A,A)),aTP_Lamp_wd(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_212
tff(fact_8394_ATP_Olambda__213,axiom,
    ! [A: $tType,Uu: list(A),Uua: nat] : aa(nat,list(A),aTP_Lamp_zk(list(A),fun(nat,list(A)),Uu),Uua) = drop(A,Uua,Uu) ).

% ATP.lambda_213
tff(fact_8395_ATP_Olambda__214,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_ach(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ).

% ATP.lambda_214
tff(fact_8396_ATP_Olambda__215,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_ade(A,fun(A,bool),Uu),Uua))
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_215
tff(fact_8397_ATP_Olambda__216,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_abg(A,fun(list(A),list(A))),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ).

% ATP.lambda_216
tff(fact_8398_ATP_Olambda__217,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(list(A)),aa(A,fun(list(A),list(list(A))),aTP_Lamp_abh(A,fun(list(A),list(list(A)))),Uu),Uua) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Uua),nil(list(A))) ).

% ATP.lambda_217
tff(fact_8399_ATP_Olambda__218,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_ue(fun(A,real),fun(A,bool),Uu),Uua))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),zero_zero(real)),aa(A,real,Uu,Uua))) ) ).

% ATP.lambda_218
tff(fact_8400_ATP_Olambda__219,axiom,
    ! [B: $tType,Uu: fun(B,real),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_uc(fun(B,real),fun(B,bool),Uu),Uua))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(B,real,Uu,Uua))) ) ).

% ATP.lambda_219
tff(fact_8401_ATP_Olambda__220,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_wp(fun(A,real),fun(A,bool),Uu),Uua))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),one_one(real)),aa(A,real,Uu,Uua))) ) ) ).

% ATP.lambda_220
tff(fact_8402_ATP_Olambda__221,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_um(fun(A,real),fun(A,bool),Uu),Uua))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,Uu,Uua))) ) ).

% ATP.lambda_221
tff(fact_8403_ATP_Olambda__222,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A] : aa(A,set(B),aTP_Lamp_adx(set(product_prod(A,B)),fun(A,set(B)),Uu),Uua) = aa(set(product_prod(A,B)),set(B),image(product_prod(A,B),B,product_snd(A,B)),Uu) ).

% ATP.lambda_222
tff(fact_8404_ATP_Olambda__223,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_fa(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),aa(num,num,bit0,one2))),Uua)),one_one(nat))) ).

% ATP.lambda_223
tff(fact_8405_ATP_Olambda__224,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ez(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),aa(num,num,bit0,one2))),Uua)) ).

% ATP.lambda_224
tff(fact_8406_ATP_Olambda__225,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_ve(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,Uu,divide_divide(A,one_one(A),Uua)) ) ).

% ATP.lambda_225
tff(fact_8407_ATP_Olambda__226,axiom,
    ! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_acm(fun(nat,bool),fun(product_prod(A,nat),bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(nat,nat,suc,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua)))) ) ).

% ATP.lambda_226
tff(fact_8408_ATP_Olambda__227,axiom,
    ! [A: $tType,Uu: fun(nat,bool),Uua: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aTP_Lamp_acn(fun(nat,bool),fun(product_prod(A,nat),bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(product_prod(A,nat),nat,product_snd(A,nat),Uua))) ) ).

% ATP.lambda_227
tff(fact_8409_ATP_Olambda__228,axiom,
    ! [Uu: fun(nat,bool),Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_tg(fun(nat,bool),fun(nat,bool),Uu),Uua))
    <=> pp(aa(nat,bool,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_228
tff(fact_8410_ATP_Olambda__229,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_as(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_229
tff(fact_8411_ATP_Olambda__230,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ry(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_230
tff(fact_8412_ATP_Olambda__231,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_bk(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_231
tff(fact_8413_ATP_Olambda__232,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_gi(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_232
tff(fact_8414_ATP_Olambda__233,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_cp(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_233
tff(fact_8415_ATP_Olambda__234,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_rx(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_234
tff(fact_8416_ATP_Olambda__235,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_adz(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),ord_min(A),Uu),Uua)) ) ).

% ATP.lambda_235
tff(fact_8417_ATP_Olambda__236,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_aec(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),ord_max(A),Uu),Uua)) ) ).

% ATP.lambda_236
tff(fact_8418_ATP_Olambda__237,axiom,
    ! [A: $tType] :
      ( semilattice_sup(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_aed(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),sup_sup(A),Uu),Uua)) ) ).

% ATP.lambda_237
tff(fact_8419_ATP_Olambda__238,axiom,
    ! [A: $tType] :
      ( semilattice_inf(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_aee(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,aa(A,fun(A,A),inf_inf(A),Uu),Uua)) ) ).

% ATP.lambda_238
tff(fact_8420_ATP_Olambda__239,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_lx(nat,fun(num,option(num)),Uu),Uua) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_239
tff(fact_8421_ATP_Olambda__240,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_ls(num,fun(nat,option(num)),Uu),Uua) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(Uua,Uu))) ).

% ATP.lambda_240
tff(fact_8422_ATP_Olambda__241,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),aTP_Lamp_abp(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),Uu),Uua) = product_case_prod(A,list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_abo(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua)) ).

% ATP.lambda_241
tff(fact_8423_ATP_Olambda__242,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_le(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_ld(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_242
tff(fact_8424_ATP_Olambda__243,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_lc(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_lb(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_243
tff(fact_8425_ATP_Olambda__244,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),bool),aa(nat,fun(nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_la(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = product_case_prod(nat,nat,bool,aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_kz(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).

% ATP.lambda_244
tff(fact_8426_ATP_Olambda__245,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,fun(product_prod(nat,nat),bool),aa(nat,fun(nat,fun(product_prod(nat,nat),bool)),aTP_Lamp_ky(nat,fun(nat,fun(product_prod(nat,nat),bool))),Uu),Uua) = product_case_prod(nat,nat,bool,aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_kx(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua)) ).

% ATP.lambda_245
tff(fact_8427_ATP_Olambda__246,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_kw(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_kv(nat,fun(nat,fun(nat,fun(nat,product_prod(nat,nat)))),Uu),Uua)) ).

% ATP.lambda_246
tff(fact_8428_ATP_Olambda__247,axiom,
    ! [Uu: fun(nat,real),Uua: real] : aa(real,real,aTP_Lamp_ox(fun(nat,real),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),aTP_Lamp_ow(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua)) ).

% ATP.lambda_247
tff(fact_8429_ATP_Olambda__248,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A] : aa(A,A,aTP_Lamp_om(fun(nat,A),fun(A,A),Uu),Uua) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_bt(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua)) ) ).

% ATP.lambda_248
tff(fact_8430_ATP_Olambda__249,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: real,Uua: A] : aa(A,set(A),aTP_Lamp_vp(real,fun(A,set(A)),Uu),Uua) = collect(A,aa(A,fun(A,bool),aTP_Lamp_vo(real,fun(A,fun(A,bool)),Uu),Uua)) ) ).

% ATP.lambda_249
tff(fact_8431_ATP_Olambda__250,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: B] : aa(B,set(A),aTP_Lamp_nv(fun(A,fun(B,bool)),fun(B,set(A)),Uu),Uua) = collect(A,aa(B,fun(A,bool),aTP_Lamp_nu(fun(A,fun(B,bool)),fun(B,fun(A,bool)),Uu),Uua)) ).

% ATP.lambda_250
tff(fact_8432_ATP_Olambda__251,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,complex,aTP_Lamp_fk(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),aa(num,num,bit0,one2))),pi)),aa(nat,real,semiring_1_of_nat(real),Uua)),aa(nat,real,semiring_1_of_nat(real),Uu))) ).

% ATP.lambda_251
tff(fact_8433_ATP_Olambda__252,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_adm(fun(A,option(B)),fun(A,bool),Uu),Uua))
    <=> ( aa(A,option(B),Uu,Uua) != none(B) ) ) ).

% ATP.lambda_252
tff(fact_8434_ATP_Olambda__253,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_adi(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_adh(product_prod(A,C),fun(product_prod(C,B),list(product_prod(A,B))),Uua)),Uu)) ).

% ATP.lambda_253
tff(fact_8435_ATP_Olambda__254,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_cc(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,power_power(A,Uu),Uua))) ) ).

% ATP.lambda_254
tff(fact_8436_ATP_Olambda__255,axiom,
    ! [Uu: nat,Uua: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_ah(nat,fun(nat,bool),Uu),Uua))
    <=> ~ pp(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),divide_divide(nat,Uu,aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)))) ) ).

% ATP.lambda_255
tff(fact_8437_ATP_Olambda__256,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_cd(A,fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),Uu)),Uua))) ) ).

% ATP.lambda_256
tff(fact_8438_ATP_Olambda__257,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_ca(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,power_power(A,Uu),Uua))) ) ).

% ATP.lambda_257
tff(fact_8439_ATP_Olambda__258,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_cb(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,power_power(A,Uu),Uua))) ) ).

% ATP.lambda_258
tff(fact_8440_ATP_Olambda__259,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat] : aa(nat,set(A),aTP_Lamp_nw(fun(nat,set(A)),fun(nat,set(A)),Uu),Uua) = aa(set(set(A)),set(A),complete_Sup_Sup(set(A)),aa(set(nat),set(set(A)),image(nat,set(A),Uu),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua))) ).

% ATP.lambda_259
tff(fact_8441_ATP_Olambda__260,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_aj(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_260
tff(fact_8442_ATP_Olambda__261,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_ai(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_261
tff(fact_8443_ATP_Olambda__262,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_ace(list(A),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),aa(list(A),set(A),set2(A),Uu))) ) ).

% ATP.lambda_262
tff(fact_8444_ATP_Olambda__263,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,set(nat),aTP_Lamp_aeb(nat,fun(nat,set(nat)),Uu),Uua) = aa(nat,set(nat),set_ord_atMost(nat),aa(nat,nat,minus_minus(nat,Uu),Uua)) ).

% ATP.lambda_263
tff(fact_8445_ATP_Olambda__264,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_sq(real,fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,power_power(real,Uu),Uua)) ).

% ATP.lambda_264
tff(fact_8446_ATP_Olambda__265,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_oh(A,fun(A,A),Uu),Uua) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu)) ) ).

% ATP.lambda_265
tff(fact_8447_ATP_Olambda__266,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: A] : aa(A,filter(A),aTP_Lamp_yq(A,fun(A,filter(A)),Uu),Uua) = principal(A,set_or5935395276787703475ssThan(A,Uu,Uua)) ) ).

% ATP.lambda_266
tff(fact_8448_ATP_Olambda__267,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: A,Uua: A] : aa(A,filter(A),aTP_Lamp_yr(A,fun(A,filter(A)),Uu),Uua) = principal(A,set_or5935395276787703475ssThan(A,Uua,Uu)) ) ).

% ATP.lambda_267
tff(fact_8449_ATP_Olambda__268,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_zl(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_268
tff(fact_8450_ATP_Olambda__269,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_zm(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_269
tff(fact_8451_ATP_Olambda__270,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_il(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_270
tff(fact_8452_ATP_Olambda__271,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ik(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_271
tff(fact_8453_ATP_Olambda__272,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_aeu(set(A),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uua),Uu)) ) ).

% ATP.lambda_272
tff(fact_8454_ATP_Olambda__273,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_acb(A,fun(A,bool),Uu),Uua))
    <=> ( Uu != Uua ) ) ).

% ATP.lambda_273
tff(fact_8455_ATP_Olambda__274,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_acd(A,fun(A,bool),Uu),Uua))
    <=> ( Uua != Uu ) ) ).

% ATP.lambda_274
tff(fact_8456_ATP_Olambda__275,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_bb(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_275
tff(fact_8457_ATP_Olambda__276,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_bd(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_276
tff(fact_8458_ATP_Olambda__277,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_cm(fun(B,A),fun(B,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_277
tff(fact_8459_ATP_Olambda__278,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_gf(fun(B,A),fun(B,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_278
tff(fact_8460_ATP_Olambda__279,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_sh(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_279
tff(fact_8461_ATP_Olambda__280,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_oo(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_280
tff(fact_8462_ATP_Olambda__281,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_va(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_281
tff(fact_8463_ATP_Olambda__282,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_pw(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_282
tff(fact_8464_ATP_Olambda__283,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_jm(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_283
tff(fact_8465_ATP_Olambda__284,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_gr(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_284
tff(fact_8466_ATP_Olambda__285,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_fq(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_285
tff(fact_8467_ATP_Olambda__286,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_xm(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,artanh(real),aa(real,real,Uu,Uua)) ).

% ATP.lambda_286
tff(fact_8468_ATP_Olambda__287,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_wy(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_287
tff(fact_8469_ATP_Olambda__288,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_qv(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_288
tff(fact_8470_ATP_Olambda__289,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_qe(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_289
tff(fact_8471_ATP_Olambda__290,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ph(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_290
tff(fact_8472_ATP_Olambda__291,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_xl(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_291
tff(fact_8473_ATP_Olambda__292,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_xg(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,arcosh(real),aa(real,real,Uu,Uua)) ).

% ATP.lambda_292
tff(fact_8474_ATP_Olambda__293,axiom,
    ! [B: $tType,Uu: fun(B,real),Uua: B] : aa(B,real,aTP_Lamp_ra(fun(B,real),fun(B,real),Uu),Uua) = aa(real,real,arcosh(real),aa(B,real,Uu,Uua)) ).

% ATP.lambda_293
tff(fact_8475_ATP_Olambda__294,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_wq(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_294
tff(fact_8476_ATP_Olambda__295,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_pj(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_295
tff(fact_8477_ATP_Olambda__296,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_xk(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_296
tff(fact_8478_ATP_Olambda__297,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_bc(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_297
tff(fact_8479_ATP_Olambda__298,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere166539214618696060dd_abs(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ef(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,abs_abs(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_298
tff(fact_8480_ATP_Olambda__299,axiom,
    ! [A9: $tType] :
      ( ( real_Vector_banach(A9)
        & real_V3459762299906320749_field(A9) )
     => ! [Uu: fun(A9,A9),Uua: A9] : aa(A9,A9,aTP_Lamp_ou(fun(A9,A9),fun(A9,A9),Uu),Uua) = aa(A9,A9,tanh(A9),aa(A9,A9,Uu,Uua)) ) ).

% ATP.lambda_299
tff(fact_8481_ATP_Olambda__300,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_qg(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,tan(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_300
tff(fact_8482_ATP_Olambda__301,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_tb(fun(nat,real),fun(nat,real),Uu),Uua) = cos(real,aa(nat,real,Uu,Uua)) ).

% ATP.lambda_301
tff(fact_8483_ATP_Olambda__302,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,option(A),aTP_Lamp_mc(fun(B,A),fun(B,option(A)),Uu),Uua) = aa(A,option(A),some(A),aa(B,A,Uu,Uua)) ).

% ATP.lambda_302
tff(fact_8484_ATP_Olambda__303,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(C,A),Uua: C] : aa(C,fun(B,product_prod(A,B)),aTP_Lamp_aap(fun(C,A),fun(C,fun(B,product_prod(A,B))),Uu),Uua) = product_Pair(A,B,aa(C,A,Uu,Uua)) ).

% ATP.lambda_303
tff(fact_8485_ATP_Olambda__304,axiom,
    ! [C: $tType,D: $tType,Uu: fun(D,set(C)),Uua: D] : aa(D,filter(C),aTP_Lamp_ys(fun(D,set(C)),fun(D,filter(C)),Uu),Uua) = principal(C,aa(D,set(C),Uu,Uua)) ).

% ATP.lambda_304
tff(fact_8486_ATP_Olambda__305,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,filter(B),aTP_Lamp_yt(fun(A,set(B)),fun(A,filter(B)),Uu),Uua) = principal(B,aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_305
tff(fact_8487_ATP_Olambda__306,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,set(B)),Uua: A] : aa(A,nat,aTP_Lamp_nl(fun(A,set(B)),fun(A,nat),Uu),Uua) = aa(set(B),nat,finite_card(B),aa(A,set(B),Uu,Uua)) ).

% ATP.lambda_306
tff(fact_8488_ATP_Olambda__307,axiom,
    ! [Uu: fun(real,fun(nat,real)),Uua: real] : aa(real,real,aTP_Lamp_os(fun(real,fun(nat,real)),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),Uu,Uua)) ).

% ATP.lambda_307
tff(fact_8489_ATP_Olambda__308,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_rt(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_308
tff(fact_8490_ATP_Olambda__309,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,list(A)),Uua: B] : aa(B,set(A),aTP_Lamp_yp(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_309
tff(fact_8491_ATP_Olambda__310,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_qc(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_310
tff(fact_8492_ATP_Olambda__311,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: B] : aa(B,set(set(A)),aTP_Lamp_nf(fun(B,set(A)),fun(B,set(set(A))),Uu),Uua) = pow2(A,aa(B,set(A),Uu,Uua)) ).

% ATP.lambda_311
tff(fact_8493_ATP_Olambda__312,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,nat,aTP_Lamp_kg(fun(A,nat),fun(A,nat),Uu),Uua) = aa(nat,nat,suc,aa(A,nat,Uu,Uua)) ).

% ATP.lambda_312
tff(fact_8494_ATP_Olambda__313,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_aca(fun(A,bool),fun(A,bool),Uu),Uua))
    <=> ~ pp(aa(A,bool,Uu,Uua)) ) ).

% ATP.lambda_313
tff(fact_8495_ATP_Olambda__314,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_nt(fun(A,fun(B,bool)),fun(A,bool),Uu),Uua))
    <=> ? [X_1: B] : pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uua),X_1)) ) ).

% ATP.lambda_314
tff(fact_8496_ATP_Olambda__315,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] : aa(set(A),filter(set(A)),aTP_Lamp_aah(set(A),fun(set(A),filter(set(A))),Uu),Uua) = principal(set(A),collect(set(A),aa(set(A),fun(set(A),bool),aTP_Lamp_aag(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua))) ).

% ATP.lambda_315
tff(fact_8497_ATP_Olambda__316,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real] : aa(real,filter(A),aTP_Lamp_yx(A,fun(real,filter(A)),Uu),Uua) = principal(A,collect(A,aa(real,fun(A,bool),aTP_Lamp_yw(A,fun(real,fun(A,bool)),Uu),Uua))) ) ).

% ATP.lambda_316
tff(fact_8498_ATP_Olambda__317,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(B,fun(A,bool)),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_uo(fun(B,fun(A,bool)),fun(A,bool),Uu),Uua))
        <=> ? [I3: B] : pp(aa(A,bool,aa(B,fun(A,bool),Uu,I3),Uua)) ) ) ).

% ATP.lambda_317
tff(fact_8499_ATP_Olambda__318,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_ag(list(A),fun(A,bool),Uu),Uua))
    <=> ? [I3: nat] :
          ( ( Uua = aa(nat,A,nth(A,Uu),I3) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Uu))) ) ) ).

% ATP.lambda_318
tff(fact_8500_ATP_Olambda__319,axiom,
    ! [A: $tType] :
      ( comple592849572758109894attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_nk(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F6: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image(set(A),A,F6),Uu) )
              & ! [X4: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),Uu))
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,F6,X4)),X4)) ) ) ) ) ).

% ATP.lambda_319
tff(fact_8501_ATP_Olambda__320,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_nj(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F6: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image(set(A),A,F6),Uu) )
              & ! [X4: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),Uu))
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,F6,X4)),X4)) ) ) ) ) ).

% ATP.lambda_320
tff(fact_8502_ATP_Olambda__321,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu: set(set(A)),Uua: set(A)] :
          ( pp(aa(set(A),bool,aTP_Lamp_ng(set(set(A)),fun(set(A),bool),Uu),Uua))
        <=> ? [F6: fun(set(A),A)] :
              ( ( Uua = aa(set(set(A)),set(A),image(set(A),A,F6),Uu) )
              & ! [X4: set(A)] :
                  ( pp(aa(set(set(A)),bool,aa(set(A),fun(set(set(A)),bool),member(set(A)),X4),Uu))
                 => pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),aa(set(A),A,F6,X4)),X4)) ) ) ) ) ).

% ATP.lambda_321
tff(fact_8503_ATP_Olambda__322,axiom,
    ! [A: $tType,Uu: set(filter(A)),Uua: filter(A)] :
      ( pp(aa(filter(A),bool,aTP_Lamp_no(set(filter(A)),fun(filter(A),bool),Uu),Uua))
    <=> ! [X4: filter(A)] :
          ( pp(aa(set(filter(A)),bool,aa(filter(A),fun(set(filter(A)),bool),member(filter(A)),X4),Uu))
         => pp(aa(filter(A),bool,aa(filter(A),fun(filter(A),bool),ord_less_eq(filter(A)),Uua),X4)) ) ) ).

% ATP.lambda_322
tff(fact_8504_ATP_Olambda__323,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_lk(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),X4)) ) ) ) ).

% ATP.lambda_323
tff(fact_8505_ATP_Olambda__324,axiom,
    ! [A: $tType] :
      ( comple6319245703460814977attice(A)
     => ! [Uu: set(A),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_lj(set(A),fun(A,bool),Uu),Uua))
        <=> ! [X4: A] :
              ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),Uu))
             => pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),X4),Uua)) ) ) ) ).

% ATP.lambda_324
tff(fact_8506_ATP_Olambda__325,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(A,bool),Uua: A] :
          ( pp(aa(A,bool,aTP_Lamp_to(fun(A,bool),fun(A,bool),Uu),Uua))
        <=> ! [Y5: A] :
              ( pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),Y5))
             => pp(aa(A,bool,Uu,Y5)) ) ) ) ).

% ATP.lambda_325
tff(fact_8507_ATP_Olambda__326,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: B] :
      ( pp(aa(B,bool,aTP_Lamp_aab(fun(A,option(B)),fun(B,bool),Uu),Uua))
    <=> ? [A6: A] : aa(A,option(B),Uu,A6) = aa(B,option(B),some(B),Uua) ) ).

% ATP.lambda_326
tff(fact_8508_ATP_Olambda__327,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: A] :
      ( pp(aa(A,bool,aTP_Lamp_ns(fun(B,A),fun(A,bool),Uu),Uua))
    <=> ? [X4: B] : Uua = aa(B,A,Uu,X4) ) ).

% ATP.lambda_327
tff(fact_8509_ATP_Olambda__328,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aTP_Lamp_adw(fun(A,option(B)),fun(product_prod(A,B),bool),Uu),Uua))
    <=> ? [A6: A,B6: B] :
          ( ( Uua = aa(B,product_prod(A,B),product_Pair(A,B,A6),B6) )
          & ( aa(A,option(B),Uu,A6) = aa(B,option(B),some(B),B6) ) ) ) ).

% ATP.lambda_328
tff(fact_8510_ATP_Olambda__329,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),option(A),aa(A,fun(list(A),option(A)),aTP_Lamp_adf(A,fun(list(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),Uu) ).

% ATP.lambda_329
tff(fact_8511_ATP_Olambda__330,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,set(B),aTP_Lamp_adp(list(B),fun(A,set(B)),Uu),Uua) = aa(list(B),set(B),set2(B),Uu) ).

% ATP.lambda_330
tff(fact_8512_ATP_Olambda__331,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(list(A)),aTP_Lamp_aep(set(A),fun(list(A),set(list(A))),Uu),Uua) = lists(A,Uu) ).

% ATP.lambda_331
tff(fact_8513_ATP_Olambda__332,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_bp(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = if(real,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uub),aa(nat,real,Uua,divide_divide(nat,Uub,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),aa(nat,real,Uu,divide_divide(nat,aa(nat,nat,minus_minus(nat,Uub),one_one(nat)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ).

% ATP.lambda_332
tff(fact_8514_ATP_Olambda__333,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_ey(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = if(real,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uub),aa(nat,real,Uu,Uub),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_333
tff(fact_8515_ATP_Olambda__334,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_if(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),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),aa(num,num,bit0,one2))),Uua)),one_one(code_integer))),aa(code_integer,code_integer,minus_minus(code_integer,Uub),aa(num,code_integer,numeral_numeral(code_integer),Uu))),aa(code_integer,product_prod(code_integer,code_integer),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),aa(num,num,bit0,one2))),Uua)),Uub)) ).

% ATP.lambda_334
tff(fact_8516_ATP_Olambda__335,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] : aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_ha(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = if(product_prod(nat,nat),aa(nat,bool,aa(nat,fun(nat,bool),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),aa(num,num,bit0,one2))),Uua)),one_one(nat))),aa(nat,nat,minus_minus(nat,Uub),aa(num,nat,numeral_numeral(nat),Uu))),aa(nat,product_prod(nat,nat),product_Pair(nat,nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua)),Uub)) ).

% ATP.lambda_335
tff(fact_8517_ATP_Olambda__336,axiom,
    ! [Uu: num,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_hb(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = if(product_prod(int,int),aa(int,bool,aa(int,fun(int,bool),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),aa(num,num,bit0,one2))),Uua)),one_one(int))),aa(int,int,minus_minus(int,Uub),aa(num,int,numeral_numeral(int),Uu))),aa(int,product_prod(int,int),product_Pair(int,int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),aa(num,num,bit0,one2))),Uua)),Uub)) ).

% ATP.lambda_336
tff(fact_8518_ATP_Olambda__337,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_he(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = if(product_prod(A,A),aa(A,bool,aa(A,fun(A,bool),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),aa(num,num,bit0,one2))),Uua)),one_one(A))),aa(A,A,minus_minus(A,Uub),aa(num,A,numeral_numeral(A),Uu))),aa(A,product_prod(A,A),product_Pair(A,A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2))),Uua)),Uub)) ) ).

% ATP.lambda_337
tff(fact_8519_ATP_Olambda__338,axiom,
    ! [A: $tType,Uu: A,Uua: A,Uub: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_zt(A,fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = if(list(A),aa(A,bool,aa(A,fun(A,bool),fequal(A),Uu),Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub))) ).

% ATP.lambda_338
tff(fact_8520_ATP_Olambda__339,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] : aa(B,A,aa(set(B),fun(B,A),aTP_Lamp_mp(fun(B,A),fun(set(B),fun(B,A)),Uu),Uua),Uub) = if(A,aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uua),aa(B,A,Uu,Uub),one_one(A)) ) ).

% ATP.lambda_339
tff(fact_8521_ATP_Olambda__340,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_is(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uu),Uub),aa(B,A,Uua,Uub),one_one(A)) ) ).

% ATP.lambda_340
tff(fact_8522_ATP_Olambda__341,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ir(B,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uub),Uu),aa(B,A,Uua,Uub),one_one(A)) ) ).

% ATP.lambda_341
tff(fact_8523_ATP_Olambda__342,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_ip(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),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,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uu)),Uub))) ).

% ATP.lambda_342
tff(fact_8524_ATP_Olambda__343,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_io(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = if(product_prod(code_integer,code_integer),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),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,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,minus_minus(code_integer,Uu),Uub))) ).

% ATP.lambda_343
tff(fact_8525_ATP_Olambda__344,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,A,aa(fun(B,bool),fun(B,A),aTP_Lamp_jd(fun(B,A),fun(fun(B,bool),fun(B,A)),Uu),Uua),Uub) = if(A,aa(B,bool,Uua,Uub),aa(B,A,Uu,Uub),one_one(A)) ) ).

% ATP.lambda_344
tff(fact_8526_ATP_Olambda__345,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,A)),Uua: fun(B,bool),Uub: B] : aa(B,fun(A,A),aa(fun(B,bool),fun(B,fun(A,A)),aTP_Lamp_aex(fun(B,fun(A,A)),fun(fun(B,bool),fun(B,fun(A,A))),Uu),Uua),Uub) = if(fun(A,A),aa(B,bool,Uua,Uub),aa(B,fun(A,A),Uu,Uub),id(A)) ).

% ATP.lambda_345
tff(fact_8527_ATP_Olambda__346,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: fun(B,bool),Uub: B] : aa(B,option(A),aa(fun(B,bool),fun(B,option(A)),aTP_Lamp_ack(fun(B,A),fun(fun(B,bool),fun(B,option(A))),Uu),Uua),Uub) = if(option(A),aa(B,bool,Uua,Uub),aa(A,option(A),some(A),aa(B,A,Uu,Uub)),none(A)) ).

% ATP.lambda_346
tff(fact_8528_ATP_Olambda__347,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_yk(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_347
tff(fact_8529_ATP_Olambda__348,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_hj(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,minus_minus(nat,Uua),Uub)) ) ).

% ATP.lambda_348
tff(fact_8530_ATP_Olambda__349,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_hm(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,minus_minus(nat,Uua),Uub)) ) ).

% ATP.lambda_349
tff(fact_8531_ATP_Olambda__350,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(A,B),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_ti(fun(A,fun(B,bool)),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uub),aa(A,B,Uua,Uub))) ) ).

% ATP.lambda_350
tff(fact_8532_ATP_Olambda__351,axiom,
    ! [Uu: fun(real,fun(nat,real)),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_or(fun(real,fun(nat,real)),fun(nat,fun(real,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),Uu,Uub),Uua) ).

% ATP.lambda_351
tff(fact_8533_ATP_Olambda__352,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_gp(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_352
tff(fact_8534_ATP_Olambda__353,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_fo(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_353
tff(fact_8535_ATP_Olambda__354,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,bool)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_jr(fun(B,fun(A,bool)),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(A,bool,aa(B,fun(A,bool),Uu,Uub),Uua)) ) ).

% ATP.lambda_354
tff(fact_8536_ATP_Olambda__355,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,bool)),Uua: B,Uub: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_nu(fun(A,fun(B,bool)),fun(B,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uub),Uua)) ) ).

% ATP.lambda_355
tff(fact_8537_ATP_Olambda__356,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_rc(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_356
tff(fact_8538_ATP_Olambda__357,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_ada(fun(A,fun(A,bool)),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(A,bool,aa(A,fun(A,bool),Uu,Uub),Uua)) ) ).

% ATP.lambda_357
tff(fact_8539_ATP_Olambda__358,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_ee(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ed(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_358
tff(fact_8540_ATP_Olambda__359,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ec(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_eb(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_359
tff(fact_8541_ATP_Olambda__360,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_cl(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_ck(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uub)) ) ).

% ATP.lambda_360
tff(fact_8542_ATP_Olambda__361,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_bm(nat,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uub),Uu),one_one(A),zero_zero(A))),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_361
tff(fact_8543_ATP_Olambda__362,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] : aa(code_integer,product_prod(code_integer,bool),aa(code_integer,fun(code_integer,product_prod(code_integer,bool)),aTP_Lamp_in(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,bool))),Uu),Uua),Uub) = aa(bool,product_prod(code_integer,bool),product_Pair(code_integer,bool,if(code_integer,aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),ord_less(code_integer),zero_zero(code_integer)),Uu),Uua,aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),Uub))),aa(code_integer,bool,aa(code_integer,fun(code_integer,bool),fequal(code_integer),Uub),one_one(code_integer))) ).

% ATP.lambda_362
tff(fact_8544_ATP_Olambda__363,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_gq(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_gp(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_363
tff(fact_8545_ATP_Olambda__364,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_fp(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_fo(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_364
tff(fact_8546_ATP_Olambda__365,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_wt(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_at(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) ) ).

% ATP.lambda_365
tff(fact_8547_ATP_Olambda__366,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_oy(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,power_power(real,Uua),Uub)) ).

% ATP.lambda_366
tff(fact_8548_ATP_Olambda__367,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_oz(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,power_power(real,Uu),Uub)) ).

% ATP.lambda_367
tff(fact_8549_ATP_Olambda__368,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_ew(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,power_power(real,Uu),Uub)) ) ).

% ATP.lambda_368
tff(fact_8550_ATP_Olambda__369,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] :
          ( pp(aa(A,bool,aa(nat,fun(A,bool),aTP_Lamp_jn(fun(nat,A),fun(nat,fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_cr(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),aa(nat,set(nat),set_ord_atMost(nat),Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_369
tff(fact_8551_ATP_Olambda__370,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_rq(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(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_370
tff(fact_8552_ATP_Olambda__371,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_rn(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(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_371
tff(fact_8553_ATP_Olambda__372,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_iq(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,power_power(A,zero_zero(A)),Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_372
tff(fact_8554_ATP_Olambda__373,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_qy(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,Uu,Uub)),aa(A,A,Uu,Uua)),aa(A,A,minus_minus(A,Uub),Uua)) ) ).

% ATP.lambda_373
tff(fact_8555_ATP_Olambda__374,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_ro(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,Uu,Uub)),aa(A,A,Uu,Uua)),aa(A,A,minus_minus(A,Uub),Uua)) ) ).

% ATP.lambda_374
tff(fact_8556_ATP_Olambda__375,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_ov(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uub)))),aa(nat,real,power_power(real,Uua),Uub)) ).

% ATP.lambda_375
tff(fact_8557_ATP_Olambda__376,axiom,
    ! [Uu: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_ex(real,fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),divide_divide(real,aa(nat,real,Uua,Uub),semiring_char_0_fact(real,Uub))),aa(nat,real,power_power(real,Uu),Uub)) ).

% ATP.lambda_376
tff(fact_8558_ATP_Olambda__377,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_abt(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = aa(nat,nat,minus_minus(nat,Uu),one_one(nat)) )
        & ( groups8242544230860333062m_list(nat,Uub) = Uua ) ) ) ).

% ATP.lambda_377
tff(fact_8559_ATP_Olambda__378,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_bs(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),aa(nat,A,Uu,Uub))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% ATP.lambda_378
tff(fact_8560_ATP_Olambda__379,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_qr(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),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) )
          & pp(aa(set(product_prod(list(A),list(A))),bool,aa(product_prod(list(A),list(A)),fun(set(product_prod(list(A),list(A))),bool),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_379
tff(fact_8561_ATP_Olambda__380,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_qq(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
        & ? [Xys2: list(A),X4: A,Y5: A,Xs5: list(A),Ys6: list(A)] :
            ( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs5)) )
            & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys6)) )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X4),Y5)),Uu)) ) ) ) ).

% ATP.lambda_380
tff(fact_8562_ATP_Olambda__381,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_abu(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),groups8242544230860333062m_list(nat,Uub)),one_one(nat)) = Uua ) ) ) ).

% ATP.lambda_381
tff(fact_8563_ATP_Olambda__382,axiom,
    ! [A: $tType,Uu: nat,Uua: set(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(set(A),fun(list(A),bool),aTP_Lamp_kk(nat,fun(set(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & distinct(A,Uub)
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uua)) ) ) ).

% ATP.lambda_382
tff(fact_8564_ATP_Olambda__383,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_kj(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & distinct(A,Uub)
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)) ) ) ).

% ATP.lambda_383
tff(fact_8565_ATP_Olambda__384,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_adn(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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_384
tff(fact_8566_ATP_Olambda__385,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_hh(nat,fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),aa(list(A),set(A),set2(A),Uua))) ) ) ).

% ATP.lambda_385
tff(fact_8567_ATP_Olambda__386,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_ix(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua)) ) ) ).

% ATP.lambda_386
tff(fact_8568_ATP_Olambda__387,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(nat,fun(list(A),bool),aTP_Lamp_iw(set(A),fun(nat,fun(list(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),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_387
tff(fact_8569_ATP_Olambda__388,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(nat)] :
      ( pp(aa(list(nat),bool,aa(nat,fun(list(nat),bool),aTP_Lamp_abs(nat,fun(nat,fun(list(nat),bool)),Uu),Uua),Uub))
    <=> ( ( aa(list(nat),nat,size_size(list(nat)),Uub) = Uu )
        & ( groups8242544230860333062m_list(nat,Uub) = Uua ) ) ) ).

% ATP.lambda_388
tff(fact_8570_ATP_Olambda__389,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: fun(A,option(B))] :
      ( pp(aa(fun(A,option(B)),bool,aa(set(B),fun(fun(A,option(B)),bool),aTP_Lamp_afi(set(A),fun(set(B),fun(fun(A,option(B)),bool)),Uu),Uua),Uub))
    <=> ( ( dom(A,B,Uub) = Uu )
        & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),ran(A,B,Uub)),Uua)) ) ) ).

% ATP.lambda_389
tff(fact_8571_ATP_Olambda__390,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_ot(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,diffs(A,Uu)),Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_390
tff(fact_8572_ATP_Olambda__391,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_kf(set(nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),aa(nat,nat,suc,Uub)),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uua)) ) ) ).

% ATP.lambda_391
tff(fact_8573_ATP_Olambda__392,axiom,
    ! [A: $tType,Uu: set(nat),Uua: nat,Uub: product_prod(A,nat)] :
      ( pp(aa(product_prod(A,nat),bool,aa(nat,fun(product_prod(A,nat),bool),aTP_Lamp_acq(set(nat),fun(nat,fun(product_prod(A,nat),bool)),Uu),Uua),Uub))
    <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),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_392
tff(fact_8574_ATP_Olambda__393,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_me(list(A),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uu)))
        & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Uub),Uua)) ) ) ).

% ATP.lambda_393
tff(fact_8575_ATP_Olambda__394,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: nat] :
      ( pp(aa(nat,bool,aa(list(A),fun(nat,bool),aTP_Lamp_acg(fun(A,bool),fun(list(A),fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(list(A),nat,size_size(list(A)),Uua)))
        & pp(aa(A,bool,Uu,aa(nat,A,nth(A,Uua),Uub))) ) ) ).

% ATP.lambda_394
tff(fact_8576_ATP_Olambda__395,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: fun(A,bool),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_add(list(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),aa(list(A),set(A),set2(A),Uu)))
            & pp(aa(A,bool,Uua,Uub)) ) ) ) ).

% ATP.lambda_395
tff(fact_8577_ATP_Olambda__396,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: A] :
      ( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_aby(fun(A,bool),fun(list(A),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),aa(list(A),set(A),set2(A),Uua)))
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_396
tff(fact_8578_ATP_Olambda__397,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,option(B)),Uua: A,Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(A,fun(product_prod(A,B),bool),aTP_Lamp_ady(fun(A,option(B)),fun(A,fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),Uub),graph(A,B,Uu)))
        & ( aa(product_prod(A,B),A,product_fst(A,B),Uub) != Uua ) ) ) ).

% ATP.lambda_397
tff(fact_8579_ATP_Olambda__398,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hz(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uub)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uua),Uub))) ) ).

% ATP.lambda_398
tff(fact_8580_ATP_Olambda__399,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: nat] :
      ( pp(aa(nat,bool,aa(set(nat),fun(nat,bool),aTP_Lamp_qk(list(A),fun(set(nat),fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),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_399
tff(fact_8581_ATP_Olambda__400,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_aaj(set(A),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
            & ( aa(A,B,Uua,Uub) = aa(set(B),B,lattic643756798350308766er_Min(B),aa(set(A),set(B),image(A,B,Uua),Uu)) ) ) ) ) ).

% ATP.lambda_400
tff(fact_8582_ATP_Olambda__401,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(nat,fun(set(A),bool),aTP_Lamp_kc(set(A),fun(nat,fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uub),Uu))
        & ( aa(set(A),nat,finite_card(A),Uub) = Uua ) ) ) ).

% ATP.lambda_401
tff(fact_8583_ATP_Olambda__402,axiom,
    ! [Uu: set(nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_ke(set(nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Uub),Uu))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),aa(nat,nat,suc,Uua))) ) ) ).

% ATP.lambda_402
tff(fact_8584_ATP_Olambda__403,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_br(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu),Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_403
tff(fact_8585_ATP_Olambda__404,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_bu(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu),Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_404
tff(fact_8586_ATP_Olambda__405,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_bv(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uua),Uub)),aa(nat,A,power_power(A,Uu),Uub)) ) ).

% ATP.lambda_405
tff(fact_8587_ATP_Olambda__406,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_cx(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Uua),Uub)),aa(nat,nat,minus_minus(nat,Uu),Uub)) ).

% ATP.lambda_406
tff(fact_8588_ATP_Olambda__407,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ie(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uua))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uua),Uub)) ) ) ).

% ATP.lambda_407
tff(fact_8589_ATP_Olambda__408,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_js(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uub),Uua)) ) ) ).

% ATP.lambda_408
tff(fact_8590_ATP_Olambda__409,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_id(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uua),Uub)) ) ) ).

% ATP.lambda_409
tff(fact_8591_ATP_Olambda__410,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_ju(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uub),Uua)) ) ) ).

% ATP.lambda_410
tff(fact_8592_ATP_Olambda__411,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_jv(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less_eq(int),Uub),Uua)) ) ) ).

% ATP.lambda_411
tff(fact_8593_ATP_Olambda__412,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( pp(aa(int,bool,aa(int,fun(int,bool),aTP_Lamp_jt(int,fun(int,fun(int,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uu),Uub))
        & pp(aa(int,bool,aa(int,fun(int,bool),ord_less(int),Uub),Uua)) ) ) ).

% ATP.lambda_412
tff(fact_8594_ATP_Olambda__413,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_aal(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(dvd_dvd(nat,Uub,Uua))
        & pp(dvd_dvd(nat,Uub,Uu)) ) ) ).

% ATP.lambda_413
tff(fact_8595_ATP_Olambda__414,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: fun(A,real),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,real),fun(A,bool),aTP_Lamp_xh(set(A),fun(fun(A,real),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),zero_zero(real)),aa(A,real,Uua,Uub))) ) ) ) ).

% ATP.lambda_414
tff(fact_8596_ATP_Olambda__415,axiom,
    ! [A: $tType,Uu: list(A),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_aaz(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_415
tff(fact_8597_ATP_Olambda__416,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: list(A),Uub: A] : aa(A,nat,aa(list(A),fun(A,nat),aTP_Lamp_abv(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_416
tff(fact_8598_ATP_Olambda__417,axiom,
    ! [B: $tType,Uu: set(B),Uua: fun(B,bool),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,bool),fun(B,bool),aTP_Lamp_jc(set(B),fun(fun(B,bool),fun(B,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uu))
        & pp(aa(B,bool,Uua,Uub)) ) ) ).

% ATP.lambda_417
tff(fact_8599_ATP_Olambda__418,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_ad(set(A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
        & pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_418
tff(fact_8600_ATP_Olambda__419,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_iy(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uu))
            & ( aa(B,A,Uua,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_419
tff(fact_8601_ATP_Olambda__420,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_km(set(A),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),Uu))
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_420
tff(fact_8602_ATP_Olambda__421,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_ja(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uu))
            & ( aa(B,A,Uua,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_421
tff(fact_8603_ATP_Olambda__422,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( pp(aa(B,bool,aa(set(B),fun(B,bool),aTP_Lamp_lu(fun(B,A),fun(set(B),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uua))
            & ( aa(B,A,Uu,Uub) != one_one(A) ) ) ) ) ).

% ATP.lambda_422
tff(fact_8604_ATP_Olambda__423,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_parity(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_ka(set(B),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uub),Uu))
            & ~ pp(dvd_dvd(A,aa(num,A,numeral_numeral(A),aa(num,num,bit0,one2)),aa(B,A,Uua,Uub))) ) ) ) ).

% ATP.lambda_423
tff(fact_8605_ATP_Olambda__424,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_aak(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uu)),Uua)) ) ).

% ATP.lambda_424
tff(fact_8606_ATP_Olambda__425,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_hi(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu)) ) ).

% ATP.lambda_425
tff(fact_8607_ATP_Olambda__426,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real,Uub: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aTP_Lamp_vg(A,fun(real,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_426
tff(fact_8608_ATP_Olambda__427,axiom,
    ! [Uu: real,Uua: complex,Uub: complex] :
      ( pp(aa(complex,bool,aa(complex,fun(complex,bool),aTP_Lamp_zc(real,fun(complex,fun(complex,bool)),Uu),Uua),Uub))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(complex,Uua,Uub)),Uu)) ) ).

% ATP.lambda_427
tff(fact_8609_ATP_Olambda__428,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),aTP_Lamp_za(real,fun(real,fun(real,bool)),Uu),Uua),Uub))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(real,Uua,Uub)),Uu)) ) ).

% ATP.lambda_428
tff(fact_8610_ATP_Olambda__429,axiom,
    ! [A: $tType] :
      ( real_V768167426530841204y_dist(A)
     => ! [Uu: real,Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_yy(real,fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Uua,Uub)),Uu)) ) ) ).

% ATP.lambda_429
tff(fact_8611_ATP_Olambda__430,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: real,Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_vo(real,fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Uua,Uub)),Uu)) ) ) ).

% ATP.lambda_430
tff(fact_8612_ATP_Olambda__431,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: A,Uua: real,Uub: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aTP_Lamp_yw(A,fun(real,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,Uub,Uu)),Uua)) ) ) ).

% ATP.lambda_431
tff(fact_8613_ATP_Olambda__432,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_hl(nat,fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu)) ) ).

% ATP.lambda_432
tff(fact_8614_ATP_Olambda__433,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mz(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,divide_divide(A,Uub,Uu)),Uua) ) ).

% ATP.lambda_433
tff(fact_8615_ATP_Olambda__434,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mx(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_434
tff(fact_8616_ATP_Olambda__435,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_my(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),divide_divide(A,Uub,Uu)),Uua) ) ).

% ATP.lambda_435
tff(fact_8617_ATP_Olambda__436,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_mw(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_436
tff(fact_8618_ATP_Olambda__437,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_ho(set(product_prod(A,B)),fun(A,fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(product_prod(A,B)),bool,aa(product_prod(A,B),fun(set(product_prod(A,B)),bool),member(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,Uua),Uub)),Uu)) ) ).

% ATP.lambda_437
tff(fact_8619_ATP_Olambda__438,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aax(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_438
tff(fact_8620_ATP_Olambda__439,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( pp(aa(complex,bool,aa(complex,fun(complex,bool),aTP_Lamp_ep(nat,fun(complex,fun(complex,bool)),Uu),Uua),Uub))
    <=> ( aa(nat,complex,power_power(complex,Uub),Uu) = Uua ) ) ).

% ATP.lambda_439
tff(fact_8621_ATP_Olambda__440,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( pp(aa(complex,bool,aa(nat,fun(complex,bool),aTP_Lamp_jy(complex,fun(nat,fun(complex,bool)),Uu),Uua),Uub))
    <=> ( aa(nat,complex,power_power(complex,Uub),Uua) = Uu ) ) ).

% ATP.lambda_440
tff(fact_8622_ATP_Olambda__441,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_je(A,fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),ring_1_Ints(A)))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uu),Uub))
            & pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uub),Uua)) ) ) ) ).

% ATP.lambda_441
tff(fact_8623_ATP_Olambda__442,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ax(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,suc,Uub))),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_442
tff(fact_8624_ATP_Olambda__443,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_az(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,suc,Uub))),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_443
tff(fact_8625_ATP_Olambda__444,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,int),Uub: nat] : aa(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_ta(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,minus_minus(real,aa(nat,real,Uu,Uub)),aa(real,real,aa(real,fun(real,real),times_times(real),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),aa(num,num,bit0,one2))),pi))) ).

% ATP.lambda_444
tff(fact_8626_ATP_Olambda__445,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_ow(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,power_power(real,Uua),aa(nat,nat,suc,Uub))) ).

% ATP.lambda_445
tff(fact_8627_ATP_Olambda__446,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_bf(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,power_power(real,Uua),Uub)) ).

% ATP.lambda_446
tff(fact_8628_ATP_Olambda__447,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_dg(fun(nat,nat),fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uub)),aa(nat,nat,power_power(nat,Uua),Uub)) ).

% ATP.lambda_447
tff(fact_8629_ATP_Olambda__448,axiom,
    ! [Aa: $tType] :
      ( ( real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa) )
     => ! [Uu: fun(nat,Aa),Uua: Aa,Uub: nat] : aa(nat,Aa,aa(Aa,fun(nat,Aa),aTP_Lamp_wh(fun(nat,Aa),fun(Aa,fun(nat,Aa)),Uu),Uua),Uub) = aa(Aa,Aa,aa(Aa,fun(Aa,Aa),times_times(Aa),aa(nat,Aa,Uu,Uub)),aa(nat,Aa,power_power(Aa,Uua),Uub)) ) ).

% ATP.lambda_448
tff(fact_8630_ATP_Olambda__449,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_cr(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_449
tff(fact_8631_ATP_Olambda__450,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_at(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_450
tff(fact_8632_ATP_Olambda__451,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_ay(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_451
tff(fact_8633_ATP_Olambda__452,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_bt(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_452
tff(fact_8634_ATP_Olambda__453,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_cv(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_453
tff(fact_8635_ATP_Olambda__454,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_db(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub)) ) ).

% ATP.lambda_454
tff(fact_8636_ATP_Olambda__455,axiom,
    ! [Uu: fun(nat,bool),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_iv(fun(nat,bool),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(nat,bool,Uu,Uub))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uub),Uua)) ) ) ).

% ATP.lambda_455
tff(fact_8637_ATP_Olambda__456,axiom,
    ! [C: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(D)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(D,real),Uua: fun(D,C),Uub: D] : aa(D,C,aa(fun(D,C),fun(D,C),aTP_Lamp_pl(fun(D,real),fun(fun(D,C),fun(D,C)),Uu),Uua),Uub) = aa(C,C,real_V8093663219630862766scaleR(C,aa(D,real,Uu,Uub)),aa(D,C,Uua,Uub)) ) ).

% ATP.lambda_456
tff(fact_8638_ATP_Olambda__457,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: B,Uub: B] :
          ( pp(aa(B,bool,aa(B,fun(B,bool),aTP_Lamp_acx(fun(B,A),fun(B,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uua)),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_457
tff(fact_8639_ATP_Olambda__458,axiom,
    ! [B: $tType,Uu: fun(B,real),Uua: fun(B,real),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,real),fun(B,bool),aTP_Lamp_ts(fun(B,real),fun(fun(B,real),fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),aa(B,real,Uu,Uub)),aa(B,real,Uua,Uub))) ) ).

% ATP.lambda_458
tff(fact_8640_ATP_Olambda__459,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_tl(fun(B,A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub))) ) ) ).

% ATP.lambda_459
tff(fact_8641_ATP_Olambda__460,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_tq(fun(B,A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub))) ) ) ).

% ATP.lambda_460
tff(fact_8642_ATP_Olambda__461,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_xz(fun(A,B),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub))) ) ) ).

% ATP.lambda_461
tff(fact_8643_ATP_Olambda__462,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_tx(fun(B,A),fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uua,Uub)),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_462
tff(fact_8644_ATP_Olambda__463,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C] : aa(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_py(fun(C,A),fun(fun(C,A),fun(C,A)),Uu),Uua),Uub) = divide_divide(A,aa(C,A,Uu,Uub),aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_463
tff(fact_8645_ATP_Olambda__464,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C] : aa(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_ps(fun(C,A),fun(fun(C,A),fun(C,A)),Uu),Uua),Uub) = divide_divide(A,aa(C,A,Uu,Uub),aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_464
tff(fact_8646_ATP_Olambda__465,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C] : aa(C,A,aa(fun(C,A),fun(C,A),aTP_Lamp_ux(fun(C,A),fun(fun(C,A),fun(C,A)),Uu),Uua),Uub) = divide_divide(A,aa(C,A,Uu,Uub),aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_465
tff(fact_8647_ATP_Olambda__466,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_rf(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = divide_divide(A,aa(B,A,Uu,Uub),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_466
tff(fact_8648_ATP_Olambda__467,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_gg(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = divide_divide(A,aa(B,A,Uu,Uub),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_467
tff(fact_8649_ATP_Olambda__468,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_ww(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_468
tff(fact_8650_ATP_Olambda__469,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_wz(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_469
tff(fact_8651_ATP_Olambda__470,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_og(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_470
tff(fact_8652_ATP_Olambda__471,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_wn(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_471
tff(fact_8653_ATP_Olambda__472,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_uz(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = divide_divide(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_472
tff(fact_8654_ATP_Olambda__473,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_xe(fun(A,B),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub))) ) ) ).

% ATP.lambda_473
tff(fact_8655_ATP_Olambda__474,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_ic(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_474
tff(fact_8656_ATP_Olambda__475,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_ib(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_475
tff(fact_8657_ATP_Olambda__476,axiom,
    ! [A: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(D)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(D,A),Uua: fun(D,A),Uub: D] : aa(D,A,aa(fun(D,A),fun(D,A),aTP_Lamp_po(fun(D,A),fun(fun(D,A),fun(D,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(D,A,Uu,Uub)),aa(D,A,Uua,Uub)) ) ).

% ATP.lambda_476
tff(fact_8658_ATP_Olambda__477,axiom,
    ! [B: $tType,D: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu: fun(D,B),Uua: fun(D,B),Uub: D] : aa(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_rg(fun(D,B),fun(fun(D,B),fun(D,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(D,B,Uu,Uub)),aa(D,B,Uua,Uub)) ) ).

% ATP.lambda_477
tff(fact_8659_ATP_Olambda__478,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_lv(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_478
tff(fact_8660_ATP_Olambda__479,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_oc(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_479
tff(fact_8661_ATP_Olambda__480,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_uu(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_480
tff(fact_8662_ATP_Olambda__481,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_uv(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),aa(A,real,Uu,Uub)) ).

% ATP.lambda_481
tff(fact_8663_ATP_Olambda__482,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_eg(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_482
tff(fact_8664_ATP_Olambda__483,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_sg(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,minus_minus(real,aa(nat,real,Uu,Uub)),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_483
tff(fact_8665_ATP_Olambda__484,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_jj(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(nat,A,Uua,Uub)),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_484
tff(fact_8666_ATP_Olambda__485,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_ej(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,minus_minus(nat,aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_485
tff(fact_8667_ATP_Olambda__486,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_el(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,minus_minus(nat,aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_486
tff(fact_8668_ATP_Olambda__487,axiom,
    ! [B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo1898628316856586783d_mult(B) )
     => ! [Uu: fun(C,B),Uua: fun(C,nat),Uub: C] : aa(C,B,aa(fun(C,nat),fun(C,B),aTP_Lamp_xa(fun(C,B),fun(fun(C,nat),fun(C,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(C,B,Uu,Uub)),aa(C,nat,Uua,Uub)) ) ).

% ATP.lambda_487
tff(fact_8669_ATP_Olambda__488,axiom,
    ! [B: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & topolo1898628316856586783d_mult(B) )
     => ! [Uu: fun(C,B),Uua: fun(C,nat),Uub: C] : aa(C,B,aa(fun(C,nat),fun(C,B),aTP_Lamp_wl(fun(C,B),fun(fun(C,nat),fun(C,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(C,B,Uu,Uub)),aa(C,nat,Uua,Uub)) ) ).

% ATP.lambda_488
tff(fact_8670_ATP_Olambda__489,axiom,
    ! [B: $tType,C: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu: fun(C,B),Uua: fun(C,nat),Uub: C] : aa(C,B,aa(fun(C,nat),fun(C,B),aTP_Lamp_rl(fun(C,B),fun(fun(C,nat),fun(C,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(C,B,Uu,Uub)),aa(C,nat,Uua,Uub)) ) ).

% ATP.lambda_489
tff(fact_8671_ATP_Olambda__490,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_ar(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_490
tff(fact_8672_ATP_Olambda__491,axiom,
    ! [B: $tType,D: $tType] :
      ( ( topolo4958980785337419405_space(D)
        & topolo6943815403480290642id_add(B) )
     => ! [Uu: fun(D,B),Uua: fun(D,B),Uub: D] : aa(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_xc(fun(D,B),fun(fun(D,B),fun(D,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(D,B,Uu,Uub)),aa(D,B,Uua,Uub)) ) ).

% ATP.lambda_491
tff(fact_8673_ATP_Olambda__492,axiom,
    ! [B: $tType,D: $tType] :
      ( ( topological_t2_space(D)
        & topolo6943815403480290642id_add(B) )
     => ! [Uu: fun(D,B),Uua: fun(D,B),Uub: D] : aa(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_wm(fun(D,B),fun(fun(D,B),fun(D,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(D,B,Uu,Uub)),aa(D,B,Uua,Uub)) ) ).

% ATP.lambda_492
tff(fact_8674_ATP_Olambda__493,axiom,
    ! [B: $tType,D: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [Uu: fun(D,B),Uua: fun(D,B),Uub: D] : aa(D,B,aa(fun(D,B),fun(D,B),aTP_Lamp_rj(fun(D,B),fun(fun(D,B),fun(D,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(D,B,Uu,Uub)),aa(D,B,Uua,Uub)) ) ).

% ATP.lambda_493
tff(fact_8675_ATP_Olambda__494,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo6943815403480290642id_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_rh(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_494
tff(fact_8676_ATP_Olambda__495,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_fb(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_495
tff(fact_8677_ATP_Olambda__496,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_pn(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_496
tff(fact_8678_ATP_Olambda__497,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_oe(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_497
tff(fact_8679_ATP_Olambda__498,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_wo(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_498
tff(fact_8680_ATP_Olambda__499,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_ut(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_499
tff(fact_8681_ATP_Olambda__500,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_oq(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),aa(real,real,Uua,Uub)) ).

% ATP.lambda_500
tff(fact_8682_ATP_Olambda__501,axiom,
    ! [C: $tType] :
      ( topolo4958980785337419405_space(C)
     => ! [Uu: fun(C,real),Uua: fun(C,real),Uub: C] : aa(C,real,aa(fun(C,real),fun(C,real),aTP_Lamp_xi(fun(C,real),fun(fun(C,real),fun(C,real)),Uu),Uua),Uub) = powr(real,aa(C,real,Uu,Uub),aa(C,real,Uua,Uub)) ) ).

% ATP.lambda_501
tff(fact_8683_ATP_Olambda__502,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_qa(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_502
tff(fact_8684_ATP_Olambda__503,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_uf(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_503
tff(fact_8685_ATP_Olambda__504,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_xj(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_504
tff(fact_8686_ATP_Olambda__505,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_wr(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_505
tff(fact_8687_ATP_Olambda__506,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_rb(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_506
tff(fact_8688_ATP_Olambda__507,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_np(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uu,Uub))
        & pp(aa(A,bool,Uua,Uub)) ) ) ).

% ATP.lambda_507
tff(fact_8689_ATP_Olambda__508,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: fun(A,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_abx(fun(A,bool),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ( pp(aa(A,bool,Uua,Uub))
        & pp(aa(A,bool,Uu,Uub)) ) ) ).

% ATP.lambda_508
tff(fact_8690_ATP_Olambda__509,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_abz(fun(A,B),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ).

% ATP.lambda_509
tff(fact_8691_ATP_Olambda__510,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
      ( pp(aa(A,bool,aa(fun(A,B),fun(A,bool),aTP_Lamp_tk(fun(A,B),fun(fun(A,B),fun(A,bool)),Uu),Uua),Uub))
    <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ).

% ATP.lambda_510
tff(fact_8692_ATP_Olambda__511,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_acz(B,fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> ( aa(B,A,Uua,Uu) = aa(B,A,Uua,Uub) ) ) ) ).

% ATP.lambda_511
tff(fact_8693_ATP_Olambda__512,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_uh(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,Uu,Uub)),ring_1_of_int(B,archimedean_ceiling(B,Uua)))) ) ) ).

% ATP.lambda_512
tff(fact_8694_ATP_Olambda__513,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_oj(fun(A,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(nat,A,power_power(A,aa(A,A,Uu,Uub)),aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_513
tff(fact_8695_ATP_Olambda__514,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: A] : aa(A,C,aa(fun(A,fun(B,C)),fun(A,C),aTP_Lamp_rd(set(B),fun(fun(A,fun(B,C)),fun(A,C)),Uu),Uua),Uub) = aa(set(B),C,aa(fun(B,C),fun(set(B),C),groups7121269368397514597t_prod(B,C),aa(A,fun(B,C),Uua,Uub)),Uu) ) ).

% ATP.lambda_514
tff(fact_8696_ATP_Olambda__515,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu: fun(nat,set(A)),Uua: set(A),Uub: nat] :
          ( pp(aa(nat,bool,aa(set(A),fun(nat,bool),aTP_Lamp_ur(fun(nat,set(A)),fun(set(A),fun(nat,bool)),Uu),Uua),Uub))
        <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(nat,set(A),Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_515
tff(fact_8697_ATP_Olambda__516,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_iu(fun(nat,nat),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(nat,nat,Uu,Uub)),Uua)) ) ).

% ATP.lambda_516
tff(fact_8698_ATP_Olambda__517,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,set(A)),Uua: set(A),Uub: B] :
      ( pp(aa(B,bool,aa(set(A),fun(B,bool),aTP_Lamp_nm(fun(B,set(A)),fun(set(A),fun(B,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),aa(B,set(A),Uu,Uub)),Uua)) ) ).

% ATP.lambda_517
tff(fact_8699_ATP_Olambda__518,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_tz(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_518
tff(fact_8700_ATP_Olambda__519,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_tu(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),aa(B,A,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_519
tff(fact_8701_ATP_Olambda__520,axiom,
    ! [A: $tType,B: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_vw(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_520
tff(fact_8702_ATP_Olambda__521,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_vt(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_521
tff(fact_8703_ATP_Olambda__522,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_co(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = modulo_modulo(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_522
tff(fact_8704_ATP_Olambda__523,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_aq(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(nat,A,Uu,Uub),Uua) ) ).

% ATP.lambda_523
tff(fact_8705_ATP_Olambda__524,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_ms(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = divide_divide(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_524
tff(fact_8706_ATP_Olambda__525,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_re(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = divide_divide(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_525
tff(fact_8707_ATP_Olambda__526,axiom,
    ! [B: $tType,A: $tType] :
      ( field(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_fc(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = divide_divide(A,aa(B,A,Uu,Uub),Uua) ) ).

% ATP.lambda_526
tff(fact_8708_ATP_Olambda__527,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_of(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = divide_divide(A,aa(A,A,Uu,Uub),Uua) ) ).

% ATP.lambda_527
tff(fact_8709_ATP_Olambda__528,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_rw(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(nat,A,Uua,Uub),Uu) ) ).

% ATP.lambda_528
tff(fact_8710_ATP_Olambda__529,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_tm(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),aa(B,A,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_529
tff(fact_8711_ATP_Olambda__530,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_ua(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_530
tff(fact_8712_ATP_Olambda__531,axiom,
    ! [A: $tType,B: $tType] :
      ( ( dense_linorder(B)
        & no_bot(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_vu(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),aa(A,B,Uu,Uub)),Uua)) ) ) ).

% ATP.lambda_531
tff(fact_8713_ATP_Olambda__532,axiom,
    ! [Uu: fun(real,real),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_oi(fun(real,real),fun(nat,fun(real,real)),Uu),Uua),Uub) = aa(nat,real,power_power(real,aa(real,real,Uu,Uub)),Uua) ).

% ATP.lambda_532
tff(fact_8714_ATP_Olambda__533,axiom,
    ! [C: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(C,B),Uua: nat,Uub: C] : aa(C,B,aa(nat,fun(C,B),aTP_Lamp_xb(fun(C,B),fun(nat,fun(C,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(C,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_533
tff(fact_8715_ATP_Olambda__534,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_pu(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_534
tff(fact_8716_ATP_Olambda__535,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_ok(fun(A,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(nat,A,power_power(A,aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_535
tff(fact_8717_ATP_Olambda__536,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_wk(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_536
tff(fact_8718_ATP_Olambda__537,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_uy(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_537
tff(fact_8719_ATP_Olambda__538,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_qz(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_538
tff(fact_8720_ATP_Olambda__539,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(B)
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_gh(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_539
tff(fact_8721_ATP_Olambda__540,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_rk(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_540
tff(fact_8722_ATP_Olambda__541,axiom,
    ! [Uu: nat,Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_vq(nat,fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = aa(nat,real,power_power(real,aa(real,real,Uua,Uub)),Uu) ).

% ATP.lambda_541
tff(fact_8723_ATP_Olambda__542,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_qu(nat,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(nat,real,power_power(real,aa(A,real,Uua,Uub)),Uu) ).

% ATP.lambda_542
tff(fact_8724_ATP_Olambda__543,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_vh(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_543
tff(fact_8725_ATP_Olambda__544,axiom,
    ! [B: $tType,A: $tType] :
      ( linord4140545234300271783up_add(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_aaf(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_544
tff(fact_8726_ATP_Olambda__545,axiom,
    ! [Uu: fun(real,real),Uua: real,Uub: real] : aa(real,real,aa(real,fun(real,real),aTP_Lamp_op(fun(real,real),fun(real,fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),Uua) ).

% ATP.lambda_545
tff(fact_8727_ATP_Olambda__546,axiom,
    ! [A: $tType,Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_uw(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uua,Uub),Uu) ).

% ATP.lambda_546
tff(fact_8728_ATP_Olambda__547,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_aes(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_547
tff(fact_8729_ATP_Olambda__548,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_hy(nat,fun(nat,fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uua),Uub)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uu),Uub))) ) ).

% ATP.lambda_548
tff(fact_8730_ATP_Olambda__549,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real,Uua: fun(nat,A),Uub: nat] : aa(nat,real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_bg(real,fun(fun(nat,A),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uua,Uub))),aa(nat,real,power_power(real,Uu),Uub)) ) ).

% ATP.lambda_549
tff(fact_8731_ATP_Olambda__550,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_ub(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uub))),aa(nat,real,Uua,Uub))) ) ) ).

% ATP.lambda_550
tff(fact_8732_ATP_Olambda__551,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,B),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,B),fun(nat,bool),aTP_Lamp_vl(fun(nat,A),fun(fun(nat,B),fun(nat,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uub))),real_V7770717601297561774m_norm(B,aa(nat,B,Uua,Uub)))) ) ) ).

% ATP.lambda_551
tff(fact_8733_ATP_Olambda__552,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aTP_Lamp_vf(fun(A,B),fun(real,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uub))),Uua)) ) ) ).

% ATP.lambda_552
tff(fact_8734_ATP_Olambda__553,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_fg(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)))),aa(nat,A,power_power(A,Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).

% ATP.lambda_553
tff(fact_8735_ATP_Olambda__554,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_ug(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),ring_1_of_int(B,archim6421214686448440834_floor(B,Uua))),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_554
tff(fact_8736_ATP_Olambda__555,axiom,
    ! [A: $tType] :
      ( comple9053668089753744459l_ccpo(A)
     => ! [Uu: fun(A,A),Uua: fun(A,bool),Uub: A] :
          ( pp(aa(A,bool,aa(fun(A,bool),fun(A,bool),aTP_Lamp_afh(fun(A,A),fun(fun(A,bool),fun(A,bool)),Uu),Uua),Uub))
        <=> ( ? [X4: A] :
                ( ( Uub = aa(A,A,Uu,X4) )
                & pp(aa(A,bool,Uua,X4)) )
            | ? [M8: set(A)] :
                ( ( Uub = aa(set(A),A,complete_Sup_Sup(A),M8) )
                & comple1602240252501008431_chain(A,ord_less_eq(A),M8)
                & ! [X4: A] :
                    ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),M8))
                   => pp(aa(A,bool,Uua,X4)) ) ) ) ) ) ).

% ATP.lambda_555
tff(fact_8737_ATP_Olambda__556,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(list(A),fun(list(A),bool)),Uua: list(A),Uub: list(A)] :
          ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)),aTP_Lamp_yc(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Uu),Uua),Uub))
        <=> ( ? [Y5: A,Ys4: list(A)] :
                ( ( Uua = nil(A) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) ) )
            | ? [X4: A,Y5: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) )
                & pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y5)) )
            | ? [X4: A,Y5: A,Xs3: list(A),Ys4: list(A)] :
                ( ( Uua = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
                & ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) )
                & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),X4),Y5))
                & ~ pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Y5),X4))
                & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),Uu,Xs3),Ys4)) ) ) ) ) ).

% ATP.lambda_556
tff(fact_8738_ATP_Olambda__557,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),aTP_Lamp_aag(set(A),fun(set(A),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ( pp(aa(set(A),bool,finite_finite(A),Uub))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uua),Uub))
        & pp(aa(set(A),bool,aa(set(A),fun(set(A),bool),ord_less_eq(set(A)),Uub),Uu)) ) ) ).

% ATP.lambda_557
tff(fact_8739_ATP_Olambda__558,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_hv(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_558
tff(fact_8740_ATP_Olambda__559,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_hw(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_559
tff(fact_8741_ATP_Olambda__560,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_hu(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_560
tff(fact_8742_ATP_Olambda__561,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fx(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_561
tff(fact_8743_ATP_Olambda__562,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_fv(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_562
tff(fact_8744_ATP_Olambda__563,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_ev(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,power_power(A,Uu),aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_563
tff(fact_8745_ATP_Olambda__564,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_sp(real,fun(real,fun(nat,real)),Uu),Uua),Uub) = divide_divide(real,Uua,aa(nat,real,power_power(real,Uu),Uub)) ).

% ATP.lambda_564
tff(fact_8746_ATP_Olambda__565,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_dl(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,power_power(A,Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ) ).

% ATP.lambda_565
tff(fact_8747_ATP_Olambda__566,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_fw(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_566
tff(fact_8748_ATP_Olambda__567,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: A,Uua: B,Uub: C] : aa(C,product_prod(A,product_prod(B,C)),aa(B,fun(C,product_prod(A,product_prod(B,C))),aa(A,fun(B,fun(C,product_prod(A,product_prod(B,C)))),aTP_Lamp_aba(A,fun(B,fun(C,product_prod(A,product_prod(B,C))))),Uu),Uua),Uub) = aa(product_prod(B,C),product_prod(A,product_prod(B,C)),product_Pair(A,product_prod(B,C),Uu),aa(C,product_prod(B,C),product_Pair(B,C,Uua),Uub)) ).

% ATP.lambda_567
tff(fact_8749_ATP_Olambda__568,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: B,Uua: A,Uub: C] : aa(C,product_prod(A,product_prod(B,C)),aa(A,fun(C,product_prod(A,product_prod(B,C))),aTP_Lamp_abj(B,fun(A,fun(C,product_prod(A,product_prod(B,C)))),Uu),Uua),Uub) = aa(product_prod(B,C),product_prod(A,product_prod(B,C)),product_Pair(A,product_prod(B,C),Uua),aa(C,product_prod(B,C),product_Pair(B,C,Uu),Uub)) ).

% ATP.lambda_568
tff(fact_8750_ATP_Olambda__569,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: nat] : aa(nat,list(A),aa(list(A),fun(nat,list(A)),aTP_Lamp_zj(A,fun(list(A),fun(nat,list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),take(A,Uub,Uua)) ).

% ATP.lambda_569
tff(fact_8751_ATP_Olambda__570,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,option(A)),Uua: list(B),Uub: A] : aa(A,list(A),aa(list(B),fun(A,list(A)),aTP_Lamp_acl(fun(B,option(A)),fun(list(B),fun(A,list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uub),map_filter(B,A,Uu,Uua)) ).

% ATP.lambda_570
tff(fact_8752_ATP_Olambda__571,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( pp(aa(real,bool,aa(real,fun(real,bool),aTP_Lamp_us(real,fun(real,fun(real,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(real),bool,aa(real,fun(set(real),bool),member(real),Uub),set_or5935395276787703475ssThan(real,Uu,Uua))) ) ).

% ATP.lambda_571
tff(fact_8753_ATP_Olambda__572,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aTP_Lamp_tv(A,fun(fun(B,A),fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uu),aa(B,A,Uua,Uub))) ) ) ).

% ATP.lambda_572
tff(fact_8754_ATP_Olambda__573,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_ty(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_573
tff(fact_8755_ATP_Olambda__574,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_tw(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_574
tff(fact_8756_ATP_Olambda__575,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_tp(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less_eq(B),Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_575
tff(fact_8757_ATP_Olambda__576,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_yl(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = divide_divide(A,Uua,aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_576
tff(fact_8758_ATP_Olambda__577,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_tn(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_577
tff(fact_8759_ATP_Olambda__578,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_adb(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,aa(A,fun(A,bool),ord_less(A),Uua),aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_578
tff(fact_8760_ATP_Olambda__579,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_tr(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_579
tff(fact_8761_ATP_Olambda__580,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( pp(aa(A,bool,aa(B,fun(A,bool),aTP_Lamp_vz(fun(A,B),fun(B,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(B,bool,aa(B,fun(B,bool),ord_less(B),Uua),aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_580
tff(fact_8762_ATP_Olambda__581,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_bj(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_581
tff(fact_8763_ATP_Olambda__582,axiom,
    ! [A: $tType,B: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_vy(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(B,A,Uu,Uub)) ) ).

% ATP.lambda_582
tff(fact_8764_ATP_Olambda__583,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: fun(B,nat),Uub: B] : aa(B,A,aa(fun(B,nat),fun(B,A),aTP_Lamp_gj(A,fun(fun(B,nat),fun(B,A)),Uu),Uua),Uub) = aa(nat,A,power_power(A,Uu),aa(B,nat,Uua,Uub)) ) ).

% ATP.lambda_583
tff(fact_8765_ATP_Olambda__584,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(B,nat),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_su(fun(B,nat),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(nat,A,power_power(A,Uua),aa(B,nat,Uu,Uub)) ) ).

% ATP.lambda_584
tff(fact_8766_ATP_Olambda__585,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1633459387980952147up_add(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ri(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_585
tff(fact_8767_ATP_Olambda__586,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_mt(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> pp(dvd_dvd(A,Uua,aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_586
tff(fact_8768_ATP_Olambda__587,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(C,B),Uua: A,Uub: C] : aa(C,product_prod(A,B),aa(A,fun(C,product_prod(A,B)),aTP_Lamp_aaq(fun(C,B),fun(A,fun(C,product_prod(A,B))),Uu),Uua),Uub) = aa(B,product_prod(A,B),product_Pair(A,B,Uua),aa(C,B,Uu,Uub)) ).

% ATP.lambda_587
tff(fact_8769_ATP_Olambda__588,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,set(A)),Uub: C] : aa(C,set(B),aa(fun(C,set(A)),fun(C,set(B)),aTP_Lamp_yh(fun(A,B),fun(fun(C,set(A)),fun(C,set(B))),Uu),Uua),Uub) = aa(set(A),set(B),image(A,B,Uu),aa(C,set(A),Uua,Uub)) ).

% ATP.lambda_588
tff(fact_8770_ATP_Olambda__589,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(list(A),A),Uua: list(A),Uub: A] :
          ( pp(aa(A,bool,aa(list(A),fun(A,bool),aTP_Lamp_acv(fun(list(A),A),fun(list(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ( Uub = aa(list(A),A,Uu,Uua) ) ) ) ).

% ATP.lambda_589
tff(fact_8771_ATP_Olambda__590,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_gv(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,minus_minus(nat,Uua),Uub))) ) ).

% ATP.lambda_590
tff(fact_8772_ATP_Olambda__591,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( pp(aa(A,bool,aa(set(nat),fun(A,bool),aTP_Lamp_acc(list(A),fun(set(nat),fun(A,bool)),Uu),Uua),Uub))
    <=> pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),aa(list(A),set(A),set2(A),nths(A,Uu,Uua)))) ) ).

% ATP.lambda_591
tff(fact_8773_ATP_Olambda__592,axiom,
    ! [A: $tType,C: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(C,A),Uua: real,Uub: C] :
          ( pp(aa(C,bool,aa(real,fun(C,bool),aTP_Lamp_vd(fun(C,A),fun(real,fun(C,bool)),Uu),Uua),Uub))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Uua),real_V7770717601297561774m_norm(A,aa(C,A,Uu,Uub)))) ) ) ).

% ATP.lambda_592
tff(fact_8774_ATP_Olambda__593,axiom,
    ! [A: $tType,Uu: list(A),Uua: A,Uub: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_zs(list(A),fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uub),Uu) ).

% ATP.lambda_593
tff(fact_8775_ATP_Olambda__594,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] : aa(A,set(A),aa(set(B),fun(A,set(A)),aTP_Lamp_ael(fun(B,A),fun(set(B),fun(A,set(A))),Uu),Uua),Uub) = aa(set(B),set(A),image(B,A,Uu),Uua) ).

% ATP.lambda_594
tff(fact_8776_ATP_Olambda__595,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_gl(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_595
tff(fact_8777_ATP_Olambda__596,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_em(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_596
tff(fact_8778_ATP_Olambda__597,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_sd(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_597
tff(fact_8779_ATP_Olambda__598,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_qx(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_598
tff(fact_8780_ATP_Olambda__599,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_oa(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_599
tff(fact_8781_ATP_Olambda__600,axiom,
    ! [Uu: fun(nat,bool),Uua: nat,Uub: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aTP_Lamp_th(fun(nat,bool),fun(nat,fun(nat,bool)),Uu),Uua),Uub))
    <=> pp(aa(nat,bool,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).

% ATP.lambda_600
tff(fact_8782_ATP_Olambda__601,axiom,
    ! [A: $tType,Uu: fun(nat,set(A)),Uua: nat,Uub: nat] : aa(nat,set(A),aa(nat,fun(nat,set(A)),aTP_Lamp_nh(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_601
tff(fact_8783_ATP_Olambda__602,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ap(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_602
tff(fact_8784_ATP_Olambda__603,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_rz(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_603
tff(fact_8785_ATP_Olambda__604,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_vj(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_604
tff(fact_8786_ATP_Olambda__605,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_gk(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_605
tff(fact_8787_ATP_Olambda__606,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_fl(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_606
tff(fact_8788_ATP_Olambda__607,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,bool),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_tt(fun(A,bool),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua))) ) ) ).

% ATP.lambda_607
tff(fact_8789_ATP_Olambda__608,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_qw(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_608
tff(fact_8790_ATP_Olambda__609,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_rs(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_609
tff(fact_8791_ATP_Olambda__610,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_ob(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_610
tff(fact_8792_ATP_Olambda__611,axiom,
    ! [D: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(D) )
     => ! [Uu: A,Uua: fun(A,D),Uub: A] : aa(A,D,aa(fun(A,D),fun(A,D),aTP_Lamp_rm(A,fun(fun(A,D),fun(A,D)),Uu),Uua),Uub) = aa(A,D,Uua,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uub)) ) ).

% ATP.lambda_611
tff(fact_8793_ATP_Olambda__612,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_ws(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_612
tff(fact_8794_ATP_Olambda__613,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_bl(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_613
tff(fact_8795_ATP_Olambda__614,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,nat),Uub: nat] : aa(nat,real,aa(fun(nat,nat),fun(nat,real),aTP_Lamp_yd(fun(nat,real),fun(fun(nat,nat),fun(nat,real)),Uu),Uua),Uub) = aa(nat,real,Uu,aa(nat,nat,Uua,Uub)) ).

% ATP.lambda_614
tff(fact_8796_ATP_Olambda__615,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,nat),Uub: nat] : aa(nat,A,aa(fun(nat,nat),fun(nat,A),aTP_Lamp_ye(fun(nat,A),fun(fun(nat,nat),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,Uua,Uub)) ) ).

% ATP.lambda_615
tff(fact_8797_ATP_Olambda__616,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(C,fun(B,bool)),Uua: fun(A,C),Uub: A] : aa(A,fun(B,bool),aa(fun(A,C),fun(A,fun(B,bool)),aTP_Lamp_afc(fun(C,fun(B,bool)),fun(fun(A,C),fun(A,fun(B,bool))),Uu),Uua),Uub) = aa(C,fun(B,bool),Uu,aa(A,C,Uua,Uub)) ).

% ATP.lambda_616
tff(fact_8798_ATP_Olambda__617,axiom,
    ! [C: $tType,B: $tType] :
      ( ( topolo3112930676232923870pology(B)
        & topolo1944317154257567458pology(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(B,C),Uua: fun(nat,B),Uub: nat] : aa(nat,C,aa(fun(nat,B),fun(nat,C),aTP_Lamp_wb(fun(B,C),fun(fun(nat,B),fun(nat,C)),Uu),Uua),Uub) = aa(B,C,Uu,aa(nat,B,Uua,Uub)) ) ).

% ATP.lambda_617
tff(fact_8799_ATP_Olambda__618,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo3112930676232923870pology(B)
        & topolo1944317154257567458pology(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(B,A),Uua: fun(nat,B),Uub: nat] : aa(nat,A,aa(fun(nat,B),fun(nat,A),aTP_Lamp_xs(fun(B,A),fun(fun(nat,B),fun(nat,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(nat,B,Uua,Uub)) ) ).

% ATP.lambda_618
tff(fact_8800_ATP_Olambda__619,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( counta4013691401010221786attice(A)
        & counta3822494911875563373attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aeq(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_619
tff(fact_8801_ATP_Olambda__620,axiom,
    ! [A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Uu: fun(A,bool),Uua: fun(nat,A),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,A),fun(nat,bool),aTP_Lamp_wa(fun(A,bool),fun(fun(nat,A),fun(nat,bool)),Uu),Uua),Uub))
        <=> pp(aa(A,bool,Uu,aa(nat,A,Uua,Uub))) ) ) ).

% ATP.lambda_620
tff(fact_8802_ATP_Olambda__621,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_pq(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_621
tff(fact_8803_ATP_Olambda__622,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_xd(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_622
tff(fact_8804_ATP_Olambda__623,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( comple6319245703460814977attice(A)
        & comple6319245703460814977attice(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_xp(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_623
tff(fact_8805_ATP_Olambda__624,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(C,A),Uub: C] : aa(C,fun(B,bool),aa(fun(C,A),fun(C,fun(B,bool)),aTP_Lamp_afd(fun(A,fun(B,bool)),fun(fun(C,A),fun(C,fun(B,bool))),Uu),Uua),Uub) = aa(A,fun(B,bool),Uu,aa(C,A,Uua,Uub)) ).

% ATP.lambda_624
tff(fact_8806_ATP_Olambda__625,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(num,A),Uub: num] : aa(num,B,aa(fun(num,A),fun(num,B),aTP_Lamp_mb(fun(A,B),fun(fun(num,A),fun(num,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(num,A,Uua,Uub)) ).

% ATP.lambda_625
tff(fact_8807_ATP_Olambda__626,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_ig(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ).

% ATP.lambda_626
tff(fact_8808_ATP_Olambda__627,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,C),Uua: fun(C,A),Uub: B] : aa(B,A,aa(fun(C,A),fun(B,A),aTP_Lamp_jl(fun(B,C),fun(fun(C,A),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uua,aa(B,C,Uu,Uub)) ) ).

% ATP.lambda_627
tff(fact_8809_ATP_Olambda__628,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_rp(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_628
tff(fact_8810_ATP_Olambda__629,axiom,
    ! [D: $tType,C: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(D) )
     => ! [Uu: fun(A,C),Uua: fun(C,D),Uub: A] : aa(A,D,aa(fun(C,D),fun(A,D),aTP_Lamp_wv(fun(A,C),fun(fun(C,D),fun(A,D)),Uu),Uua),Uub) = aa(C,D,Uua,aa(A,C,Uu,Uub)) ) ).

% ATP.lambda_629
tff(fact_8811_ATP_Olambda__630,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_vn(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_630
tff(fact_8812_ATP_Olambda__631,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_na(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_631
tff(fact_8813_ATP_Olambda__632,axiom,
    ! [Aa: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa) )
     => ! [Uu: fun(A,Aa),Uua: fun(nat,Aa),Uub: A] : aa(A,Aa,aa(fun(nat,Aa),fun(A,Aa),aTP_Lamp_wj(fun(A,Aa),fun(fun(nat,Aa),fun(A,Aa)),Uu),Uua),Uub) = suminf(Aa,aa(A,fun(nat,Aa),aa(fun(nat,Aa),fun(A,fun(nat,Aa)),aTP_Lamp_wi(fun(A,Aa),fun(fun(nat,Aa),fun(A,fun(nat,Aa))),Uu),Uua),Uub)) ) ).

% ATP.lambda_632
tff(fact_8814_ATP_Olambda__633,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_qt(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_qs(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub)) ) ).

% ATP.lambda_633
tff(fact_8815_ATP_Olambda__634,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_au(fun(nat,A),fun(A,fun(nat,real)),Uu),Uua),Uub) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,power_power(A,Uua),Uub))) ) ).

% ATP.lambda_634
tff(fact_8816_ATP_Olambda__635,axiom,
    ! [A: $tType,I6: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: fun(I6,A),Uua: fun(I6,A),Uub: I6] : aa(I6,real,aa(fun(I6,A),fun(I6,real),aTP_Lamp_gs(fun(I6,A),fun(fun(I6,A),fun(I6,real)),Uu),Uua),Uub) = real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(I6,A,Uu,Uub)),aa(I6,A,Uua,Uub))) ) ).

% ATP.lambda_635
tff(fact_8817_ATP_Olambda__636,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_sz(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = cos(real,aa(real,real,minus_minus(real,aa(nat,real,Uu,Uub)),Uua)) ).

% ATP.lambda_636
tff(fact_8818_ATP_Olambda__637,axiom,
    ! [A: $tType,B: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( pp(aa(B,bool,aa(A,fun(B,bool),aTP_Lamp_mu(fun(B,A),fun(A,fun(B,bool)),Uu),Uua),Uub))
        <=> ~ pp(dvd_dvd(A,Uua,aa(B,A,Uu,Uub))) ) ) ).

% ATP.lambda_637
tff(fact_8819_ATP_Olambda__638,axiom,
    ! [A: $tType,B: $tType,Uu: list(A),Uua: list(B),Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(list(B),fun(product_prod(A,B),bool),aTP_Lamp_zn(list(A),fun(list(B),fun(product_prod(A,B),bool)),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)) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),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_638
tff(fact_8820_ATP_Olambda__639,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(A,B),fun(product_prod(A,B),bool),aTP_Lamp_aea(set(A),fun(fun(A,B),fun(product_prod(A,B),bool)),Uu),Uua),Uub))
    <=> ? [A6: A] :
          ( ( Uub = aa(B,product_prod(A,B),product_Pair(A,B,A6),aa(A,B,Uua,A6)) )
          & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A6),Uu)) ) ) ).

% ATP.lambda_639
tff(fact_8821_ATP_Olambda__640,axiom,
    ! [A: $tType,Uu: list(A),Uua: set(nat),Uub: A] :
      ( pp(aa(A,bool,aa(set(nat),fun(A,bool),aTP_Lamp_md(list(A),fun(set(nat),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [I3: nat] :
          ( ( Uub = aa(nat,A,nth(A,Uu),I3) )
          & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(list(A),nat,size_size(list(A)),Uu)))
          & pp(aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),I3),Uua)) ) ) ).

% ATP.lambda_640
tff(fact_8822_ATP_Olambda__641,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aeh(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ? [A6: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),Uua),A6) )
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A6),Uu)) ) ) ) ).

% ATP.lambda_641
tff(fact_8823_ATP_Olambda__642,axiom,
    ! [A: $tType] :
      ( finite8700451911770168679attice(A)
     => ! [Uu: A,Uua: set(A),Uub: A] :
          ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_ne(A,fun(set(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ? [B6: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uu),B6) )
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B6),Uua)) ) ) ) ).

% ATP.lambda_642
tff(fact_8824_ATP_Olambda__643,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: A,Uub: A] :
          ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_aeg(set(A),fun(A,fun(A,bool)),Uu),Uua),Uub))
        <=> ? [A6: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),Uua),A6) )
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A6),Uu)) ) ) ) ).

% ATP.lambda_643
tff(fact_8825_ATP_Olambda__644,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aTP_Lamp_mk(fun(B,A),fun(set(B),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X4: B] :
          ( ( Uub = aa(B,A,Uu,X4) )
          & pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),X4),Uua)) ) ) ).

% ATP.lambda_644
tff(fact_8826_ATP_Olambda__645,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,bool),Uub: set(A)] :
      ( pp(aa(set(A),bool,aa(fun(B,bool),fun(set(A),bool),aTP_Lamp_ks(fun(B,set(A)),fun(fun(B,bool),fun(set(A),bool)),Uu),Uua),Uub))
    <=> ? [X4: B] :
          ( ( Uub = aa(B,set(A),Uu,X4) )
          & pp(aa(B,bool,Uua,X4)) ) ) ).

% ATP.lambda_645
tff(fact_8827_ATP_Olambda__646,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(B,bool),fun(A,bool),aTP_Lamp_ml(fun(B,A),fun(fun(B,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X4: B] :
          ( ( Uub = aa(B,A,Uu,X4) )
          & pp(aa(B,bool,Uua,X4)) ) ) ).

% ATP.lambda_646
tff(fact_8828_ATP_Olambda__647,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(A,B),Uub: B] :
      ( pp(aa(B,bool,aa(fun(A,B),fun(B,bool),aTP_Lamp_jw(fun(A,bool),fun(fun(A,B),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [X4: A] :
          ( ( Uub = aa(A,B,Uua,X4) )
          & pp(aa(A,bool,Uu,X4)) ) ) ).

% ATP.lambda_647
tff(fact_8829_ATP_Olambda__648,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_uk(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),Uu),Uua),Uub))
        <=> ! [N5: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uub),N5))
             => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or7035219750837199246ssThan(nat,Uub,N5)))),aa(nat,real,Uua,Uub))) ) ) ) ).

% ATP.lambda_648
tff(fact_8830_ATP_Olambda__649,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( pp(aa(nat,bool,aa(fun(nat,real),fun(nat,bool),aTP_Lamp_te(fun(nat,A),fun(fun(nat,real),fun(nat,bool)),Uu),Uua),Uub))
        <=> ! [A6: nat] :
              ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uub),A6))
             => ! [B6: nat] :
                  ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),A6),B6))
                 => pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or3652927894154168847AtMost(nat,A6,B6)))),aa(nat,real,Uua,A6))) ) ) ) ) ).

% ATP.lambda_649
tff(fact_8831_ATP_Olambda__650,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,bool),Uua: fun(B,fun(A,bool)),Uub: B] :
      ( pp(aa(B,bool,aa(fun(B,fun(A,bool)),fun(B,bool),aTP_Lamp_jq(fun(A,bool),fun(fun(B,fun(A,bool)),fun(B,bool)),Uu),Uua),Uub))
    <=> ? [Y5: A] :
          ( pp(aa(A,bool,Uu,Y5))
          & pp(aa(A,bool,aa(B,fun(A,bool),Uua,Uub),Y5)) ) ) ).

% ATP.lambda_650
tff(fact_8832_ATP_Olambda__651,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,set(A)),Uua: fun(B,bool),Uub: A] :
      ( pp(aa(A,bool,aa(fun(B,bool),fun(A,bool),aTP_Lamp_kt(fun(B,set(A)),fun(fun(B,bool),fun(A,bool)),Uu),Uua),Uub))
    <=> ? [X4: B] :
          ( pp(aa(B,bool,Uua,X4))
          & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uub),aa(B,set(A),Uu,X4))) ) ) ).

% ATP.lambda_651
tff(fact_8833_ATP_Olambda__652,axiom,
    ! [A: $tType,Uu: fun(A,A),Uua: A,Uub: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_yi(fun(A,A),fun(A,fun(A,bool)),Uu),Uua),Uub))
    <=> ? [N5: nat] : Uub = aa(A,A,aa(fun(A,A),fun(A,A),aa(nat,fun(fun(A,A),fun(A,A)),compow(fun(A,A)),N5),Uu),Uua) ) ).

% ATP.lambda_652
tff(fact_8834_ATP_Olambda__653,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,B)),Uua: fun(B,A),Uub: product_prod(A,A)] :
      ( pp(aa(product_prod(A,A),bool,aa(fun(B,A),fun(product_prod(A,A),bool),aTP_Lamp_aem(set(product_prod(B,B)),fun(fun(B,A),fun(product_prod(A,A),bool)),Uu),Uua),Uub))
    <=> ? [A19: B,A26: B] :
          ( ( Uub = aa(A,product_prod(A,A),product_Pair(A,A,aa(B,A,Uua,A19)),aa(B,A,Uua,A26)) )
          & pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),product_Pair(B,B,A19),A26)),Uu)) ) ) ).

% ATP.lambda_653
tff(fact_8835_ATP_Olambda__654,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_wc(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ? [A6: A,V6: list(A)] :
          ( ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A6),V6)) )
          | ? [U4: list(A),Aa2: A,B6: A,Va4: list(A),W3: list(A)] :
              ( pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Aa2),B6)),Uu))
              & ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Aa2),Va4)) )
              & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),U4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B6),W3)) ) ) ) ) ).

% ATP.lambda_654
tff(fact_8836_ATP_Olambda__655,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_aei(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ? [A6: A,B6: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),sup_sup(A),A6),B6) )
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A6),Uu))
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B6),Uua)) ) ) ) ).

% ATP.lambda_655
tff(fact_8837_ATP_Olambda__656,axiom,
    ! [A: $tType] :
      ( distrib_lattice(A)
     => ! [Uu: set(A),Uua: set(A),Uub: A] :
          ( pp(aa(A,bool,aa(set(A),fun(A,bool),aTP_Lamp_aef(set(A),fun(set(A),fun(A,bool)),Uu),Uua),Uub))
        <=> ? [A6: A,B6: A] :
              ( ( Uub = aa(A,A,aa(A,fun(A,A),inf_inf(A),A6),B6) )
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A6),Uu))
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),B6),Uua)) ) ) ) ).

% ATP.lambda_656
tff(fact_8838_ATP_Olambda__657,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(list(A)),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(set(list(A)),fun(list(A),bool),aTP_Lamp_qm(set(A),fun(set(list(A)),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ? [X4: A,Xs3: list(A)] :
          ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
          & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),X4),Uu))
          & pp(aa(set(list(A)),bool,aa(list(A),fun(set(list(A)),bool),member(list(A)),Xs3),Uua)) ) ) ).

% ATP.lambda_657
tff(fact_8839_ATP_Olambda__658,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aTP_Lamp_xn(set(product_prod(A,A)),fun(list(A),fun(list(A),bool)),Uu),Uua),Uub))
    <=> ? [Us2: list(A),Z5: A,Z8: A,Vs3: list(A)] :
          ( ( Uua = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z5),Vs3)) )
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Z5),Z8)),Uu))
          & ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z8),Vs3)) ) ) ) ).

% ATP.lambda_658
tff(fact_8840_ATP_Olambda__659,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_eb(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,fconj(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uub),dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uuc)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,ring_1_of_int(real,aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),one_one(int))),divide_divide(nat,Uub,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc)))),semiring_char_0_fact(real,Uub))),aa(nat,A,power_power(A,Uu),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_659
tff(fact_8841_ATP_Olambda__660,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_ck(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,fconj(dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uub),aa(bool,bool,fNot,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uuc))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),divide_divide(real,ring_1_of_int(real,aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),one_one(int))),divide_divide(nat,Uub,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc)))),semiring_char_0_fact(real,Uub)))),aa(nat,A,power_power(A,Uu),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_660
tff(fact_8842_ATP_Olambda__661,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_ed(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)),Uub),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,divide_divide(real,ring_1_of_int(real,aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,power_power(int,aa(int,int,uminus_uminus(int),one_one(int))),divide_divide(nat,Uub,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc)))),semiring_char_0_fact(real,Uub))),aa(nat,A,power_power(A,Uu),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_661
tff(fact_8843_ATP_Olambda__662,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: set(A),Uub: A,Uuc: B] : aa(B,A,aa(A,fun(B,A),aa(set(A),fun(A,fun(B,A)),aTP_Lamp_yj(fun(A,B),fun(set(A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),aa(set(A),set(B),image(A,B,Uu),Uua)),the_inv_into(A,B,Uua,Uu,Uuc),Uub) ).

% ATP.lambda_662
tff(fact_8844_ATP_Olambda__663,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_dm(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),Uu),zero_zero(A),aa(nat,A,Uub,aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat)))))) ) ).

% ATP.lambda_663
tff(fact_8845_ATP_Olambda__664,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_gx(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),Uu),one_one(A),aa(nat,A,Uub,aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat)))))) ) ).

% ATP.lambda_664
tff(fact_8846_ATP_Olambda__665,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_wu(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = if(B,aa(A,bool,aa(A,fun(A,bool),ord_less_eq(A),Uuc),Uu),aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_665
tff(fact_8847_ATP_Olambda__666,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_gy(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_666
tff(fact_8848_ATP_Olambda__667,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_dn(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_667
tff(fact_8849_ATP_Olambda__668,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_jk(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = if(A,aa(set(nat),bool,aa(nat,fun(set(nat),bool),member(nat),Uuc),Uua),aa(nat,A,Uub,Uuc),aa(nat,A,Uu,Uuc)) ) ).

% ATP.lambda_668
tff(fact_8850_ATP_Olambda__669,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B,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_kl(B,fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_669
tff(fact_8851_ATP_Olambda__670,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B,Uua: fun(B,A),Uub: A,Uuc: B] : aa(B,A,aa(A,fun(B,A),aa(fun(B,A),fun(A,fun(B,A)),aTP_Lamp_ki(B,fun(fun(B,A),fun(A,fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,aa(B,fun(B,bool),fequal(B),Uuc),Uu),aa(B,A,Uua,Uuc),Uub) ) ).

% ATP.lambda_670
tff(fact_8852_ATP_Olambda__671,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: A,Uub: list(A),Uuc: list(A)] : aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),aa(A,fun(list(A),fun(list(A),product_prod(list(A),list(A)))),aTP_Lamp_adk(fun(A,bool),fun(A,fun(list(A),fun(list(A),product_prod(list(A),list(A))))),Uu),Uua),Uub),Uuc) = if(product_prod(list(A),list(A)),aa(A,bool,Uu,Uua),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub)),Uuc),aa(list(A),product_prod(list(A),list(A)),product_Pair(list(A),list(A),Uub),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uuc))) ).

% ATP.lambda_671
tff(fact_8853_ATP_Olambda__672,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,bool),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_mr(fun(B,bool),fun(fun(B,A),fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = if(A,aa(B,bool,Uu,Uuc),aa(B,A,Uua,Uuc),aa(B,A,Uub,Uuc)) ) ).

% ATP.lambda_672
tff(fact_8854_ATP_Olambda__673,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_ko(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_673
tff(fact_8855_ATP_Olambda__674,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(B,A),Uub: B,Uuc: B] :
      ( pp(aa(B,bool,aa(B,fun(B,bool),aa(fun(B,A),fun(B,fun(B,bool)),aTP_Lamp_acw(fun(A,fun(A,bool)),fun(fun(B,A),fun(B,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(A,bool,aa(A,fun(A,bool),Uu,aa(B,A,Uua,Uub)),aa(B,A,Uua,Uuc))) ) ).

% ATP.lambda_674
tff(fact_8856_ATP_Olambda__675,axiom,
    ! [B: $tType,A: $tType,C: $tType,D: $tType,Uu: fun(B,fun(C,A)),Uua: fun(D,B),Uub: fun(D,C),Uuc: D] : aa(D,A,aa(fun(D,C),fun(D,A),aa(fun(D,B),fun(fun(D,C),fun(D,A)),aTP_Lamp_aau(fun(B,fun(C,A)),fun(fun(D,B),fun(fun(D,C),fun(D,A))),Uu),Uua),Uub),Uuc) = aa(C,A,aa(B,fun(C,A),Uu,aa(D,B,Uua,Uuc)),aa(D,C,Uub,Uuc)) ).

% ATP.lambda_675
tff(fact_8857_ATP_Olambda__676,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,fun(C,bool)),Uua: fun(B,C),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(fun(B,C),fun(A,fun(B,bool)),aTP_Lamp_afb(fun(A,fun(C,bool)),fun(fun(B,C),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(C,bool,aa(A,fun(C,bool),Uu,Uub),aa(B,C,Uua,Uuc))) ) ).

% ATP.lambda_676
tff(fact_8858_ATP_Olambda__677,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,bool)),Uua: fun(C,B),Uub: A,Uuc: C] :
      ( pp(aa(C,bool,aa(A,fun(C,bool),aa(fun(C,B),fun(A,fun(C,bool)),aTP_Lamp_afe(fun(A,fun(B,bool)),fun(fun(C,B),fun(A,fun(C,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(B,bool,aa(A,fun(B,bool),Uu,Uub),aa(C,B,Uua,Uuc))) ) ).

% ATP.lambda_677
tff(fact_8859_ATP_Olambda__678,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_fj(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fi(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,Uub),aa(nat,nat,suc,zero_zero(nat)))),Uuc))) ) ).

% ATP.lambda_678
tff(fact_8860_ATP_Olambda__679,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_dr(nat,fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),zero_zero(nat)),aa(A,A,uminus_uminus(A),Uub),if(A,aa(nat,bool,aa(nat,fun(nat,bool),fequal(nat),Uuc),Uu),one_one(A),zero_zero(A)))),aa(nat,A,power_power(A,Uua),Uuc)) ) ).

% ATP.lambda_679
tff(fact_8861_ATP_Olambda__680,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_di(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_dh(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uuc))),aa(nat,nat,power_power(nat,Uub),Uuc)) ).

% ATP.lambda_680
tff(fact_8862_ATP_Olambda__681,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_dd(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_dc(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uuc))),aa(nat,A,power_power(A,Uub),Uuc)) ) ).

% ATP.lambda_681
tff(fact_8863_ATP_Olambda__682,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_pe(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,power_power(real,Uub),Uuc)) ).

% ATP.lambda_682
tff(fact_8864_ATP_Olambda__683,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_pc(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,power_power(real,Uub),Uuc)) ).

% ATP.lambda_683
tff(fact_8865_ATP_Olambda__684,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_pb(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,power_power(real,aa(real,real,minus_minus(real,Uub),Uua)),Uuc)) ).

% ATP.lambda_684
tff(fact_8866_ATP_Olambda__685,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_pa(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,power_power(real,aa(real,real,minus_minus(real,Uua),Uub)),Uuc)) ).

% ATP.lambda_685
tff(fact_8867_ATP_Olambda__686,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_er(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uua)),aa(nat,nat,minus_minus(nat,Uub),Uuc))),aa(nat,A,power_power(A,Uu),Uuc))),aa(nat,A,power_power(A,Uu),Uub)) ) ).

% ATP.lambda_686
tff(fact_8868_ATP_Olambda__687,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_kv(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_687
tff(fact_8869_ATP_Olambda__688,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_fi(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu)),Uuc)),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,Uub),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))),Uuc))) ) ).

% ATP.lambda_688
tff(fact_8870_ATP_Olambda__689,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_sw(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = divide_divide(A,aa(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_689
tff(fact_8871_ATP_Olambda__690,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: set(product_prod(A,A)),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(set(product_prod(A,A)),fun(A,fun(A,bool)),aTP_Lamp_nn(fun(A,nat),fun(set(product_prod(A,A)),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc)))
        | ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uuc)))
          & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,Uub),Uuc)),Uua)) ) ) ) ).

% ATP.lambda_690
tff(fact_8872_ATP_Olambda__691,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V7819770556892013058_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: real,Uuc: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aa(B,fun(real,fun(A,bool)),aTP_Lamp_vi(fun(A,B),fun(B,fun(real,fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V557655796197034286t_dist(B,aa(A,B,Uu,Uuc),Uua)),Uub)) ) ) ).

% ATP.lambda_691
tff(fact_8873_ATP_Olambda__692,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: real,Uuc: B] :
          ( pp(aa(B,bool,aa(real,fun(B,bool),aa(A,fun(real,fun(B,bool)),aTP_Lamp_vm(fun(B,A),fun(A,fun(real,fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),real_V557655796197034286t_dist(A,aa(B,A,Uu,Uuc),Uua)),Uub)) ) ) ).

% ATP.lambda_692
tff(fact_8874_ATP_Olambda__693,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_dk(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),comm_s3205402744901411588hammer(A,Uu,Uuc))),comm_s3205402744901411588hammer(A,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_693
tff(fact_8875_ATP_Olambda__694,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_cz(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,semiring_1_of_nat(nat),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,nat,power_power(nat,Uu),Uuc))),aa(nat,nat,power_power(nat,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))) ).

% ATP.lambda_694
tff(fact_8876_ATP_Olambda__695,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_df(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,A,power_power(A,Uu),Uuc))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_695
tff(fact_8877_ATP_Olambda__696,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_dq(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uuc))),aa(nat,A,power_power(A,Uu),Uuc))),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,Uub),Uuc)))),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc)))) ) ).

% ATP.lambda_696
tff(fact_8878_ATP_Olambda__697,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat,Uub: list(A),Uuc: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(nat,fun(list(A),fun(list(A),bool)),aTP_Lamp_qn(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),bool))),Uu),Uua),Uub),Uuc))
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & ( aa(list(A),nat,size_size(list(A)),Uuc) = Uua )
        & ? [Xys2: list(A),X4: A,Y5: A,Xs5: list(A),Ys6: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs5)) )
            & ( Uuc = aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),append(A),Xys2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys6)) )
            & pp(aa(set(product_prod(A,A)),bool,aa(product_prod(A,A),fun(set(product_prod(A,A)),bool),member(product_prod(A,A)),aa(A,product_prod(A,A),product_Pair(A,A,X4),Y5)),Uu)) ) ) ) ).

% ATP.lambda_697
tff(fact_8879_ATP_Olambda__698,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_eu(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Uub),aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uuc)))),aa(nat,A,power_power(A,Uu),Uuc)) ) ).

% ATP.lambda_698
tff(fact_8880_ATP_Olambda__699,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_es(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Uu),Uuc)),aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uua)),aa(nat,nat,minus_minus(nat,Uub),Uuc))),aa(nat,A,power_power(A,Uu),aa(nat,nat,minus_minus(nat,Uub),Uuc)))) ) ).

% ATP.lambda_699
tff(fact_8881_ATP_Olambda__700,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: set(A),Uua: fun(A,real),Uub: fun(A,real),Uuc: A] :
          ( pp(aa(A,bool,aa(fun(A,real),fun(A,bool),aa(fun(A,real),fun(fun(A,real),fun(A,bool)),aTP_Lamp_xf(set(A),fun(fun(A,real),fun(fun(A,real),fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uu))
            & pp(aa(real,bool,aa(real,fun(real,bool),ord_less(real),aa(A,real,Uua,Uuc)),aa(A,real,Uub,Uuc))) ) ) ) ).

% ATP.lambda_700
tff(fact_8882_ATP_Olambda__701,axiom,
    ! [B: $tType,C: $tType,Uu: set(B),Uua: fun(B,C),Uub: C,Uuc: B] :
      ( pp(aa(B,bool,aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_mm(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uu))
        & ( aa(B,C,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_701
tff(fact_8883_ATP_Olambda__702,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
      ( pp(aa(A,bool,aa(B,fun(A,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_nb(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),Uuc),Uu))
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_702
tff(fact_8884_ATP_Olambda__703,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_et(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,power_power(A,Uu),Uuc)),aa(nat,A,power_power(A,Uub),aa(nat,nat,minus_minus(nat,Uua),Uuc))) ) ).

% ATP.lambda_703
tff(fact_8885_ATP_Olambda__704,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_cy(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Uu),Uuc)),aa(nat,nat,binomial(Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))) ).

% ATP.lambda_704
tff(fact_8886_ATP_Olambda__705,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_kz(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),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_705
tff(fact_8887_ATP_Olambda__706,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(nat,fun(nat,fun(nat,bool)),aTP_Lamp_kx(nat,fun(nat,fun(nat,fun(nat,bool))),Uu),Uua),Uub),Uuc))
    <=> pp(aa(nat,bool,aa(nat,fun(nat,bool),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_706
tff(fact_8888_ATP_Olambda__707,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_lb(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_707
tff(fact_8889_ATP_Olambda__708,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_ld(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_708
tff(fact_8890_ATP_Olambda__709,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: list(A),Uub: B,Uuc: list(B)] : aa(list(B),list(product_prod(A,B)),aa(B,fun(list(B),list(product_prod(A,B))),aa(list(A),fun(B,fun(list(B),list(product_prod(A,B)))),aTP_Lamp_zp(A,fun(list(A),fun(B,fun(list(B),list(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,Uu),Uub)),zip(A,B,Uua,Uuc)) ).

% ATP.lambda_709
tff(fact_8891_ATP_Olambda__710,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: list(B),Uub: A,Uuc: list(A)] : aa(list(A),list(product_prod(A,B)),aa(A,fun(list(A),list(product_prod(A,B))),aa(list(B),fun(A,fun(list(A),list(product_prod(A,B)))),aTP_Lamp_zu(B,fun(list(B),fun(A,fun(list(A),list(product_prod(A,B))))),Uu),Uua),Uub),Uuc) = aa(list(product_prod(A,B)),list(product_prod(A,B)),aa(product_prod(A,B),fun(list(product_prod(A,B)),list(product_prod(A,B))),cons(product_prod(A,B)),aa(B,product_prod(A,B),product_Pair(A,B,Uub),Uu)),zip(A,B,Uuc,Uua)) ).

% ATP.lambda_710
tff(fact_8892_ATP_Olambda__711,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_jb(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uu))
            & ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) != one_one(A) ) ) ) ) ).

% ATP.lambda_711
tff(fact_8893_ATP_Olambda__712,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,A),Uub: fun(B,A),Uuc: B] :
          ( pp(aa(B,bool,aa(fun(B,A),fun(B,bool),aa(fun(B,A),fun(fun(B,A),fun(B,bool)),aTP_Lamp_iz(set(B),fun(fun(B,A),fun(fun(B,A),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( pp(aa(set(B),bool,aa(B,fun(set(B),bool),member(B),Uuc),Uu))
            & ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uua,Uuc)),aa(B,A,Uub,Uuc)) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_712
tff(fact_8894_ATP_Olambda__713,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_tf(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,divide_divide(real,one_one(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uub)))),aa(B,B,minus_minus(B,aa(A,B,Uu,Uuc)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),Uub))))) ) ).

% ATP.lambda_713
tff(fact_8895_ATP_Olambda__714,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_ba(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,power_power(A,Uub),Uuc)) ) ).

% ATP.lambda_714
tff(fact_8896_ATP_Olambda__715,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_ia(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,aa(nat,B,nth(B,Uub),Uuc))),aa(nat,A,power_power(A,Uua),Uuc)) ) ).

% ATP.lambda_715
tff(fact_8897_ATP_Olambda__716,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_qs(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(A,A,minus_minus(A,divide_divide(A,aa(A,A,minus_minus(A,aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)),Uuc)),aa(nat,A,power_power(A,Uua),Uuc)),Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uuc)),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat))))))) ) ).

% ATP.lambda_716
tff(fact_8898_ATP_Olambda__717,axiom,
    ! [A: $tType,Aa: $tType] :
      ( ( real_Vector_banach(Aa)
        & real_V3459762299906320749_field(Aa)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,Aa),Uua: fun(nat,Aa),Uub: A,Uuc: nat] : aa(nat,Aa,aa(A,fun(nat,Aa),aa(fun(nat,Aa),fun(A,fun(nat,Aa)),aTP_Lamp_wi(fun(A,Aa),fun(fun(nat,Aa),fun(A,fun(nat,Aa))),Uu),Uua),Uub),Uuc) = aa(Aa,Aa,aa(Aa,fun(Aa,Aa),times_times(Aa),aa(nat,Aa,Uua,Uuc)),aa(nat,Aa,power_power(Aa,aa(A,Aa,Uu,Uub)),Uuc)) ) ).

% ATP.lambda_717
tff(fact_8899_ATP_Olambda__718,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_qd(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),aa(num,num,bit0,one2)))) ) ).

% ATP.lambda_718
tff(fact_8900_ATP_Olambda__719,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_fy(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,power_power(A,Uua),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,Uuc),Uub)),one_one(nat)))) ) ).

% ATP.lambda_719
tff(fact_8901_ATP_Olambda__720,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_aco(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_720
tff(fact_8902_ATP_Olambda__721,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_dh(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uuc)),aa(nat,nat,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ).

% ATP.lambda_721
tff(fact_8903_ATP_Olambda__722,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_dc(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_722
tff(fact_8904_ATP_Olambda__723,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [Uu: fun(A,B),Uua: fun(C,B),Uub: A,Uuc: C] : aa(C,B,aa(A,fun(C,B),aa(fun(C,B),fun(A,fun(C,B)),aTP_Lamp_aev(fun(A,B),fun(fun(C,B),fun(A,fun(C,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(C,B,Uua,Uuc)) ) ).

% ATP.lambda_723
tff(fact_8905_ATP_Olambda__724,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(C,A),Uua: fun(D,B),Uub: C,Uuc: D] : aa(D,product_prod(A,B),aa(C,fun(D,product_prod(A,B)),aa(fun(D,B),fun(C,fun(D,product_prod(A,B))),aTP_Lamp_aas(fun(C,A),fun(fun(D,B),fun(C,fun(D,product_prod(A,B)))),Uu),Uua),Uub),Uuc) = aa(B,product_prod(A,B),product_Pair(A,B,aa(C,A,Uu,Uub)),aa(D,B,Uua,Uuc)) ).

% ATP.lambda_724
tff(fact_8906_ATP_Olambda__725,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(A)
     => ! [Uu: fun(B,A),Uua: fun(list(B),A),Uub: list(B),Uuc: B] :
          ( pp(aa(B,bool,aa(list(B),fun(B,bool),aa(fun(list(B),A),fun(list(B),fun(B,bool)),aTP_Lamp_acu(fun(B,A),fun(fun(list(B),A),fun(list(B),fun(B,bool))),Uu),Uua),Uub),Uuc))
        <=> ( aa(B,A,Uu,Uuc) = aa(list(B),A,Uua,Uub) ) ) ) ).

% ATP.lambda_725
tff(fact_8907_ATP_Olambda__726,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_qh(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(nat,real,power_power(real,cos(real,aa(A,real,Uu,Uua))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))) ) ).

% ATP.lambda_726
tff(fact_8908_ATP_Olambda__727,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_qf(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,power_power(real,aa(A,real,Uu,Uub)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))) ) ).

% ATP.lambda_727
tff(fact_8909_ATP_Olambda__728,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_px(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_728
tff(fact_8910_ATP_Olambda__729,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_pi(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))))))) ) ).

% ATP.lambda_729
tff(fact_8911_ATP_Olambda__730,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_pk(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,uminus_uminus(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,power_power(real,aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))) ) ).

% ATP.lambda_730
tff(fact_8912_ATP_Olambda__731,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_uj(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,minus_minus(B,aa(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_731
tff(fact_8913_ATP_Olambda__732,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,set(B)),Uua: set(B),Uub: set(B),Uuc: A] :
      ( pp(aa(A,bool,aa(set(B),fun(A,bool),aa(set(B),fun(set(B),fun(A,bool)),aTP_Lamp_aad(fun(A,set(B)),fun(set(B),fun(set(B),fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( pp(aa(set(B),bool,finite_finite(B),aa(A,set(B),Uu,Uuc)))
        & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),Uub),aa(A,set(B),Uu,Uuc)))
        & pp(aa(set(B),bool,aa(set(B),fun(set(B),bool),ord_less_eq(set(B)),aa(A,set(B),Uu,Uuc)),Uua)) ) ) ).

% ATP.lambda_732
tff(fact_8914_ATP_Olambda__733,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] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aa(fun(A,C),fun(real,fun(A,bool)),aTP_Lamp_ud(fun(A,B),fun(fun(A,C),fun(real,fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),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_733
tff(fact_8915_ATP_Olambda__734,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_nc(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,bool),aa(fun(A,B),fun(B,fun(A,bool)),aTP_Lamp_nb(set(A),fun(fun(A,B),fun(B,fun(A,bool))),Uu),Uua),Uuc))))),aa(B,C,Uub,Uuc)) ) ).

% ATP.lambda_734
tff(fact_8916_ATP_Olambda__735,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(A,fun(A,bool)),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aTP_Lamp_xt(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ? [A6: A] :
            ( ( Uub = A6 )
            & ( Uuc = A6 ) )
        | ? [A6: A,B6: A,C3: A] :
            ( ( Uub = A6 )
            & ( Uuc = C3 )
            & pp(aa(A,bool,aa(A,fun(A,bool),Uua,A6),B6))
            & pp(aa(A,bool,aa(A,fun(A,bool),Uu,B6),C3)) ) ) ) ).

% ATP.lambda_735
tff(fact_8917_ATP_Olambda__736,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(A,fun(A,bool)),Uub: A,Uuc: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aa(fun(A,fun(A,bool)),fun(A,fun(A,bool)),aTP_Lamp_xu(fun(A,fun(A,bool)),fun(fun(A,fun(A,bool)),fun(A,fun(A,bool))),Uu),Uua),Uub),Uuc))
    <=> ( ? [A6: A,B6: A] :
            ( ( Uub = A6 )
            & ( Uuc = B6 )
            & pp(aa(A,bool,aa(A,fun(A,bool),Uu,A6),B6)) )
        | ? [A6: A,B6: A,C3: A] :
            ( ( Uub = A6 )
            & ( Uuc = C3 )
            & pp(aa(A,bool,aa(A,fun(A,bool),Uua,A6),B6))
            & pp(aa(A,bool,aa(A,fun(A,bool),Uu,B6),C3)) ) ) ) ).

% ATP.lambda_736
tff(fact_8918_ATP_Olambda__737,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,bool)),Uua: fun(list(A),fun(list(A),bool)),Uub: list(A),Uuc: list(A)] :
      ( pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),aa(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool)),aTP_Lamp_ya(fun(A,fun(A,bool)),fun(fun(list(A),fun(list(A),bool)),fun(list(A),fun(list(A),bool))),Uu),Uua),Uub),Uuc))
    <=> ( ? [Y5: A,Ys4: list(A)] :
            ( ( Uub = nil(A) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) ) )
        | ? [X4: A,Y5: A,Xs3: list(A),Ys4: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) )
            & pp(aa(A,bool,aa(A,fun(A,bool),Uu,X4),Y5)) )
        | ? [X4: A,Y5: A,Xs3: list(A),Ys4: list(A)] :
            ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X4),Xs3) )
            & ( Uuc = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) )
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),Uu,X4),Y5))
            & ~ pp(aa(A,bool,aa(A,fun(A,bool),Uu,Y5),X4))
            & pp(aa(list(A),bool,aa(list(A),fun(list(A),bool),Uua,Xs3),Ys4)) ) ) ) ).

% ATP.lambda_737
tff(fact_8919_ATP_Olambda__738,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: A,Uuc: nat] : aa(nat,list(A),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_qi(A,fun(list(A),fun(A,fun(nat,list(A)))),Uu),Uua),Uub),Uuc) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),list_update(A,Uua,Uuc,Uub)) ).

% ATP.lambda_738
tff(fact_8920_ATP_Olambda__739,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: fun(B,A),Uuc: C] : aa(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_mo(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,aa(fun(B,A),fun(set(B),A),groups7121269368397514597t_prod(B,A),Uub),collect(B,aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_mm(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_739
tff(fact_8921_ATP_Olambda__740,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: set(B),Uua: fun(B,C),Uub: fun(B,A),Uuc: C] : aa(C,A,aa(fun(B,A),fun(C,A),aa(fun(B,C),fun(fun(B,A),fun(C,A)),aTP_Lamp_mn(set(B),fun(fun(B,C),fun(fun(B,A),fun(C,A))),Uu),Uua),Uub),Uuc) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,Uub),collect(B,aa(C,fun(B,bool),aa(fun(B,C),fun(C,fun(B,bool)),aTP_Lamp_mm(set(B),fun(fun(B,C),fun(C,fun(B,bool))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_740
tff(fact_8922_ATP_Olambda__741,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: real,Uuc: A] :
          ( pp(aa(A,bool,aa(real,fun(A,bool),aa(nat,fun(real,fun(A,bool)),aTP_Lamp_ul(fun(nat,A),fun(nat,fun(real,fun(A,bool))),Uu),Uua),Uub),Uuc))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),Uub),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_at(fun(nat,A),fun(A,fun(nat,A)),Uu),Uuc)),aa(nat,set(nat),set_ord_atMost(nat),Uua))))) ) ) ).

% ATP.lambda_741
tff(fact_8923_ATP_Olambda__742,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_zv(A,fun(list(A),fun(A,fun(list(A),A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),ord_min(A),Uu),min_list(A,Uua)) ) ).

% ATP.lambda_742
tff(fact_8924_ATP_Olambda__743,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: fun(A,fun(list(A),B)),Uub: A,Uuc: list(A)] : aa(list(A),C,aa(A,fun(list(A),C),aa(fun(A,fun(list(A),B)),fun(A,fun(list(A),C)),aTP_Lamp_zq(fun(B,C),fun(fun(A,fun(list(A),B)),fun(A,fun(list(A),C))),Uu),Uua),Uub),Uuc) = aa(B,C,Uu,aa(list(A),B,aa(A,fun(list(A),B),Uua,Uub),Uuc)) ).

% ATP.lambda_743
tff(fact_8925_ATP_Olambda__744,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(bool,fun(bool,A)),Uub: bool,Uuc: bool] : aa(bool,B,aa(bool,fun(bool,B),aa(fun(bool,fun(bool,A)),fun(bool,fun(bool,B)),aTP_Lamp_ac(fun(A,B),fun(fun(bool,fun(bool,A)),fun(bool,fun(bool,B))),Uu),Uua),Uub),Uuc) = aa(A,B,Uu,aa(bool,A,aa(bool,fun(bool,A),Uua,Uub),Uuc)) ).

% ATP.lambda_744
tff(fact_8926_ATP_Olambda__745,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_hs(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_745
tff(fact_8927_ATP_Olambda__746,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_hq(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_746
tff(fact_8928_ATP_Olambda__747,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_gm(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_747
tff(fact_8929_ATP_Olambda__748,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_fm(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_748
tff(fact_8930_ATP_Olambda__749,axiom,
    ! [A: $tType,D: $tType,B: $tType,C: $tType,Uu: fun(product_prod(B,C),A),Uua: fun(D,B),Uub: D,Uuc: C] : aa(C,A,aa(D,fun(C,A),aa(fun(D,B),fun(D,fun(C,A)),aTP_Lamp_aar(fun(product_prod(B,C),A),fun(fun(D,B),fun(D,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu,aa(C,product_prod(B,C),product_Pair(B,C,aa(D,B,Uua,Uub)),Uuc)) ).

% ATP.lambda_749
tff(fact_8931_ATP_Olambda__750,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Uu: fun(product_prod(B,C),A),Uua: fun(D,C),Uub: B,Uuc: D] : aa(D,A,aa(B,fun(D,A),aa(fun(D,C),fun(B,fun(D,A)),aTP_Lamp_aat(fun(product_prod(B,C),A),fun(fun(D,C),fun(B,fun(D,A))),Uu),Uua),Uub),Uuc) = aa(product_prod(B,C),A,Uu,aa(C,product_prod(B,C),product_Pair(B,C,Uub),aa(D,C,Uua,Uuc))) ).

% ATP.lambda_750
tff(fact_8932_ATP_Olambda__751,axiom,
    ! [A: $tType,Uu: fun(A,bool),Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_adc(fun(A,bool),fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = 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)),takeWhile(A,aa(fun(A,bool),fun(A,bool),comp(bool,bool,A,fNot),Uu),Uua)),aa(list(A),product_prod(A,list(A)),product_Pair(A,list(A),Uub),Uuc))) ).

% ATP.lambda_751
tff(fact_8933_ATP_Olambda__752,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_abo(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = 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)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),Uua)),aa(list(A),product_prod(A,list(A)),product_Pair(A,list(A),Uub),Uuc))) ).

% ATP.lambda_752
tff(fact_8934_ATP_Olambda__753,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: set(product_prod(A,C)),Uua: set(product_prod(C,B)),Uub: A,Uuc: B] :
      ( pp(aa(B,bool,aa(A,fun(B,bool),aa(set(product_prod(C,B)),fun(A,fun(B,bool)),aTP_Lamp_adj(set(product_prod(A,C)),fun(set(product_prod(C,B)),fun(A,fun(B,bool))),Uu),Uua),Uub),Uuc))
    <=> ? [Y5: C] :
          ( pp(aa(set(product_prod(A,C)),bool,aa(product_prod(A,C),fun(set(product_prod(A,C)),bool),member(product_prod(A,C)),aa(C,product_prod(A,C),product_Pair(A,C,Uub),Y5)),Uu))
          & pp(aa(set(product_prod(C,B)),bool,aa(product_prod(C,B),fun(set(product_prod(C,B)),bool),member(product_prod(C,B)),aa(B,product_prod(C,B),product_Pair(C,B,Y5),Uuc)),Uua)) ) ) ).

% ATP.lambda_753
tff(fact_8935_ATP_Olambda__754,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: set(C),Uua: fun(C,A),Uub: fun(C,B),Uuc: product_prod(A,B)] :
      ( pp(aa(product_prod(A,B),bool,aa(fun(C,B),fun(product_prod(A,B),bool),aa(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool)),aTP_Lamp_kn(set(C),fun(fun(C,A),fun(fun(C,B),fun(product_prod(A,B),bool))),Uu),Uua),Uub),Uuc))
    <=> ? [A6: C] :
          ( ( Uuc = aa(B,product_prod(A,B),product_Pair(A,B,aa(C,A,Uua,A6)),aa(C,B,Uub,A6)) )
          & pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),A6),Uu)) ) ) ).

% ATP.lambda_754
tff(fact_8936_ATP_Olambda__755,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,bool),Uua: fun(B,bool),Uub: fun(A,fun(B,C)),Uuc: C] :
      ( pp(aa(C,bool,aa(fun(A,fun(B,C)),fun(C,bool),aa(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool)),aTP_Lamp_jx(fun(A,bool),fun(fun(B,bool),fun(fun(A,fun(B,C)),fun(C,bool))),Uu),Uua),Uub),Uuc))
    <=> ? [X4: A,Y5: B] :
          ( ( Uuc = aa(B,C,aa(A,fun(B,C),Uub,X4),Y5) )
          & pp(aa(A,bool,Uu,X4))
          & pp(aa(B,bool,Uua,Y5)) ) ) ).

% ATP.lambda_755
tff(fact_8937_ATP_Olambda__756,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(product_prod(B,B)),Uub: fun(A,B),Uuc: product_prod(A,A)] :
      ( pp(aa(product_prod(A,A),bool,aa(fun(A,B),fun(product_prod(A,A),bool),aa(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool)),aTP_Lamp_aek(set(A),fun(set(product_prod(B,B)),fun(fun(A,B),fun(product_prod(A,A),bool))),Uu),Uua),Uub),Uuc))
    <=> ? [A19: A,A26: A] :
          ( ( Uuc = aa(A,product_prod(A,A),product_Pair(A,A,A19),A26) )
          & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A19),Uu))
          & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A26),Uu))
          & pp(aa(set(product_prod(B,B)),bool,aa(product_prod(B,B),fun(set(product_prod(B,B)),bool),member(product_prod(B,B)),aa(B,product_prod(B,B),product_Pair(B,B,aa(A,B,Uub,A19)),aa(A,B,Uub,A26))),Uua)) ) ) ).

% ATP.lambda_756
tff(fact_8938_ATP_Olambda__757,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,fun(A,B)),Uua: fun(C,A),Uub: C,Uuc: fun(C,A),Uud: C] : aa(C,B,aa(fun(C,A),fun(C,B),aa(C,fun(fun(C,A),fun(C,B)),aa(fun(C,A),fun(C,fun(fun(C,A),fun(C,B))),aTP_Lamp_pr(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,B,aa(A,fun(A,B),Uu,aa(C,A,Uua,Uub)),aa(C,A,Uuc,Uud)) ) ).

% ATP.lambda_757
tff(fact_8939_ATP_Olambda__758,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_ff(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aa(A,fun(A,fun(nat,fun(nat,A))),aTP_Lamp_fe(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uua),Uub),Uuc),Uud)),aa(nat,set(nat),set_ord_lessThan(nat),aa(nat,nat,minus_minus(nat,Uu),Uud))) ) ).

% ATP.lambda_758
tff(fact_8940_ATP_Olambda__759,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_pd(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Uu),Uua),Uub),Uuc),Uud) = aa(real,real,minus_minus(real,aa(real,real,aa(nat,fun(real,real),Uua,Uuc),Uud)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_pc(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uua),Uuc),Uud)),aa(nat,set(nat),set_ord_lessThan(nat),aa(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,power_power(real,Uud),aa(nat,nat,minus_minus(nat,Uu),Uuc)),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,Uu),Uuc)))))) ).

% ATP.lambda_759
tff(fact_8941_ATP_Olambda__760,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_fz(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_fy(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uua),Uuc),Uud)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uud),Uu))),aa(nat,A,power_power(A,Uub),Uud)) ) ).

% ATP.lambda_760
tff(fact_8942_ATP_Olambda__761,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_du(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Uud)),Uua)),one_one(A))),Uud)),aa(nat,A,power_power(A,Uub),Uud))),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),aa(nat,nat,minus_minus(nat,Uu),Uud))) ) ).

% ATP.lambda_761
tff(fact_8943_ATP_Olambda__762,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_do(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,power_power(A,Uub),Uud))),aa(nat,A,power_power(A,Uuc),aa(nat,nat,minus_minus(nat,Uu),Uud))) ) ).

% ATP.lambda_762
tff(fact_8944_ATP_Olambda__763,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_dp(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),Uua)),Uud)),aa(nat,A,power_power(A,aa(A,A,uminus_uminus(A),Uub)),Uud))),aa(nat,A,power_power(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),aa(nat,nat,minus_minus(nat,Uu),Uud))) ) ).

% ATP.lambda_763
tff(fact_8945_ATP_Olambda__764,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_fe(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,power_power(A,Uub),Uud))),aa(nat,A,power_power(A,Uua),Uuc)) ) ).

% ATP.lambda_764
tff(fact_8946_ATP_Olambda__765,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V7819770556892013058_space(A)
        & real_V7819770556892013058_space(B) )
     => ! [Uu: fun(C,A),Uua: A,Uub: fun(C,B),Uuc: B,Uud: C] :
          ( pp(aa(C,bool,aa(B,fun(C,bool),aa(fun(C,B),fun(B,fun(C,bool)),aa(A,fun(fun(C,B),fun(B,fun(C,bool))),aTP_Lamp_vk(fun(C,A),fun(A,fun(fun(C,B),fun(B,fun(C,bool)))),Uu),Uua),Uub),Uuc),Uud))
        <=> pp(aa(real,bool,aa(real,fun(real,bool),ord_less_eq(real),real_V557655796197034286t_dist(B,aa(C,B,Uub,Uud),Uuc)),real_V557655796197034286t_dist(A,aa(C,A,Uu,Uud),Uua))) ) ) ).

% ATP.lambda_765
tff(fact_8947_ATP_Olambda__766,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_pv(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),Uuc)),aa(A,B,Uua,Uud))),aa(nat,B,power_power(B,aa(A,B,Uu,Uub)),aa(nat,nat,minus_minus(nat,Uuc),one_one(nat)))) ) ).

% ATP.lambda_766
tff(fact_8948_ATP_Olambda__767,axiom,
    ! [Uu: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat] :
      ( pp(aa(nat,bool,aa(nat,fun(nat,bool),aa(vEBT_VEBT,fun(nat,fun(nat,bool)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool))),aTP_Lamp_lg(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,bool)))),Uu),Uua),Uub),Uuc),Uud))
    <=> ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),Uuc),Uud))
        & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uud),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uu)))
        & ! [I3: nat] :
            ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),I3),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),aa(nat,nat,minus_minus(nat,Uu),divide_divide(nat,Uu,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2)))))))
           => ( ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),I3)),X_1))
            <=> pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,Uub),I3)) ) )
        & ( ( Uuc = Uud )
         => ! [X4: vEBT_VEBT] :
              ( pp(aa(set(vEBT_VEBT),bool,aa(vEBT_VEBT,fun(set(vEBT_VEBT),bool),member(vEBT_VEBT),X4),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Uua)))
             => ~ ? [X_1: nat] : pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,X4),X_1)) ) )
        & ( ( Uuc != Uud )
         => ( vEBT_V5917875025757280293ildren(divide_divide(nat,Uu,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua,Uud)
            & ! [X4: nat] :
                ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),X4),aa(nat,nat,power_power(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uu)))
               => ( vEBT_V5917875025757280293ildren(divide_divide(nat,Uu,aa(num,nat,numeral_numeral(nat),aa(num,num,bit0,one2))),Uua,X4)
                 => ( pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less(nat),Uuc),X4))
                    & pp(aa(nat,bool,aa(nat,fun(nat,bool),ord_less_eq(nat),X4),Uud)) ) ) ) ) ) ) ) ).

% ATP.lambda_767
tff(fact_8949_ATP_Olambda__768,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( semiring_0(B)
     => ! [Uu: fun(A,B),Uua: fun(C,B),Uub: set(A),Uuc: set(C),Uud: B] :
          ( pp(aa(B,bool,aa(set(C),fun(B,bool),aa(set(A),fun(set(C),fun(B,bool)),aa(fun(C,B),fun(set(A),fun(set(C),fun(B,bool))),aTP_Lamp_aew(fun(A,B),fun(fun(C,B),fun(set(A),fun(set(C),fun(B,bool)))),Uu),Uua),Uub),Uuc),Uud))
        <=> ? [A6: A,B6: C] :
              ( ( Uud = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,A6)),aa(C,B,Uua,B6)) )
              & pp(aa(set(A),bool,aa(A,fun(set(A),bool),member(A),A6),Uub))
              & pp(aa(set(C),bool,aa(C,fun(set(C),bool),member(C),B6),Uuc)) ) ) ) ).

% ATP.lambda_768
tff(fact_8950_ATP_Olambda__769,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C,Uuc: fun(C,A),Uud: fun(C,A),Uue: C] : aa(C,A,aa(fun(C,A),fun(C,A),aa(fun(C,A),fun(fun(C,A),fun(C,A)),aa(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))),aa(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A)))),aTP_Lamp_pt(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = divide_divide(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,A,Uua,Uue)),aa(C,A,Uuc,Uub))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,A,Uu,Uub)),aa(C,A,Uud,Uue))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(C,A,Uuc,Uub)),aa(C,A,Uuc,Uub))) ) ).

% ATP.lambda_769
tff(fact_8951_ATP_Olambda__770,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: fun(A,real),Uud: fun(A,real),Uue: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(fun(A,real),fun(fun(A,real),fun(A,real)),aa(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))),aa(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real)))),aTP_Lamp_qb(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(A,real,Uu,Uub),aa(A,real,Uuc,Uub))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uud,Uue)),aa(real,real,ln_ln(real),aa(A,real,Uu,Uub)))),divide_divide(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uue)),aa(A,real,Uuc,Uub)),aa(A,real,Uu,Uub)))) ) ).

% ATP.lambda_770
tff(fact_8952_ATP_Olambda__771,axiom,
    ! [C: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(D)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(D,real),Uua: fun(D,real),Uub: D,Uuc: fun(D,C),Uud: fun(D,C),Uue: D] : aa(D,C,aa(fun(D,C),fun(D,C),aa(fun(D,C),fun(fun(D,C),fun(D,C)),aa(D,fun(fun(D,C),fun(fun(D,C),fun(D,C))),aa(fun(D,real),fun(D,fun(fun(D,C),fun(fun(D,C),fun(D,C)))),aTP_Lamp_pm(fun(D,real),fun(fun(D,real),fun(D,fun(fun(D,C),fun(fun(D,C),fun(D,C))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(C,C,aa(C,fun(C,C),plus_plus(C),aa(C,C,real_V8093663219630862766scaleR(C,aa(D,real,Uu,Uub)),aa(D,C,Uud,Uue))),aa(C,C,real_V8093663219630862766scaleR(C,aa(D,real,Uua,Uue)),aa(D,C,Uuc,Uub))) ) ).

% ATP.lambda_771
tff(fact_8953_ATP_Olambda__772,axiom,
    ! [A: $tType,D: $tType] :
      ( ( real_V822414075346904944vector(D)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(D,A),Uua: fun(D,A),Uub: D,Uuc: fun(D,A),Uud: fun(D,A),Uue: D] : aa(D,A,aa(fun(D,A),fun(D,A),aa(fun(D,A),fun(fun(D,A),fun(D,A)),aa(D,fun(fun(D,A),fun(fun(D,A),fun(D,A))),aa(fun(D,A),fun(D,fun(fun(D,A),fun(fun(D,A),fun(D,A)))),aTP_Lamp_pp(fun(D,A),fun(fun(D,A),fun(D,fun(fun(D,A),fun(fun(D,A),fun(D,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(D,A,Uu,Uub)),aa(D,A,Uud,Uue))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(D,A,Uua,Uue)),aa(D,A,Uuc,Uub))) ) ).

% ATP.lambda_772
tff(fact_8954_ATP_Olambda__773,axiom,
    ! [A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(C,A),Uua: fun(C,A),Uub: C,Uuc: fun(C,A),Uud: fun(C,A),Uue: C] : aa(C,A,aa(fun(C,A),fun(C,A),aa(fun(C,A),fun(fun(C,A),fun(C,A)),aa(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))),aa(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A)))),aTP_Lamp_pz(fun(C,A),fun(fun(C,A),fun(C,fun(fun(C,A),fun(fun(C,A),fun(C,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = 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,uminus_uminus(A),aa(C,A,Uu,Uub))),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(C,A,Uuc,Uub))),aa(C,A,Uud,Uue))),aa(A,A,inverse_inverse(A),aa(C,A,Uuc,Uub))))),divide_divide(A,aa(C,A,Uua,Uue),aa(C,A,Uuc,Uub))) ) ).

% ATP.lambda_773
tff(fact_8955_ATP_Olambda__774,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT] : 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))),aa(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)))),aTP_Lamp_ab(fun(A,B),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))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(A,B,Uu,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))),Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_774
tff(fact_8956_ATP_Olambda__775,axiom,
    ! [B: $tType,C: $tType,Uu: set(C),Uua: B] : aa(B,set(C),aTP_Lamp_adr(set(C),fun(B,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_775
tff(fact_8957_ATP_Olambda__776,axiom,
    ! [A: $tType,C: $tType,Uu: set(C),Uua: A] : aa(A,set(C),aTP_Lamp_ads(set(C),fun(A,set(C)),Uu),Uua) = Uu ).

% ATP.lambda_776
tff(fact_8958_ATP_Olambda__777,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_adv(set(B),fun(A,set(B)),Uu),Uua) = Uu ) ).

% ATP.lambda_777
tff(fact_8959_ATP_Olambda__778,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: A] : aa(A,set(B),aTP_Lamp_adq(set(B),fun(A,set(B)),Uu),Uua) = Uu ).

% ATP.lambda_778
tff(fact_8960_ATP_Olambda__779,axiom,
    ! [A: $tType,Uu: set(A),Uua: list(A)] : aa(list(A),set(A),aTP_Lamp_ado(set(A),fun(list(A),set(A)),Uu),Uua) = Uu ).

% ATP.lambda_779
tff(fact_8961_ATP_Olambda__780,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: B] : aa(B,set(A),aTP_Lamp_adu(set(A),fun(B,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_780
tff(fact_8962_ATP_Olambda__781,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] : aa(A,set(A),aTP_Lamp_adt(set(A),fun(A,set(A)),Uu),Uua) = Uu ).

% ATP.lambda_781
tff(fact_8963_ATP_Olambda__782,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_aer(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_782
tff(fact_8964_ATP_Olambda__783,axiom,
    ! [C: $tType,B: $tType] :
      ( semiring_1(B)
     => ! [Uu: B,Uua: C] : aa(C,B,aTP_Lamp_abn(B,fun(C,B),Uu),Uua) = Uu ) ).

% ATP.lambda_783
tff(fact_8965_ATP_Olambda__784,axiom,
    ! [A: $tType] :
      ( counta3822494911875563373attice(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_nd(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_784
tff(fact_8966_ATP_Olambda__785,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_kb(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_785
tff(fact_8967_ATP_Olambda__786,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_aet(A,fun(list(A),A)),Uu),Uua) = Uu ) ).

% ATP.lambda_786
tff(fact_8968_ATP_Olambda__787,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_aae(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_787
tff(fact_8969_ATP_Olambda__788,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_zz(A,fun(list(A),A)),Uu),Uua) = Uu ).

% ATP.lambda_788
tff(fact_8970_ATP_Olambda__789,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aaw(A,fun(nat,A),Uu),Uua) = Uu ).

% ATP.lambda_789
tff(fact_8971_ATP_Olambda__790,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aTP_Lamp_aav(A,fun(B,A),Uu),Uua) = Uu ).

% ATP.lambda_790
tff(fact_8972_ATP_Olambda__791,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_zr(A,fun(list(A),list(A))),Uu),Uua) = Uua ).

% ATP.lambda_791
tff(fact_8973_ATP_Olambda__792,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_zw(A,fun(list(A),bool)),Uu),Uua))
    <=> $false ) ).

% ATP.lambda_792
tff(fact_8974_ATP_Olambda__793,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( pp(aa(list(A),bool,aa(A,fun(list(A),bool),aTP_Lamp_zx(A,fun(list(A),bool)),Uu),Uua))
    <=> $true ) ).

% ATP.lambda_793
tff(fact_8975_ATP_Olambda__794,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( pp(aa(A,bool,aa(A,fun(A,bool),aTP_Lamp_acy(A,fun(A,bool)),Uu),Uua))
    <=> $true ) ).

% ATP.lambda_794
tff(fact_8976_ATP_Olambda__795,axiom,
    ! [Uu: complex] : aa(complex,complex,aTP_Lamp_eo(complex,complex),Uu) = Uu ).

% ATP.lambda_795
tff(fact_8977_ATP_Olambda__796,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_fu(nat,nat),Uu) = Uu ).

% ATP.lambda_796
tff(fact_8978_ATP_Olambda__797,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_hd(int,int),Uu) = Uu ).

% ATP.lambda_797
tff(fact_8979_ATP_Olambda__798,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_od(A,A),Uu) = Uu ) ).

% ATP.lambda_798
tff(fact_8980_ATP_Olambda__799,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_wx(A,A),Uu) = Uu ) ).

% ATP.lambda_799
tff(fact_8981_ATP_Olambda__800,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_abm(A,A),Uu) = Uu ) ).

% ATP.lambda_800
tff(fact_8982_ATP_Olambda__801,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_ma(A,A),Uu) = Uu ) ).

% ATP.lambda_801
tff(fact_8983_ATP_Olambda__802,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_af(A,A),Uu) = Uu ) ).

% ATP.lambda_802
tff(fact_8984_ATP_Olambda__803,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_aam(A,A),Uu) = Uu ).

% ATP.lambda_803
tff(fact_8985_ATP_Olambda__804,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_aa(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_804
tff(fact_8986_ATP_Olambda__805,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_gd(B,A),Uu) = one_one(A) ) ).

% ATP.lambda_805
tff(fact_8987_ATP_Olambda__806,axiom,
    ! [A: $tType,Uu: A] : aa(A,real,aTP_Lamp_kh(A,real),Uu) = one_one(real) ).

% ATP.lambda_806
tff(fact_8988_ATP_Olambda__807,axiom,
    ! [A: $tType,Uu: A] : aa(A,nat,aTP_Lamp_kd(A,nat),Uu) = one_one(nat) ).

% ATP.lambda_807
tff(fact_8989_ATP_Olambda__808,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_aaa(B,option(A)),Uu) = none(A) ).

% ATP.lambda_808
tff(fact_8990_ATP_Olambda__809,axiom,
    ! [A: $tType,C: $tType,Uu: A] : aa(A,option(C),aTP_Lamp_yo(A,option(C)),Uu) = none(C) ).

% ATP.lambda_809
tff(fact_8991_ATP_Olambda__810,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_zy(A,option(B)),Uu) = none(B) ).

% ATP.lambda_810
tff(fact_8992_ATP_Olambda__811,axiom,
    ! [Uu: real] :
      ( pp(aa(real,bool,aTP_Lamp_gn(real,bool),Uu))
    <=> $false ) ).

% ATP.lambda_811
tff(fact_8993_ATP_Olambda__812,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_ih(nat,bool),Uu))
    <=> $false ) ).

% ATP.lambda_812
tff(fact_8994_ATP_Olambda__813,axiom,
    ! [A: $tType,Uu: A] :
      ( pp(aa(A,bool,aTP_Lamp_lh(A,bool),Uu))
    <=> $false ) ).

% ATP.lambda_813
tff(fact_8995_ATP_Olambda__814,axiom,
    ! [Uu: nat] :
      ( pp(aa(nat,bool,aTP_Lamp_ii(nat,bool),Uu))
    <=> $true ) ).

% ATP.lambda_814
tff(fact_8996_ATP_Olambda__815,axiom,
    ! [A: $tType,Uu: A] :
      ( pp(aa(A,bool,aTP_Lamp_li(A,bool),Uu))
    <=> $true ) ).

% ATP.lambda_815
tff(fact_8997_ATP_Olambda__816,axiom,
    ! [B: $tType,Uu: B] : aa(B,fun(nat,nat),aTP_Lamp_zi(B,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_816
tff(fact_8998_ATP_Olambda__817,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_zh(A,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_817

% Type constructors (783)
tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice,axiom,
    ! [A9: $tType,A20: $tType] :
      ( comple592849572758109894attice(A20)
     => counta4013691401010221786attice(fun(A9,A20)) ) ).

tff(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
    ! [A9: $tType,A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => condit1219197933456340205attice(fun(A9,A20)) ) ).

tff(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
    ! [A9: $tType,A20: $tType] :
      ( counta3822494911875563373attice(A20)
     => counta3822494911875563373attice(fun(A9,A20)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
    ! [A9: $tType,A20: $tType] :
      ( comple592849572758109894attice(A20)
     => comple592849572758109894attice(fun(A9,A20)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
    ! [A9: $tType,A20: $tType] :
      ( bounded_lattice(A20)
     => bounde4967611905675639751up_bot(fun(A9,A20)) ) ).

tff(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
    ! [A9: $tType,A20: $tType] :
      ( bounded_lattice(A20)
     => bounde4346867609351753570nf_top(fun(A9,A20)) ) ).

tff(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
    ! [A9: $tType,A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => comple6319245703460814977attice(fun(A9,A20)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A9: $tType,A20: $tType] :
      ( boolea8198339166811842893lgebra(A20)
     => boolea8198339166811842893lgebra(fun(A9,A20)) ) ).

tff(tcon_fun___Complete__Partial__Order_Occpo,axiom,
    ! [A9: $tType,A20: $tType] :
      ( comple6319245703460814977attice(A20)
     => comple9053668089753744459l_ccpo(fun(A9,A20)) ) ).

tff(tcon_fun___Lattices_Osemilattice__sup,axiom,
    ! [A9: $tType,A20: $tType] :
      ( semilattice_sup(A20)
     => semilattice_sup(fun(A9,A20)) ) ).

tff(tcon_fun___Lattices_Osemilattice__inf,axiom,
    ! [A9: $tType,A20: $tType] :
      ( semilattice_inf(A20)
     => semilattice_inf(fun(A9,A20)) ) ).

tff(tcon_fun___Lattices_Odistrib__lattice,axiom,
    ! [A9: $tType,A20: $tType] :
      ( distrib_lattice(A20)
     => distrib_lattice(fun(A9,A20)) ) ).

tff(tcon_fun___Lattices_Obounded__lattice,axiom,
    ! [A9: $tType,A20: $tType] :
      ( bounded_lattice(A20)
     => bounded_lattice(fun(A9,A20)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A9: $tType,A20: $tType] :
      ( order_top(A20)
     => order_top(fun(A9,A20)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A9: $tType,A20: $tType] :
      ( order_bot(A20)
     => order_bot(fun(A9,A20)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A9: $tType,A20: $tType] :
      ( preorder(A20)
     => preorder(fun(A9,A20)) ) ).

tff(tcon_fun___Lattices_Olattice,axiom,
    ! [A9: $tType,A20: $tType] :
      ( lattice(A20)
     => lattice(fun(A9,A20)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A9: $tType,A20: $tType] :
      ( order(A20)
     => order(fun(A9,A20)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A9: $tType,A20: $tType] :
      ( ord(A20)
     => ord(fun(A9,A20)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A9: $tType,A20: $tType] :
      ( uminus(A20)
     => uminus(fun(A9,A20)) ) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
    condit6923001295902523014norder(int) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1,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___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_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_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_2,axiom,
    semilattice_sup(int) ).

tff(tcon_Int_Oint___Lattices_Osemilattice__inf_3,axiom,
    semilattice_inf(int) ).

tff(tcon_Int_Oint___Lattices_Odistrib__lattice_4,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_5,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_6,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_7,axiom,
    order(int) ).

tff(tcon_Int_Oint___Num_Oneg__numeral,axiom,
    neg_numeral(int) ).

tff(tcon_Int_Oint___Nat_Oring__char__0,axiom,
    ring_char_0(int) ).

tff(tcon_Int_Oint___Rings_Osemiring,axiom,
    semiring(int) ).

tff(tcon_Int_Oint___Orderings_Oord_8,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_9,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___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_10,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_11,axiom,
    condit1219197933456340205attice(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_12,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_13,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_14,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_15,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_16,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_17,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_18,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_19,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_20,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_21,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_22,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_23,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_24,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_25,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_26,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_27,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_28,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_29,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__mult_30,axiom,
    topolo4987421752381908075d_mult(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_31,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_32,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_33,axiom,
    topolo2564578578187576103pology(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_34,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_35,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_36,axiom,
    topolo4211221413907600880p_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_37,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_38,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_39,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_40,axiom,
    semiri6843258321239162965malize(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__mult_41,axiom,
    topolo1898628316856586783d_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_42,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_43,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__add_44,axiom,
    topolo6943815403480290642id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_45,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_46,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_47,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_48,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_49,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_50,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_51,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_52,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_53,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__sup_54,axiom,
    semilattice_sup(nat) ).

tff(tcon_Nat_Onat___Lattices_Osemilattice__inf_55,axiom,
    semilattice_inf(nat) ).

tff(tcon_Nat_Onat___Lattices_Odistrib__lattice_56,axiom,
    distrib_lattice(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_57,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_58,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_59,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_60,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_61,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_62,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_63,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_64,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_65,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_66,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_67,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_68,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_69,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_70,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_71,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_72,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_73,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_74,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_75,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_76,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_77,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_78,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_79,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_80,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_81,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_82,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_83,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_84,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___Lattices_Olattice_85,axiom,
    lattice(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_86,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__Gcd_87,axiom,
    semiring_Gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_88,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_89,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_90,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_91,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Power_Opower_92,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_93,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_94,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_95,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_96,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_97,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Orderings_Opreorder_98,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_99,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_100,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_101,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Oplus_102,axiom,
    plus(num) ).

tff(tcon_Num_Onum___Nat_Osize_103,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_104,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_105,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_106,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_107,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_108,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_109,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_110,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_111,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_112,axiom,
    ordere8940638589300402666id_add(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
    archim462609752435547400_field(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_113,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_114,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_115,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_116,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_117,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_118,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_119,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_120,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_121,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_122,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_123,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_124,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_125,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_126,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_127,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_128,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_129,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_130,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_131,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_132,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_133,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_134,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__sup_135,axiom,
    semilattice_sup(rat) ).

tff(tcon_Rat_Orat___Lattices_Osemilattice__inf_136,axiom,
    semilattice_inf(rat) ).

tff(tcon_Rat_Orat___Lattices_Odistrib__lattice_137,axiom,
    distrib_lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_138,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_139,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_140,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_141,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_142,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_143,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_144,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_145,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_146,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_147,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_148,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_149,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_150,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_151,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_152,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_153,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_154,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_155,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_156,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_157,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_158,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_159,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_160,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_161,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_162,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_163,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_164,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_165,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_166,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_167,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_168,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_169,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Lattices_Olattice_170,axiom,
    lattice(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_171,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_172,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_173,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_174,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_175,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_176,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_177,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_178,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_179,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_180,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_181,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_182,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_183,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_184,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_185,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_186,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_187,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_188,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_189,axiom,
    dvd(rat) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_190,axiom,
    ! [A9: $tType] : counta4013691401010221786attice(set(A9)) ).

tff(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_191,axiom,
    ! [A9: $tType] : condit1219197933456340205attice(set(A9)) ).

tff(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_192,axiom,
    ! [A9: $tType] : counta3822494911875563373attice(set(A9)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_193,axiom,
    ! [A9: $tType] : comple592849572758109894attice(set(A9)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_194,axiom,
    ! [A9: $tType] : bounde4967611905675639751up_bot(set(A9)) ).

tff(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_195,axiom,
    ! [A9: $tType] : bounde4346867609351753570nf_top(set(A9)) ).

tff(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_196,axiom,
    ! [A9: $tType] : comple6319245703460814977attice(set(A9)) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_197,axiom,
    ! [A9: $tType] : boolea8198339166811842893lgebra(set(A9)) ).

tff(tcon_Set_Oset___Complete__Partial__Order_Occpo_198,axiom,
    ! [A9: $tType] : comple9053668089753744459l_ccpo(set(A9)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__sup_199,axiom,
    ! [A9: $tType] : semilattice_sup(set(A9)) ).

tff(tcon_Set_Oset___Lattices_Osemilattice__inf_200,axiom,
    ! [A9: $tType] : semilattice_inf(set(A9)) ).

tff(tcon_Set_Oset___Lattices_Odistrib__lattice_201,axiom,
    ! [A9: $tType] : distrib_lattice(set(A9)) ).

tff(tcon_Set_Oset___Lattices_Obounded__lattice_202,axiom,
    ! [A9: $tType] : bounded_lattice(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_203,axiom,
    ! [A9: $tType] : order_top(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_204,axiom,
    ! [A9: $tType] : order_bot(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_205,axiom,
    ! [A9: $tType] : preorder(set(A9)) ).

tff(tcon_Set_Oset___Lattices_Olattice_206,axiom,
    ! [A9: $tType] : lattice(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Oorder_207,axiom,
    ! [A9: $tType] : order(set(A9)) ).

tff(tcon_Set_Oset___Orderings_Oord_208,axiom,
    ! [A9: $tType] : ord(set(A9)) ).

tff(tcon_Set_Oset___Groups_Ouminus_209,axiom,
    ! [A9: $tType] : uminus(set(A9)) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_210,axiom,
    counta4013691401010221786attice(bool) ).

tff(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_211,axiom,
    condit1219197933456340205attice(bool) ).

tff(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_212,axiom,
    counta3822494911875563373attice(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_213,axiom,
    comple592849572758109894attice(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_214,axiom,
    topolo4958980785337419405_space(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_215,axiom,
    topolo1944317154257567458pology(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_216,axiom,
    bounde4967611905675639751up_bot(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_217,axiom,
    bounde4346867609351753570nf_top(bool) ).

tff(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_218,axiom,
    comple6319245703460814977attice(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_219,axiom,
    topolo2564578578187576103pology(bool) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_220,axiom,
    boolea8198339166811842893lgebra(bool) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_221,axiom,
    topological_t2_space(bool) ).

tff(tcon_HOL_Obool___Complete__Partial__Order_Occpo_222,axiom,
    comple9053668089753744459l_ccpo(bool) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__sup_223,axiom,
    semilattice_sup(bool) ).

tff(tcon_HOL_Obool___Lattices_Osemilattice__inf_224,axiom,
    semilattice_inf(bool) ).

tff(tcon_HOL_Obool___Lattices_Odistrib__lattice_225,axiom,
    distrib_lattice(bool) ).

tff(tcon_HOL_Obool___Lattices_Obounded__lattice_226,axiom,
    bounded_lattice(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_227,axiom,
    order_top(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_228,axiom,
    order_bot(bool) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_229,axiom,
    preorder(bool) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_230,axiom,
    linorder(bool) ).

tff(tcon_HOL_Obool___Lattices_Olattice_231,axiom,
    lattice(bool) ).

tff(tcon_HOL_Obool___Orderings_Oorder_232,axiom,
    order(bool) ).

tff(tcon_HOL_Obool___Orderings_Oord_233,axiom,
    ord(bool) ).

tff(tcon_HOL_Obool___Groups_Ouminus_234,axiom,
    uminus(bool) ).

tff(tcon_List_Olist___Nat_Osize_235,axiom,
    ! [A9: $tType] : size(list(A9)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_236,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_237,axiom,
    condit1219197933456340205attice(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_238,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_239,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_240,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_241,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_242,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_243,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_244,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_245,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_246,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_247,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_248,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_249,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_250,axiom,
    linord715952674999750819strict(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ouniformity__dist,axiom,
    real_V768167426530841204y_dist(real) ).

tff(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_251,axiom,
    unboun7993243217541854897norder(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_252,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_253,axiom,
    linord4140545234300271783up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_254,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_255,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_256,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_257,axiom,
    topolo4211221413907600880p_mult(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ouniform__space,axiom,
    topolo7287701948861334536_space(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_258,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_259,axiom,
    semiri3467727345109120633visors(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
    real_V7819770556892013058_space(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__ab__group__add,axiom,
    topolo1287966508704411220up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_260,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_261,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_262,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_263,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_264,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_265,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_266,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_267,axiom,
    topolo6943815403480290642id_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_268,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_269,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_270,axiom,
    comm_s4317794764714335236cancel(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
    topolo1633459387980952147up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot2__space_271,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_272,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_273,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_274,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_275,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_276,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_277,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_278,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_279,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__sup_280,axiom,
    semilattice_sup(real) ).

tff(tcon_Real_Oreal___Lattices_Osemilattice__inf_281,axiom,
    semilattice_inf(real) ).

tff(tcon_Real_Oreal___Lattices_Odistrib__lattice_282,axiom,
    distrib_lattice(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_283,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_284,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_285,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__add_286,axiom,
    ab_semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_287,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_288,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_289,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_290,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_291,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_292,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_293,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__0_294,axiom,
    comm_semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_295,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__mult_296,axiom,
    semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_297,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_298,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_299,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__abs__sgn_300,axiom,
    field_abs_sgn(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_301,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_302,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring_303,axiom,
    comm_semiring(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_304,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_305,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_306,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_307,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_308,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_309,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_310,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_311,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_312,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__divide_313,axiom,
    idom_divide(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_314,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_315,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_316,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_317,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_318,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_319,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Lattices_Olattice_320,axiom,
    lattice(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_321,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_322,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_323,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_324,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_325,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_326,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring_327,axiom,
    semiring(real) ).

tff(tcon_Real_Oreal___Fields_Oinverse_328,axiom,
    inverse(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_329,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_330,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_331,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_332,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_333,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_334,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_335,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_336,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_337,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oring_338,axiom,
    ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_339,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_340,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_341,axiom,
    dvd(real) ).

tff(tcon_String_Ochar___Nat_Osize_342,axiom,
    size(char) ).

tff(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_343,axiom,
    ! [A9: $tType] : condit1219197933456340205attice(filter(A9)) ).

tff(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_344,axiom,
    ! [A9: $tType] : counta3822494911875563373attice(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_345,axiom,
    ! [A9: $tType] : bounde4967611905675639751up_bot(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_346,axiom,
    ! [A9: $tType] : bounde4346867609351753570nf_top(filter(A9)) ).

tff(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_347,axiom,
    ! [A9: $tType] : comple6319245703460814977attice(filter(A9)) ).

tff(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_348,axiom,
    ! [A9: $tType] : comple9053668089753744459l_ccpo(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_349,axiom,
    ! [A9: $tType] : semilattice_sup(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_350,axiom,
    ! [A9: $tType] : semilattice_inf(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_351,axiom,
    ! [A9: $tType] : distrib_lattice(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Obounded__lattice_352,axiom,
    ! [A9: $tType] : bounded_lattice(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_353,axiom,
    ! [A9: $tType] : order_top(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_354,axiom,
    ! [A9: $tType] : order_bot(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_355,axiom,
    ! [A9: $tType] : preorder(filter(A9)) ).

tff(tcon_Filter_Ofilter___Lattices_Olattice_356,axiom,
    ! [A9: $tType] : lattice(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_357,axiom,
    ! [A9: $tType] : order(filter(A9)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_358,axiom,
    ! [A9: $tType] : ord(filter(A9)) ).

tff(tcon_Option_Ooption___Nat_Osize_359,axiom,
    ! [A9: $tType] : size(option(A9)) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_360,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_361,axiom,
    topolo3112930676232923870pology(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_362,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_363,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_364,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_365,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_366,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_367,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_368,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_369,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_370,axiom,
    real_V768167426530841204y_dist(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_371,axiom,
    topolo5987344860129210374id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_372,axiom,
    semiri2026040879449505780visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_373,axiom,
    real_V2191834092415804123ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_374,axiom,
    real_V8037385150606011577_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_375,axiom,
    topolo4211221413907600880p_mult(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_376,axiom,
    topolo7287701948861334536_space(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_377,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_378,axiom,
    real_V7819770556892013058_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_379,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_380,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_381,axiom,
    cancel2418104881723323429up_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_382,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_383,axiom,
    real_V7773925162809079976_field(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_384,axiom,
    topolo6943815403480290642id_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_385,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_386,axiom,
    comm_s4317794764714335236cancel(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__group__add_387,axiom,
    topolo1633459387980952147up_add(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_388,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_389,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_390,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_391,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_392,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_393,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_394,axiom,
    ab_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_395,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_396,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_397,axiom,
    comm_semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_398,axiom,
    semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_399,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_400,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__add_401,axiom,
    semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_402,axiom,
    field_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_403,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_404,axiom,
    comm_semiring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_405,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_406,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_407,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_408,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_409,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_410,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__divide_411,axiom,
    idom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_412,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_413,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_414,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_415,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_416,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_417,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_418,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_419,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_420,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring_421,axiom,
    semiring(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Oinverse_422,axiom,
    inverse(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_423,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_424,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_425,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_426,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_427,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_428,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_429,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring_430,axiom,
    ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_431,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_432,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_433,axiom,
    dvd(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_434,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_435,axiom,
    counta4013691401010221786attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_436,axiom,
    condit1219197933456340205attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_437,axiom,
    counta3822494911875563373attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_438,axiom,
    comple592849572758109894attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_439,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_440,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_441,axiom,
    canoni5634975068530333245id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_442,axiom,
    bounde4967611905675639751up_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_443,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_444,axiom,
    linord4140545234300271783up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_445,axiom,
    comple6319245703460814977attice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_446,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_447,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_448,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_449,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_450,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_451,axiom,
    comple9053668089753744459l_ccpo(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_452,axiom,
    semilattice_sup(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_453,axiom,
    semilattice_inf(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_454,axiom,
    distrib_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_455,axiom,
    bounded_lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_456,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_457,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_458,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_459,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_460,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_461,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_462,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_463,axiom,
    semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_464,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_465,axiom,
    semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_466,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_467,axiom,
    comm_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_468,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_469,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_470,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_471,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_472,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_473,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_474,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_475,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_476,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_477,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_478,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Lattices_Olattice_479,axiom,
    lattice(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_480,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_481,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring_482,axiom,
    semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_483,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_484,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_485,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_486,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_487,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_488,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_489,axiom,
    dvd(extended_enat) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_490,axiom,
    ! [A9: $tType,A20: $tType] :
      ( ( topolo4958980785337419405_space(A9)
        & topolo4958980785337419405_space(A20) )
     => topolo4958980785337419405_space(product_prod(A9,A20)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_491,axiom,
    ! [A9: $tType,A20: $tType] :
      ( ( topological_t2_space(A9)
        & topological_t2_space(A20) )
     => topological_t2_space(product_prod(A9,A20)) ) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_492,axiom,
    ! [A9: $tType,A20: $tType] : size(product_prod(A9,A20)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_493,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_494,axiom,
    counta4013691401010221786attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_495,axiom,
    condit1219197933456340205attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_496,axiom,
    counta3822494911875563373attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_497,axiom,
    comple592849572758109894attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_498,axiom,
    bounde4967611905675639751up_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_499,axiom,
    bounde4346867609351753570nf_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_500,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_501,axiom,
    comple6319245703460814977attice(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_502,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_503,axiom,
    comple9053668089753744459l_ccpo(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_504,axiom,
    semilattice_sup(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_505,axiom,
    semilattice_inf(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_506,axiom,
    distrib_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_507,axiom,
    bounded_lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_508,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_509,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_510,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_511,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_512,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Lattices_Olattice_513,axiom,
    lattice(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_514,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_515,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_516,axiom,
    uminus(product_unit) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_517,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_518,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_519,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_520,axiom,
    euclid8789492081693882211th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_521,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_522,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_523,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_524,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_525,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_526,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_527,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_528,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_529,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_530,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_531,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_532,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_533,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_534,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_535,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_536,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_537,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_538,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_539,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_540,axiom,
    euclid5891614535332579305n_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_541,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_542,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_543,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_544,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_545,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_546,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_547,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_548,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_549,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_550,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_551,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_552,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_553,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_554,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_555,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_556,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_557,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_558,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_559,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_560,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_561,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_562,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_563,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_564,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_565,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_566,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_567,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_568,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_569,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_570,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_571,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_572,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_573,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_574,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_575,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_576,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_577,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_578,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_579,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_580,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_581,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_582,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_583,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_584,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_585,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_586,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_587,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_588,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_589,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_590,axiom,
    idom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_591,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_592,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_593,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_594,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_595,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_596,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_597,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_598,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_599,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_600,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_601,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_602,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_603,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_604,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_605,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_606,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_607,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_608,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_609,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_610,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_611,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_612,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_613,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_614,axiom,
    dvd(code_integer) ).

tff(tcon_VEBT__Definitions_OVEBT___Nat_Osize_615,axiom,
    size(vEBT_VEBT) ).

% Helper facts (24)
tff(help_If_2_1_T,axiom,
    ! [A: $tType,X: A,Y: A] : if(A,fFalse,X,Y) = Y ).

tff(help_If_1_1_T,axiom,
    ! [A: $tType,X: A,Y: A] : if(A,fTrue,X,Y) = X ).

tff(help_fEx_1_1_U,axiom,
    ! [A: $tType,P: fun(A,bool),X: A] :
      ( ~ pp(aa(A,bool,P,X))
      | pp(aa(fun(A,bool),bool,fEx(A),P)) ) ).

tff(help_fAll_1_1_U,axiom,
    ! [A: $tType,P: fun(A,bool),X: A] :
      ( ~ pp(fAll(A,P))
      | pp(aa(A,bool,P,X)) ) ).

tff(help_fNot_2_1_U,axiom,
    ! [P: bool] :
      ( pp(P)
      | pp(aa(bool,bool,fNot,P)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P: bool] :
      ( ~ pp(aa(bool,bool,fNot,P))
      | ~ pp(P) ) ).

tff(help_COMBB_1_1_U,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(B,C),Q: fun(A,B),R: A] : aa(A,C,combb(B,C,A,P,Q),R) = aa(B,C,P,aa(A,B,Q,R)) ).

tff(help_COMBC_1_1_U,axiom,
    ! [A: $tType,C: $tType,B: $tType,P: fun(A,fun(B,C)),Q: B,R: A] : aa(A,C,combc(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).

tff(help_COMBS_1_1_U,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(A,fun(B,C)),Q: fun(A,B),R: A] : aa(A,C,combs(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),aa(A,B,Q,R)) ).

tff(help_fTrue_1_1_U,axiom,
    pp(fTrue) ).

tff(help_fconj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fconj(P,Q))
      | pp(Q) ) ).

tff(help_fconj_2_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fconj(P,Q))
      | pp(P) ) ).

tff(help_fconj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | ~ pp(Q)
      | pp(fconj(P,Q)) ) ).

tff(help_fdisj_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(fdisj(P,Q))
      | pp(P)
      | pp(Q) ) ).

tff(help_fdisj_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(fdisj(P,Q)) ) ).

tff(help_fdisj_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(P)
      | pp(fdisj(P,Q)) ) ).

tff(help_fFalse_1_1_T,axiom,
    ! [P: bool] :
      ( ( P = fTrue )
      | ( P = fFalse ) ) ).

tff(help_fFalse_1_1_U,axiom,
    ~ pp(fFalse) ).

tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ( X != Y )
      | pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y)) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X: A,Y: A] :
      ( ~ pp(aa(A,bool,aa(A,fun(A,bool),fequal(A),X),Y))
      | ( X = Y ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P: fun(A,bool)] : aa(A,bool,P,fChoice(A,P)) = aa(fun(A,bool),bool,fEx(A),P) ).

tff(help_fimplies_3_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q))
      | ~ pp(P)
      | pp(Q) ) ).

tff(help_fimplies_2_1_U,axiom,
    ! [Q: bool,P: bool] :
      ( ~ pp(Q)
      | pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).

tff(help_fimplies_1_1_U,axiom,
    ! [P: bool,Q: bool] :
      ( pp(P)
      | pp(aa(bool,bool,aa(bool,fun(bool,bool),fimplies,P),Q)) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    pp(aa(nat,bool,aa(vEBT_VEBT,fun(nat,bool),vEBT_V8194947554948674370ptions,vEBT_Node(info,deg,treeList,summary)),x)) ).

%------------------------------------------------------------------------------