SET007 Axioms: SET007+628.ax


%------------------------------------------------------------------------------
% File     : SET007+628 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Asymptotic Notation. Part I: Theory
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : asympt_0 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :  113 (   2 unit)
%            Number of atoms       :  976 (  53 equality)
%            Maximal formula depth :   21 (   9 average)
%            Number of connectives :  910 (  47 ~  ;   4  |; 605  &)
%                                         (  29 <=>; 225 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   32 (   1 propositional; 0-4 arity)
%            Number of functors    :   59 (  15 constant; 0-4 arity)
%            Number of variables   :  279 (   0 singleton; 228 !;  51 ?)
%            Maximal term depth    :    4 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(rc1_asympt_0,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_numbers)
      & ~ v1_xboole_0(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & v2_xreal_0(A)
      & ~ v3_xreal_0(A) ) )).

fof(rc2_asympt_0,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_numbers)
      & ~ v1_xboole_0(A)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & ~ v2_xreal_0(A)
      & v3_xreal_0(A) ) )).

fof(rc3_asympt_0,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_numbers)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & v1_asympt_0(A) ) )).

fof(rc4_asympt_0,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_numbers)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & ~ v3_xreal_0(A) ) )).

fof(rc5_asympt_0,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_numbers)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & ~ v2_xreal_0(A) ) )).

fof(rc6_asympt_0,axiom,(
    ? [A] :
      ( m1_subset_1(A,k1_numbers)
      & v1_xcmplx_0(A)
      & v1_xreal_0(A)
      & ~ v1_asympt_0(A) ) )).

fof(rc7_asympt_0,axiom,(
    ? [A] :
      ( m1_relset_1(A,k5_numbers,k1_numbers)
      & v1_relat_1(A)
      & v1_funct_1(A)
      & v1_funct_2(A,k5_numbers,k1_numbers)
      & v1_seq_1(A)
      & v2_asympt_0(A)
      & v3_asympt_0(A)
      & v4_asympt_0(A)
      & v5_asympt_0(A)
      & v6_asympt_0(A) ) )).

fof(cc1_asympt_0,axiom,(
    ! [A] :
      ( m1_relset_1(A,k5_numbers,k1_numbers)
     => ( ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k1_numbers)
          & v3_asympt_0(A) )
       => ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k1_numbers)
          & v1_seq_1(A)
          & v4_asympt_0(A) ) ) ) )).

fof(cc2_asympt_0,axiom,(
    ! [A] :
      ( m1_relset_1(A,k5_numbers,k1_numbers)
     => ( ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k1_numbers)
          & v4_asympt_0(A) )
       => ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k1_numbers)
          & v1_seq_1(A)
          & v2_asympt_0(A)
          & v5_asympt_0(A) ) ) ) )).

fof(cc3_asympt_0,axiom,(
    ! [A] :
      ( m1_relset_1(A,k5_numbers,k1_numbers)
     => ( ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k1_numbers)
          & v2_asympt_0(A)
          & v5_asympt_0(A) )
       => ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k1_numbers)
          & v1_seq_1(A)
          & v4_asympt_0(A) ) ) ) )).

fof(fc1_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & ~ v3_xreal_0(B)
        & m1_subset_1(B,k1_numbers) )
     => ( v1_relat_1(k2_asympt_0(A,B))
        & v1_funct_1(k2_asympt_0(A,B))
        & v1_funct_2(k2_asympt_0(A,B),k5_numbers,k1_numbers)
        & v1_seq_1(k2_asympt_0(A,B))
        & v2_asympt_0(k2_asympt_0(A,B)) ) ) )).

fof(fc2_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v2_xreal_0(B)
        & m1_subset_1(B,k1_numbers) )
     => ( v1_relat_1(k2_asympt_0(A,B))
        & v1_funct_1(k2_asympt_0(A,B))
        & v1_funct_2(k2_asympt_0(A,B),k5_numbers,k1_numbers)
        & v1_seq_1(k2_asympt_0(A,B))
        & v2_asympt_0(k2_asympt_0(A,B))
        & v4_asympt_0(k2_asympt_0(A,B))
        & v5_asympt_0(k2_asympt_0(A,B)) ) ) )).

fof(fc3_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & v2_asympt_0(B)
        & m1_relset_1(B,k5_numbers,k1_numbers) )
     => ( v1_relat_1(k4_asympt_0(A,B))
        & v1_funct_1(k4_asympt_0(A,B))
        & v1_funct_2(k4_asympt_0(A,B),k5_numbers,k1_numbers)
        & v1_seq_1(k4_asympt_0(A,B))
        & v2_asympt_0(k4_asympt_0(A,B)) ) ) )).

fof(fc4_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & v4_asympt_0(B)
        & m1_relset_1(B,k5_numbers,k1_numbers) )
     => ( v1_relat_1(k4_asympt_0(A,B))
        & v1_funct_1(k4_asympt_0(A,B))
        & v1_funct_2(k4_asympt_0(A,B),k5_numbers,k1_numbers)
        & v1_seq_1(k4_asympt_0(A,B))
        & v2_asympt_0(k4_asympt_0(A,B))
        & v4_asympt_0(k4_asympt_0(A,B))
        & v5_asympt_0(k4_asympt_0(A,B)) ) ) )).

fof(d1_asympt_0,axiom,(
    $true )).

fof(d2_asympt_0,axiom,(
    $true )).

fof(d3_asympt_0,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ( v1_asympt_0(A)
      <=> ( ~ r1_xreal_0(A,np__0)
          & A != np__1 ) ) ) )).

fof(d4_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v2_asympt_0(A)
      <=> ? [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( r1_xreal_0(B,C)
                 => r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,A,C)) ) ) ) ) ) )).

fof(d5_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v3_asympt_0(A)
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,B),np__0) ) ) ) )).

fof(d6_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v4_asympt_0(A)
      <=> ? [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ~ ( r1_xreal_0(B,C)
                    & r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,C),np__0) ) ) ) ) ) )).

fof(d7_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v5_asympt_0(A)
      <=> ? [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ~ ( r1_xreal_0(B,C)
                    & k2_seq_1(k5_numbers,k1_numbers,A,C) = np__0 ) ) ) ) ) )).

fof(d8_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v6_asympt_0(A)
      <=> ? [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( r1_xreal_0(B,C)
                 => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,C),k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(C,np__1))) ) ) ) ) ) )).

fof(d9_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( v1_xreal_0(B)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( C = k2_asympt_0(A,B)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,C,D) = k2_xcmplx_0(B,k2_seq_1(k5_numbers,k1_numbers,A,D)) ) ) ) ) ) )).

fof(d10_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( C = k4_asympt_0(A,B)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,C,D) = k4_square_1(k2_seq_1(k5_numbers,k1_numbers,A,D),k2_seq_1(k5_numbers,k1_numbers,B,D)) ) ) ) ) ) )).

fof(d11_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( r1_asympt_0(A,B)
          <=> ? [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
                & ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( r1_xreal_0(C,D)
                     => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,D),k2_seq_1(k5_numbers,k1_numbers,B,D)) ) ) ) ) ) ) )).

fof(t1_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( r1_xreal_0(B,C)
                 => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,C),k2_seq_1(k5_numbers,k1_numbers,A,k1_nat_1(C,np__1))) ) )
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( ( r1_xreal_0(B,C)
                        & r1_xreal_0(C,D) )
                     => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,C),k2_seq_1(k5_numbers,k1_numbers,A,D)) ) ) ) ) ) ) )).

fof(t2_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v4_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v4_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( v4_seq_2(k19_seq_1(A,B))
           => ( k2_seq_2(k19_seq_1(A,B)) = np__0
              | ( v4_seq_2(k19_seq_1(B,A))
                & k2_seq_2(k19_seq_1(B,A)) = k2_real_1(k2_seq_2(k19_seq_1(A,B))) ) ) ) ) ) )).

fof(t3_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v4_seq_2(A)
       => r1_xreal_0(np__0,k2_seq_2(A)) ) ) )).

fof(t4_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( ( v4_seq_2(A)
              & v4_seq_2(B)
              & r1_asympt_0(A,B) )
           => r1_xreal_0(k2_seq_2(A),k2_seq_2(B)) ) ) ) )).

fof(t5_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v5_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( v1_limfunc1(k19_seq_1(A,B))
           => ( v4_seq_2(k19_seq_1(B,A))
              & k2_seq_2(k19_seq_1(B,A)) = np__0 ) ) ) ) )).

fof(t6_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & v2_asympt_0(B)
        & m2_relset_1(B,k5_numbers,k1_numbers) )
     => ( r2_hidden(A,k5_asympt_0(B))
       => ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k1_numbers)
          & v2_asympt_0(A)
          & m2_relset_1(A,k5_numbers,k1_numbers) ) ) ) )).

fof(t7_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v3_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v2_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( r2_hidden(B,k5_asympt_0(A))
          <=> ? [C] :
                ( m1_subset_1(C,k1_numbers)
                & ~ r1_xreal_0(C,np__0)
                & ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,D),k4_real_1(C,k2_seq_1(k5_numbers,k1_numbers,A,D))) ) ) ) ) ) )).

fof(t8_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v4_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v2_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ~ ( r2_hidden(B,k5_asympt_0(A))
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ~ ( r1_xreal_0(C,D)
                          & r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,A,D),np__0) ) )
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ( r1_xreal_0(C,E)
                               => r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,E),k4_real_1(D,k2_seq_1(k5_numbers,k1_numbers,A,E))) ) ) ) ) ) ) ) ) )).

fof(t9_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v2_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => k5_asympt_0(k1_asympt_0(A,B)) = k5_asympt_0(k4_asympt_0(A,B)) ) ) )).

fof(t10_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => r2_hidden(A,k5_asympt_0(A)) ) )).

fof(t11_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v2_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( r2_hidden(A,k5_asympt_0(B))
           => r1_tarski(k5_asympt_0(A),k5_asympt_0(B)) ) ) ) )).

fof(t12_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v2_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & v2_asympt_0(C)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( ( r2_hidden(A,k5_asympt_0(B))
                  & r2_hidden(B,k5_asympt_0(C)) )
               => r2_hidden(A,k5_asympt_0(C)) ) ) ) ) )).

fof(t13_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v2_xreal_0(B)
            & m1_subset_1(B,k1_numbers) )
         => k5_asympt_0(A) = k5_asympt_0(k3_asympt_0(A,B)) ) ) )).

fof(t14_asympt_0,axiom,(
    ! [A] :
      ( ( ~ v3_xreal_0(A)
        & m1_subset_1(A,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v2_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & v2_asympt_0(C)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( r2_hidden(B,k5_asympt_0(C))
               => r2_hidden(B,k5_asympt_0(k2_asympt_0(C,A))) ) ) ) ) )).

fof(t15_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v4_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v4_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( v4_seq_2(k19_seq_1(A,B))
           => ( r1_xreal_0(k2_seq_2(k19_seq_1(A,B)),np__0)
              | k5_asympt_0(A) = k5_asympt_0(B) ) ) ) ) )).

fof(t16_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v4_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v4_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( ( v4_seq_2(k19_seq_1(A,B))
              & k2_seq_2(k19_seq_1(A,B)) = np__0 )
           => ( r2_hidden(A,k5_asympt_0(B))
              & ~ r2_hidden(B,k5_asympt_0(A)) ) ) ) ) )).

fof(t17_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v4_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v4_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( v1_limfunc1(k19_seq_1(A,B))
           => ( ~ r2_hidden(A,k5_asympt_0(B))
              & r2_hidden(B,k5_asympt_0(A)) ) ) ) ) )).

fof(t18_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & v2_asympt_0(B)
        & m2_relset_1(B,k5_numbers,k1_numbers) )
     => ( r2_hidden(A,k6_asympt_0(B))
       => ( v1_funct_1(A)
          & v1_funct_2(A,k5_numbers,k1_numbers)
          & v2_asympt_0(A)
          & m2_relset_1(A,k5_numbers,k1_numbers) ) ) ) )).

fof(t19_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v2_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( r2_hidden(A,k6_asympt_0(B))
          <=> r2_hidden(B,k5_asympt_0(A)) ) ) ) )).

fof(t20_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => r2_hidden(A,k6_asympt_0(A)) ) )).

fof(t21_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v2_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & v2_asympt_0(C)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( ( r2_hidden(A,k6_asympt_0(B))
                  & r2_hidden(B,k6_asympt_0(C)) )
               => r2_hidden(A,k6_asympt_0(C)) ) ) ) ) )).

fof(t22_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v4_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v4_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( v4_seq_2(k19_seq_1(A,B))
           => ( r1_xreal_0(k2_seq_2(k19_seq_1(A,B)),np__0)
              | k6_asympt_0(A) = k6_asympt_0(B) ) ) ) ) )).

fof(t23_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v4_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v4_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( ( v4_seq_2(k19_seq_1(A,B))
              & k2_seq_2(k19_seq_1(A,B)) = np__0 )
           => ( r2_hidden(B,k6_asympt_0(A))
              & ~ r2_hidden(A,k6_asympt_0(B)) ) ) ) ) )).

fof(t24_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v4_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v4_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( v1_limfunc1(k19_seq_1(A,B))
           => ( ~ r2_hidden(B,k6_asympt_0(A))
              & r2_hidden(A,k6_asympt_0(B)) ) ) ) ) )).

fof(t25_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v3_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v3_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( r2_hidden(B,k6_asympt_0(A))
          <=> ? [C] :
                ( m1_subset_1(C,k1_numbers)
                & ~ r1_xreal_0(C,np__0)
                & ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => r1_xreal_0(k4_real_1(C,k2_seq_1(k5_numbers,k1_numbers,A,D)),k2_seq_1(k5_numbers,k1_numbers,B,D)) ) ) ) ) ) )).

fof(t26_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v2_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => k6_asympt_0(k1_asympt_0(A,B)) = k6_asympt_0(k4_asympt_0(A,B)) ) ) )).

fof(d14_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => k7_asympt_0(A) = k3_xboole_0(k5_asympt_0(A),k6_asympt_0(A)) ) )).

fof(t28_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => r2_hidden(A,k7_asympt_0(A)) ) )).

fof(t29_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v2_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( r2_hidden(A,k7_asympt_0(B))
           => r2_hidden(B,k7_asympt_0(A)) ) ) ) )).

fof(t30_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v2_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & v2_asympt_0(C)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( ( r2_hidden(A,k7_asympt_0(B))
                  & r2_hidden(B,k7_asympt_0(C)) )
               => r2_hidden(A,k7_asympt_0(C)) ) ) ) ) )).

fof(t31_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v3_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v3_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( r2_hidden(B,k7_asympt_0(A))
          <=> ? [C] :
                ( m1_subset_1(C,k1_numbers)
                & ? [D] :
                    ( m1_subset_1(D,k1_numbers)
                    & ~ r1_xreal_0(C,np__0)
                    & ~ r1_xreal_0(D,np__0)
                    & ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r1_xreal_0(k4_real_1(D,k2_seq_1(k5_numbers,k1_numbers,A,E)),k2_seq_1(k5_numbers,k1_numbers,B,E))
                          & r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,E),k4_real_1(C,k2_seq_1(k5_numbers,k1_numbers,A,E))) ) ) ) ) ) ) ) )).

fof(t32_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v2_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => k7_asympt_0(k1_asympt_0(A,B)) = k7_asympt_0(k4_asympt_0(A,B)) ) ) )).

fof(t33_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v4_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v4_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( v4_seq_2(k19_seq_1(A,B))
           => ( r1_xreal_0(k2_seq_2(k19_seq_1(A,B)),np__0)
              | r2_hidden(A,k7_asympt_0(B)) ) ) ) ) )).

fof(t34_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v4_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v4_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( ( v4_seq_2(k19_seq_1(A,B))
              & k2_seq_2(k19_seq_1(A,B)) = np__0 )
           => ( r2_hidden(A,k5_asympt_0(B))
              & ~ r2_hidden(A,k7_asympt_0(B)) ) ) ) ) )).

fof(t35_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v4_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v4_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ( v1_limfunc1(k19_seq_1(A,B))
           => ( r2_hidden(A,k6_asympt_0(B))
              & ~ r2_hidden(A,k7_asympt_0(B)) ) ) ) ) )).

fof(t36_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] : k10_asympt_0(A,B) = k3_xboole_0(k8_asympt_0(A,B),k9_asympt_0(A,B)) ) )).

fof(d18_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v1_funct_2(C,k5_numbers,k1_numbers)
                & m2_relset_1(C,k5_numbers,k1_numbers) )
             => ( C = k11_asympt_0(A,B)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => k2_seq_1(k5_numbers,k1_numbers,C,D) = k2_seq_1(k5_numbers,k1_numbers,A,k2_nat_1(B,D)) ) ) ) ) ) )).

fof(d19_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_asympt_0(A,B)
          <=> ( v6_asympt_0(A)
              & r2_hidden(k11_asympt_0(A,B),k5_asympt_0(A)) ) ) ) ) )).

fof(d20_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( v7_asympt_0(A)
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r1_xreal_0(np__2,B)
             => r2_asympt_0(A,B) ) ) ) ) )).

fof(t37_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ( ? [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
            & r1_xreal_0(np__2,B)
            & r2_asympt_0(A,B) )
       => v7_asympt_0(A) ) ) )).

fof(t41_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v2_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => k12_asympt_0(k5_numbers,k5_asympt_0(A),k5_asympt_0(B)) = k5_asympt_0(k1_asympt_0(A,B)) ) ) )).

fof(t42_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v2_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => k13_asympt_0(k5_numbers,k5_asympt_0(A),k5_asympt_0(B)) = k5_asympt_0(k4_asympt_0(A,B)) ) ) )).

fof(dt_k1_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & v2_asympt_0(B)
        & m1_relset_1(B,k5_numbers,k1_numbers) )
     => ( v1_funct_1(k1_asympt_0(A,B))
        & v1_funct_2(k1_asympt_0(A,B),k5_numbers,k1_numbers)
        & v2_asympt_0(k1_asympt_0(A,B))
        & m2_relset_1(k1_asympt_0(A,B),k5_numbers,k1_numbers) ) ) )).

fof(commutativity_k1_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & v2_asympt_0(B)
        & m1_relset_1(B,k5_numbers,k1_numbers) )
     => k1_asympt_0(A,B) = k1_asympt_0(B,A) ) )).

fof(redefinition_k1_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & v2_asympt_0(B)
        & m1_relset_1(B,k5_numbers,k1_numbers) )
     => k1_asympt_0(A,B) = k3_seq_1(A,B) ) )).

fof(dt_k2_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_xreal_0(B) )
     => ( v1_funct_1(k2_asympt_0(A,B))
        & v1_funct_2(k2_asympt_0(A,B),k5_numbers,k1_numbers)
        & m2_relset_1(k2_asympt_0(A,B),k5_numbers,k1_numbers) ) ) )).

fof(dt_k3_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v2_xreal_0(B)
        & m1_subset_1(B,k1_numbers) )
     => ( v1_funct_1(k3_asympt_0(A,B))
        & v1_funct_2(k3_asympt_0(A,B),k5_numbers,k1_numbers)
        & v2_asympt_0(k3_asympt_0(A,B))
        & m2_relset_1(k3_asympt_0(A,B),k5_numbers,k1_numbers) ) ) )).

fof(redefinition_k3_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v2_xreal_0(B)
        & m1_subset_1(B,k1_numbers) )
     => k3_asympt_0(A,B) = k12_seq_1(A,B) ) )).

fof(dt_k4_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & m1_relset_1(B,k5_numbers,k1_numbers) )
     => ( v1_funct_1(k4_asympt_0(A,B))
        & v1_funct_2(k4_asympt_0(A,B),k5_numbers,k1_numbers)
        & m2_relset_1(k4_asympt_0(A,B),k5_numbers,k1_numbers) ) ) )).

fof(commutativity_k4_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & m1_relset_1(B,k5_numbers,k1_numbers) )
     => k4_asympt_0(A,B) = k4_asympt_0(B,A) ) )).

fof(dt_k5_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m1_relset_1(A,k5_numbers,k1_numbers) )
     => m1_fraenkel(k5_asympt_0(A),k5_numbers,k1_numbers) ) )).

fof(dt_k6_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m1_relset_1(A,k5_numbers,k1_numbers) )
     => m1_fraenkel(k6_asympt_0(A),k5_numbers,k1_numbers) ) )).

fof(dt_k7_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m1_relset_1(A,k5_numbers,k1_numbers) )
     => m1_fraenkel(k7_asympt_0(A),k5_numbers,k1_numbers) ) )).

fof(dt_k8_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m1_relset_1(A,k5_numbers,k1_numbers) )
     => m1_fraenkel(k8_asympt_0(A,B),k5_numbers,k1_numbers) ) )).

fof(dt_k9_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m1_relset_1(A,k5_numbers,k1_numbers) )
     => m1_fraenkel(k9_asympt_0(A,B),k5_numbers,k1_numbers) ) )).

fof(dt_k10_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m1_relset_1(A,k5_numbers,k1_numbers) )
     => m1_fraenkel(k10_asympt_0(A,B),k5_numbers,k1_numbers) ) )).

fof(dt_k11_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & m1_relset_1(A,k5_numbers,k1_numbers)
        & m1_subset_1(B,k5_numbers) )
     => ( v1_funct_1(k11_asympt_0(A,B))
        & v1_funct_2(k11_asympt_0(A,B),k5_numbers,k1_numbers)
        & m2_relset_1(k11_asympt_0(A,B),k5_numbers,k1_numbers) ) ) )).

fof(dt_k12_asympt_0,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_fraenkel(B,A,k1_numbers)
        & m1_fraenkel(C,A,k1_numbers) )
     => m1_fraenkel(k12_asympt_0(A,B,C),A,k1_numbers) ) )).

fof(dt_k13_asympt_0,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_fraenkel(B,A,k1_numbers)
        & m1_fraenkel(C,A,k1_numbers) )
     => m1_fraenkel(k13_asympt_0(A,B,C),A,k1_numbers) ) )).

fof(dt_k14_asympt_0,axiom,(
    ! [A,B] :
      ( ( m1_fraenkel(A,k5_numbers,k1_numbers)
        & m1_fraenkel(B,k5_numbers,k1_numbers) )
     => m1_fraenkel(k14_asympt_0(A,B),k5_numbers,k1_numbers) ) )).

fof(d12_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => k5_asympt_0(A) = a_1_0_asympt_0(A) ) )).

fof(d13_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => k6_asympt_0(A) = a_1_1_asympt_0(A) ) )).

fof(t27_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => k7_asympt_0(A) = a_1_2_asympt_0(A) ) )).

fof(d15_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] : k8_asympt_0(A,B) = a_2_0_asympt_0(A,B) ) )).

fof(d16_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] : k9_asympt_0(A,B) = a_2_1_asympt_0(A,B) ) )).

fof(d17_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] : k10_asympt_0(A,B) = a_2_2_asympt_0(A,B) ) )).

fof(t38_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v2_asympt_0(B)
            & v6_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( v7_asympt_0(A)
                  & r1_xreal_0(np__2,C)
                  & r2_hidden(B,k8_asympt_0(A,a_1_3_asympt_0(C))) )
               => r2_hidden(B,k5_asympt_0(A)) ) ) ) ) )).

fof(t39_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v2_asympt_0(B)
            & v6_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( v7_asympt_0(A)
                  & r1_xreal_0(np__2,C)
                  & r2_hidden(B,k9_asympt_0(A,a_1_3_asympt_0(C))) )
               => r2_hidden(B,k6_asympt_0(A)) ) ) ) ) )).

fof(t40_asympt_0,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & v1_funct_2(A,k5_numbers,k1_numbers)
        & v2_asympt_0(A)
        & m2_relset_1(A,k5_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,k5_numbers,k1_numbers)
            & v2_asympt_0(B)
            & v6_asympt_0(B)
            & m2_relset_1(B,k5_numbers,k1_numbers) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( v7_asympt_0(A)
                  & r1_xreal_0(np__2,C)
                  & r2_hidden(B,k10_asympt_0(A,a_1_3_asympt_0(C))) )
               => r2_hidden(B,k7_asympt_0(A)) ) ) ) ) )).

fof(d21_asympt_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fraenkel(B,A,k1_numbers)
         => ! [C] :
              ( m1_fraenkel(C,A,k1_numbers)
             => k12_asympt_0(A,B,C) = a_3_0_asympt_0(A,B,C) ) ) ) )).

fof(d22_asympt_0,axiom,(
    ! [A] :
      ( ~ v1_xboole_0(A)
     => ! [B] :
          ( m1_fraenkel(B,A,k1_numbers)
         => ! [C] :
              ( m1_fraenkel(C,A,k1_numbers)
             => k13_asympt_0(A,B,C) = a_3_1_asympt_0(A,B,C) ) ) ) )).

fof(d23_asympt_0,axiom,(
    ! [A] :
      ( m1_fraenkel(A,k5_numbers,k1_numbers)
     => ! [B] :
          ( m1_fraenkel(B,k5_numbers,k1_numbers)
         => k14_asympt_0(A,B) = a_2_3_asympt_0(A,B) ) ) )).

fof(s1_asympt_0,axiom,
    ( r1_xreal_0(f1_s1_asympt_0,f2_s1_asympt_0)
   => ( ~ v1_xboole_0(a_0_0_asympt_0)
      & v1_finset_1(a_0_0_asympt_0)
      & m1_subset_1(a_0_0_asympt_0,k1_zfmisc_1(f3_s1_asympt_0)) ) )).

fof(s2_asympt_0,axiom,
    ( ~ v1_xboole_0(a_0_1_asympt_0)
    & v1_finset_1(a_0_1_asympt_0)
    & m1_subset_1(a_0_1_asympt_0,k1_zfmisc_1(f2_s2_asympt_0)) )).

fof(s3_asympt_0,axiom,
    ( ~ r1_xreal_0(f1_s3_asympt_0,np__0)
   => ( ~ v1_xboole_0(a_0_2_asympt_0)
      & v1_finset_1(a_0_2_asympt_0)
      & m1_subset_1(a_0_2_asympt_0,k1_zfmisc_1(f2_s3_asympt_0)) ) )).

fof(fraenkel_a_1_0_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & v2_asympt_0(B)
        & m2_relset_1(B,k5_numbers,k1_numbers) )
     => ( r2_hidden(A,a_1_0_asympt_0(B))
      <=> ? [C] :
            ( m2_fraenkel(C,k5_numbers,k1_numbers,k1_fraenkel(k5_numbers,k1_numbers))
            & A = C
            & ? [D] :
                ( m1_subset_1(D,k1_numbers)
                & ? [E] :
                    ( m2_subset_1(E,k1_numbers,k5_numbers)
                    & ~ r1_xreal_0(D,np__0)
                    & ! [F] :
                        ( m2_subset_1(F,k1_numbers,k5_numbers)
                       => ( r1_xreal_0(E,F)
                         => ( r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,F),k4_real_1(D,k2_seq_1(k5_numbers,k1_numbers,B,F)))
                            & r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,C,F)) ) ) ) ) ) ) ) ) )).

fof(fraenkel_a_1_1_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & v2_asympt_0(B)
        & m2_relset_1(B,k5_numbers,k1_numbers) )
     => ( r2_hidden(A,a_1_1_asympt_0(B))
      <=> ? [C] :
            ( m2_fraenkel(C,k5_numbers,k1_numbers,k1_fraenkel(k5_numbers,k1_numbers))
            & A = C
            & ? [D] :
                ( m1_subset_1(D,k1_numbers)
                & ? [E] :
                    ( m2_subset_1(E,k1_numbers,k5_numbers)
                    & ~ r1_xreal_0(D,np__0)
                    & ! [F] :
                        ( m2_subset_1(F,k1_numbers,k5_numbers)
                       => ( r1_xreal_0(E,F)
                         => ( r1_xreal_0(k4_real_1(D,k2_seq_1(k5_numbers,k1_numbers,B,F)),k2_seq_1(k5_numbers,k1_numbers,C,F))
                            & r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,C,F)) ) ) ) ) ) ) ) ) )).

fof(fraenkel_a_1_2_asympt_0,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & v2_asympt_0(B)
        & m2_relset_1(B,k5_numbers,k1_numbers) )
     => ( r2_hidden(A,a_1_2_asympt_0(B))
      <=> ? [C] :
            ( m2_fraenkel(C,k5_numbers,k1_numbers,k1_fraenkel(k5_numbers,k1_numbers))
            & A = C
            & ? [D] :
                ( m1_subset_1(D,k1_numbers)
                & ? [E] :
                    ( m1_subset_1(E,k1_numbers)
                    & ? [F] :
                        ( m2_subset_1(F,k1_numbers,k5_numbers)
                        & ~ r1_xreal_0(D,np__0)
                        & ~ r1_xreal_0(E,np__0)
                        & ! [G] :
                            ( m2_subset_1(G,k1_numbers,k5_numbers)
                           => ( r1_xreal_0(F,G)
                             => ( r1_xreal_0(k4_real_1(E,k2_seq_1(k5_numbers,k1_numbers,B,G)),k2_seq_1(k5_numbers,k1_numbers,C,G))
                                & r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,G),k4_real_1(D,k2_seq_1(k5_numbers,k1_numbers,B,G))) ) ) ) ) ) ) ) ) ) )).

fof(fraenkel_a_2_0_asympt_0,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & v2_asympt_0(B)
        & m2_relset_1(B,k5_numbers,k1_numbers) )
     => ( r2_hidden(A,a_2_0_asympt_0(B,C))
      <=> ? [D] :
            ( m2_fraenkel(D,k5_numbers,k1_numbers,k1_fraenkel(k5_numbers,k1_numbers))
            & A = D
            & ? [E] :
                ( m1_subset_1(E,k1_numbers)
                & ? [F] :
                    ( m2_subset_1(F,k1_numbers,k5_numbers)
                    & ~ r1_xreal_0(E,np__0)
                    & ! [G] :
                        ( m2_subset_1(G,k1_numbers,k5_numbers)
                       => ( ( r1_xreal_0(F,G)
                            & r2_hidden(G,C) )
                         => ( r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,D,G),k4_real_1(E,k2_seq_1(k5_numbers,k1_numbers,B,G)))
                            & r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,D,G)) ) ) ) ) ) ) ) ) )).

fof(fraenkel_a_2_1_asympt_0,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & v2_asympt_0(B)
        & m2_relset_1(B,k5_numbers,k1_numbers) )
     => ( r2_hidden(A,a_2_1_asympt_0(B,C))
      <=> ? [D] :
            ( m2_fraenkel(D,k5_numbers,k1_numbers,k1_fraenkel(k5_numbers,k1_numbers))
            & A = D
            & ? [E] :
                ( m1_subset_1(E,k1_numbers)
                & ? [F] :
                    ( m2_subset_1(F,k1_numbers,k5_numbers)
                    & ~ r1_xreal_0(E,np__0)
                    & ! [G] :
                        ( m2_subset_1(G,k1_numbers,k5_numbers)
                       => ( ( r1_xreal_0(F,G)
                            & r2_hidden(G,C) )
                         => ( r1_xreal_0(k4_real_1(E,k2_seq_1(k5_numbers,k1_numbers,B,G)),k2_seq_1(k5_numbers,k1_numbers,D,G))
                            & r1_xreal_0(np__0,k2_seq_1(k5_numbers,k1_numbers,D,G)) ) ) ) ) ) ) ) ) )).

fof(fraenkel_a_2_2_asympt_0,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & v1_funct_2(B,k5_numbers,k1_numbers)
        & v2_asympt_0(B)
        & m2_relset_1(B,k5_numbers,k1_numbers) )
     => ( r2_hidden(A,a_2_2_asympt_0(B,C))
      <=> ? [D] :
            ( m2_fraenkel(D,k5_numbers,k1_numbers,k1_fraenkel(k5_numbers,k1_numbers))
            & A = D
            & ? [E] :
                ( m1_subset_1(E,k1_numbers)
                & ? [F] :
                    ( m1_subset_1(F,k1_numbers)
                    & ? [G] :
                        ( m2_subset_1(G,k1_numbers,k5_numbers)
                        & ~ r1_xreal_0(E,np__0)
                        & ~ r1_xreal_0(F,np__0)
                        & ! [H] :
                            ( m2_subset_1(H,k1_numbers,k5_numbers)
                           => ( ( r1_xreal_0(G,H)
                                & r2_hidden(H,C) )
                             => ( r1_xreal_0(k4_real_1(F,k2_seq_1(k5_numbers,k1_numbers,B,H)),k2_seq_1(k5_numbers,k1_numbers,D,H))
                                & r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,D,H),k4_real_1(E,k2_seq_1(k5_numbers,k1_numbers,B,H))) ) ) ) ) ) ) ) ) ) )).

fof(fraenkel_a_1_3_asympt_0,axiom,(
    ! [A,B] :
      ( m2_subset_1(B,k1_numbers,k5_numbers)
     => ( r2_hidden(A,a_1_3_asympt_0(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & A = k3_newton(B,C) ) ) ) )).

fof(fraenkel_a_3_0_asympt_0,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & m1_fraenkel(C,B,k1_numbers)
        & m1_fraenkel(D,B,k1_numbers) )
     => ( r2_hidden(A,a_3_0_asympt_0(B,C,D))
      <=> ? [E] :
            ( m2_fraenkel(E,B,k1_numbers,k1_fraenkel(B,k1_numbers))
            & A = E
            & ? [F] :
                ( m2_fraenkel(F,B,k1_numbers,k1_fraenkel(B,k1_numbers))
                & ? [G] :
                    ( m2_fraenkel(G,B,k1_numbers,k1_fraenkel(B,k1_numbers))
                    & r2_hidden(F,C)
                    & r2_hidden(G,D)
                    & ! [H] :
                        ( m1_subset_1(H,B)
                       => k2_seq_1(B,k1_numbers,E,H) = k3_real_1(k2_seq_1(B,k1_numbers,F,H),k2_seq_1(B,k1_numbers,G,H)) ) ) ) ) ) ) )).

fof(fraenkel_a_3_1_asympt_0,axiom,(
    ! [A,B,C,D] :
      ( ( ~ v1_xboole_0(B)
        & m1_fraenkel(C,B,k1_numbers)
        & m1_fraenkel(D,B,k1_numbers) )
     => ( r2_hidden(A,a_3_1_asympt_0(B,C,D))
      <=> ? [E] :
            ( m2_fraenkel(E,B,k1_numbers,k1_fraenkel(B,k1_numbers))
            & A = E
            & ? [F] :
                ( m2_fraenkel(F,B,k1_numbers,k1_fraenkel(B,k1_numbers))
                & ? [G] :
                    ( m2_fraenkel(G,B,k1_numbers,k1_fraenkel(B,k1_numbers))
                    & r2_hidden(F,C)
                    & r2_hidden(G,D)
                    & ! [H] :
                        ( m1_subset_1(H,B)
                       => k2_seq_1(B,k1_numbers,E,H) = k4_square_1(k2_seq_1(B,k1_numbers,F,H),k2_seq_1(B,k1_numbers,G,H)) ) ) ) ) ) ) )).

fof(fraenkel_a_2_3_asympt_0,axiom,(
    ! [A,B,C] :
      ( ( m1_fraenkel(B,k5_numbers,k1_numbers)
        & m1_fraenkel(C,k5_numbers,k1_numbers) )
     => ( r2_hidden(A,a_2_3_asympt_0(B,C))
      <=> ? [D] :
            ( m2_fraenkel(D,k5_numbers,k1_numbers,k1_fraenkel(k5_numbers,k1_numbers))
            & A = D
            & ? [E] :
                ( m2_fraenkel(E,k5_numbers,k1_numbers,k1_fraenkel(k5_numbers,k1_numbers))
                & ? [F] :
                    ( m2_fraenkel(F,k5_numbers,k1_numbers,k1_fraenkel(k5_numbers,k1_numbers))
                    & ? [G] :
                        ( m2_subset_1(G,k1_numbers,k5_numbers)
                        & r2_hidden(E,B)
                        & r2_hidden(F,C)
                        & ! [H] :
                            ( m2_subset_1(H,k1_numbers,k5_numbers)
                           => ( r1_xreal_0(G,H)
                             => k2_seq_1(k5_numbers,k1_numbers,D,H) = k4_power(k2_seq_1(k5_numbers,k1_numbers,E,H),k2_seq_1(k5_numbers,k1_numbers,F,H)) ) ) ) ) ) ) ) ) )).

fof(fraenkel_a_0_0_asympt_0,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_asympt_0)
    <=> ? [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
          & A = f4_s1_asympt_0(B)
          & r1_xreal_0(f1_s1_asympt_0,B)
          & r1_xreal_0(B,f2_s1_asympt_0) ) ) )).

fof(fraenkel_a_0_1_asympt_0,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_1_asympt_0)
    <=> ? [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
          & A = f3_s2_asympt_0(B)
          & r1_xreal_0(B,f1_s2_asympt_0) ) ) )).

fof(fraenkel_a_0_2_asympt_0,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_2_asympt_0)
    <=> ? [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
          & A = f3_s3_asympt_0(B)
          & ~ r1_xreal_0(f1_s3_asympt_0,B) ) ) )).
%------------------------------------------------------------------------------