SET007 Axioms: SET007+111.ax


%------------------------------------------------------------------------------
% File     : SET007+111 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The One-Side Limits of a Real Function at a Point
% 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    : limfunc2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   96 (   8 unit)
%            Number of atoms       : 1294 (  74 equality)
%            Maximal formula depth :   23 (  15 average)
%            Number of connectives : 1618 ( 420 ~  ;  60  |; 588  &)
%                                         (  16 <=>; 534 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   27 (   1 propositional; 0-3 arity)
%            Number of functors    :   34 (   5 constant; 0-4 arity)
%            Number of variables   :  438 (   0 singleton; 428 !;  10 ?)
%            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(t1_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,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(B)
              & ~ r1_xreal_0(k2_seq_2(B),A)
              & ! [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,B,D),A) ) ) ) ) ) )).

fof(t2_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,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(B)
              & ~ r1_xreal_0(A,k2_seq_2(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(A,k2_seq_1(k5_numbers,k1_numbers,B,D)) ) ) ) ) ) )).

fof(t3_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r1_tarski(k2_rcomp_1(k5_real_1(B,A),B),k4_relset_1(k1_numbers,k1_numbers,C))
               => ( r1_xreal_0(A,np__0)
                  | ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(B,D)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(E,D)
                                  & ~ r1_xreal_0(B,E)
                                  & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,C)) ) ) ) ) ) ) ) ) ) )).

fof(t4_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r1_tarski(k2_rcomp_1(B,k3_real_1(B,A)),k4_relset_1(k1_numbers,k1_numbers,C))
               => ( r1_xreal_0(A,np__0)
                  | ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,B)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(D,E)
                                  & ~ r1_xreal_0(E,B)
                                  & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,C)) ) ) ) ) ) ) ) ) ) )).

fof(t5_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,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)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,D),k5_real_1(A,k6_real_1(np__1,k1_nat_1(D,np__1))))
                      & ~ r1_xreal_0(A,k2_seq_1(k5_numbers,k1_numbers,B,D))
                      & r2_hidden(k2_seq_1(k5_numbers,k1_numbers,B,D),k4_relset_1(k1_numbers,k1_numbers,C)) ) )
               => ( v4_seq_2(B)
                  & k2_seq_2(B) = A
                  & r1_tarski(k1_rfunct_2(B),k4_relset_1(k1_numbers,k1_numbers,C))
                  & r1_tarski(k1_rfunct_2(B),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k12_prob_1(A))) ) ) ) ) ) )).

fof(t6_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,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)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,B,D),A)
                      & ~ r1_xreal_0(k3_real_1(A,k6_real_1(np__1,k1_nat_1(D,np__1))),k2_seq_1(k5_numbers,k1_numbers,B,D))
                      & r2_hidden(k2_seq_1(k5_numbers,k1_numbers,B,D),k4_relset_1(k1_numbers,k1_numbers,C)) ) )
               => ( v4_seq_2(B)
                  & k2_seq_2(B) = A
                  & r1_tarski(k1_rfunct_2(B),k4_relset_1(k1_numbers,k1_numbers,C))
                  & r1_tarski(k1_rfunct_2(B),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k4_limfunc1(A))) ) ) ) ) ) )).

fof(d1_limfunc2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r1_limfunc2(A,B)
          <=> ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(B,C)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( ~ r1_xreal_0(D,C)
                              & ~ r1_xreal_0(B,D)
                              & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )
              & ? [C] :
                  ( m1_subset_1(C,k1_numbers)
                  & ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,k5_numbers,k1_numbers)
                        & m2_relset_1(D,k5_numbers,k1_numbers) )
                     => ( ( v4_seq_2(D)
                          & k2_seq_2(D) = B
                          & r1_tarski(k1_rfunct_2(D),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(B))) )
                       => ( v4_seq_2(k2_rfunct_2(A,D))
                          & k2_seq_2(k2_rfunct_2(A,D)) = C ) ) ) ) ) ) ) ) )).

fof(d2_limfunc2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r2_limfunc2(A,B)
          <=> ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(B,C)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( ~ r1_xreal_0(D,C)
                              & ~ r1_xreal_0(B,D)
                              & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )
              & ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,k1_numbers)
                    & m2_relset_1(C,k5_numbers,k1_numbers) )
                 => ( ( v4_seq_2(C)
                      & k2_seq_2(C) = B
                      & r1_tarski(k1_rfunct_2(C),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(B))) )
                   => v1_limfunc1(k2_rfunct_2(A,C)) ) ) ) ) ) ) )).

fof(d3_limfunc2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r3_limfunc2(A,B)
          <=> ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(B,C)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( ~ r1_xreal_0(D,C)
                              & ~ r1_xreal_0(B,D)
                              & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )
              & ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,k1_numbers)
                    & m2_relset_1(C,k5_numbers,k1_numbers) )
                 => ( ( v4_seq_2(C)
                      & k2_seq_2(C) = B
                      & r1_tarski(k1_rfunct_2(C),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(B))) )
                   => v2_limfunc1(k2_rfunct_2(A,C)) ) ) ) ) ) ) )).

fof(d4_limfunc2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r4_limfunc2(A,B)
          <=> ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,B)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( ~ r1_xreal_0(C,D)
                              & ~ r1_xreal_0(D,B)
                              & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )
              & ? [C] :
                  ( m1_subset_1(C,k1_numbers)
                  & ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,k5_numbers,k1_numbers)
                        & m2_relset_1(D,k5_numbers,k1_numbers) )
                     => ( ( v4_seq_2(D)
                          & k2_seq_2(D) = B
                          & r1_tarski(k1_rfunct_2(D),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(B))) )
                       => ( v4_seq_2(k2_rfunct_2(A,D))
                          & k2_seq_2(k2_rfunct_2(A,D)) = C ) ) ) ) ) ) ) ) )).

fof(d5_limfunc2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r5_limfunc2(A,B)
          <=> ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,B)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( ~ r1_xreal_0(C,D)
                              & ~ r1_xreal_0(D,B)
                              & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )
              & ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,k1_numbers)
                    & m2_relset_1(C,k5_numbers,k1_numbers) )
                 => ( ( v4_seq_2(C)
                      & k2_seq_2(C) = B
                      & r1_tarski(k1_rfunct_2(C),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(B))) )
                   => v1_limfunc1(k2_rfunct_2(A,C)) ) ) ) ) ) ) )).

fof(d6_limfunc2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r6_limfunc2(A,B)
          <=> ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,B)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( ~ r1_xreal_0(C,D)
                              & ~ r1_xreal_0(D,B)
                              & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )
              & ! [C] :
                  ( ( v1_funct_1(C)
                    & v1_funct_2(C,k5_numbers,k1_numbers)
                    & m2_relset_1(C,k5_numbers,k1_numbers) )
                 => ( ( v4_seq_2(C)
                      & k2_seq_2(C) = B
                      & r1_tarski(k1_rfunct_2(C),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(B))) )
                   => v2_limfunc1(k2_rfunct_2(A,C)) ) ) ) ) ) ) )).

fof(t7_limfunc2,axiom,(
    $true )).

fof(t8_limfunc2,axiom,(
    $true )).

fof(t9_limfunc2,axiom,(
    $true )).

fof(t10_limfunc2,axiom,(
    $true )).

fof(t11_limfunc2,axiom,(
    $true )).

fof(t12_limfunc2,axiom,(
    $true )).

fof(t13_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r1_limfunc2(B,A)
          <=> ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(A,C)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( ~ r1_xreal_0(D,C)
                              & ~ r1_xreal_0(A,D)
                              & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
              & ? [C] :
                  ( m1_subset_1(C,k1_numbers)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(A,E)
                                  & ! [F] :
                                      ( m1_subset_1(F,k1_numbers)
                                     => ~ ( ~ r1_xreal_0(F,E)
                                          & ~ r1_xreal_0(A,F)
                                          & r2_hidden(F,k4_relset_1(k1_numbers,k1_numbers,B))
                                          & r1_xreal_0(D,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,B,F),C))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t14_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r2_limfunc2(B,A)
          <=> ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(A,C)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( ~ r1_xreal_0(D,C)
                              & ~ r1_xreal_0(A,D)
                              & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ? [D] :
                      ( m1_subset_1(D,k1_numbers)
                      & ~ r1_xreal_0(A,D)
                      & ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(E,D)
                              & ~ r1_xreal_0(A,E)
                              & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,B))
                              & r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),C) ) ) ) ) ) ) ) ) )).

fof(t15_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r3_limfunc2(B,A)
          <=> ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(A,C)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( ~ r1_xreal_0(D,C)
                              & ~ r1_xreal_0(A,D)
                              & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ? [D] :
                      ( m1_subset_1(D,k1_numbers)
                      & ~ r1_xreal_0(A,D)
                      & ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(E,D)
                              & ~ r1_xreal_0(A,E)
                              & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,B))
                              & r1_xreal_0(C,k2_seq_1(k1_numbers,k1_numbers,B,E)) ) ) ) ) ) ) ) ) )).

fof(t16_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r4_limfunc2(B,A)
          <=> ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,A)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( ~ r1_xreal_0(C,D)
                              & ~ r1_xreal_0(D,A)
                              & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
              & ? [C] :
                  ( m1_subset_1(C,k1_numbers)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(E,A)
                                  & ! [F] :
                                      ( m1_subset_1(F,k1_numbers)
                                     => ~ ( ~ r1_xreal_0(E,F)
                                          & ~ r1_xreal_0(F,A)
                                          & r2_hidden(F,k4_relset_1(k1_numbers,k1_numbers,B))
                                          & r1_xreal_0(D,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,B,F),C))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t17_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r5_limfunc2(B,A)
          <=> ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,A)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( ~ r1_xreal_0(C,D)
                              & ~ r1_xreal_0(D,A)
                              & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ? [D] :
                      ( m1_subset_1(D,k1_numbers)
                      & ~ r1_xreal_0(D,A)
                      & ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(D,E)
                              & ~ r1_xreal_0(E,A)
                              & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,B))
                              & r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),C) ) ) ) ) ) ) ) ) )).

fof(t18_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r6_limfunc2(B,A)
          <=> ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,A)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( ~ r1_xreal_0(C,D)
                              & ~ r1_xreal_0(D,A)
                              & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ? [D] :
                      ( m1_subset_1(D,k1_numbers)
                      & ~ r1_xreal_0(D,A)
                      & ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(D,E)
                              & ~ r1_xreal_0(E,A)
                              & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,B))
                              & r1_xreal_0(C,k2_seq_1(k1_numbers,k1_numbers,B,E)) ) ) ) ) ) ) ) ) )).

fof(t19_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r2_limfunc2(B,A)
                  & r2_limfunc2(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(A,D)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(E,D)
                                  & ~ r1_xreal_0(A,E)
                                  & r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_relset_1(k1_numbers,k1_numbers,C))) ) ) ) ) )
               => ( r2_limfunc2(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
                  & r2_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) )).

fof(t20_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r3_limfunc2(B,A)
                  & r3_limfunc2(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(A,D)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(E,D)
                                  & ~ r1_xreal_0(A,E)
                                  & r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_relset_1(k1_numbers,k1_numbers,C))) ) ) ) ) )
               => ( r3_limfunc2(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
                  & r2_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) )).

fof(t21_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r5_limfunc2(B,A)
                  & r5_limfunc2(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,A)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(D,E)
                                  & ~ r1_xreal_0(E,A)
                                  & r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_relset_1(k1_numbers,k1_numbers,C))) ) ) ) ) )
               => ( r5_limfunc2(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
                  & r5_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) )).

fof(t22_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r6_limfunc2(B,A)
                  & r6_limfunc2(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,A)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(D,E)
                                  & ~ r1_xreal_0(E,A)
                                  & r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_relset_1(k1_numbers,k1_numbers,C))) ) ) ) ) )
               => ( r6_limfunc2(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
                  & r5_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) )).

fof(t23_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r2_limfunc2(B,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(A,D)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(E,D)
                                  & ~ r1_xreal_0(A,E)
                                  & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & r2_rfunct_1(C,k2_rcomp_1(k5_real_1(A,D),A)) ) )
                  | r2_limfunc2(k6_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) )).

fof(t24_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r2_limfunc2(B,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(A,D)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(E,D)
                                  & ~ r1_xreal_0(A,E)
                                  & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(D,np__0)
                              & ~ r1_xreal_0(E,np__0)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ( r2_hidden(F,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,D),A)))
                                   => r1_xreal_0(E,k2_seq_1(k1_numbers,k1_numbers,C,F)) ) ) ) ) )
                  | r2_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) )).

fof(t25_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r5_limfunc2(B,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,A)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(D,E)
                                  & ~ r1_xreal_0(E,A)
                                  & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & r2_rfunct_1(C,k2_rcomp_1(A,k3_real_1(A,D))) ) )
                  | r5_limfunc2(k6_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) )).

fof(t26_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r5_limfunc2(B,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,A)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(D,E)
                                  & ~ r1_xreal_0(E,A)
                                  & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(D,np__0)
                              & ~ r1_xreal_0(E,np__0)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ( r2_hidden(F,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,D))))
                                   => r1_xreal_0(E,k2_seq_1(k1_numbers,k1_numbers,C,F)) ) ) ) ) )
                  | r5_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) )).

fof(t27_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r2_limfunc2(C,A)
                 => ( r1_xreal_0(B,np__0)
                    | r2_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) )
                & ( r2_limfunc2(C,A)
                 => ( r1_xreal_0(np__0,B)
                    | r3_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) )
                & ( r3_limfunc2(C,A)
                 => ( r1_xreal_0(B,np__0)
                    | r3_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) )
                & ( r3_limfunc2(C,A)
                 => ( r1_xreal_0(np__0,B)
                    | r2_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) ) ) ) ) ) )).

fof(t28_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r5_limfunc2(C,A)
                 => ( r1_xreal_0(B,np__0)
                    | r5_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) )
                & ( r5_limfunc2(C,A)
                 => ( r1_xreal_0(np__0,B)
                    | r6_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) )
                & ( r6_limfunc2(C,A)
                 => ( r1_xreal_0(B,np__0)
                    | r6_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) )
                & ( r6_limfunc2(C,A)
                 => ( r1_xreal_0(np__0,B)
                    | r5_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) ) ) ) ) ) )).

fof(t29_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r2_limfunc2(B,A)
              | r3_limfunc2(B,A) )
           => r2_limfunc2(k21_seq_1(k1_numbers,k1_numbers,B),A) ) ) ) )).

fof(t30_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r5_limfunc2(B,A)
              | r6_limfunc2(B,A) )
           => r5_limfunc2(k21_seq_1(k1_numbers,k1_numbers,B),A) ) ) ) )).

fof(t31_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ~ ( ~ r1_xreal_0(A,C)
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( ~ r1_xreal_0(D,C)
                            & ~ r1_xreal_0(A,D)
                            & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
           => ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & r3_rfunct_2(B,k2_rcomp_1(k5_real_1(A,C),A))
                      & ~ r1_rfunct_1(B,k2_rcomp_1(k5_real_1(A,C),A)) ) )
              | r2_limfunc2(B,A) ) ) ) ) )).

fof(t32_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ~ ( ~ r1_xreal_0(A,C)
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( ~ r1_xreal_0(D,C)
                            & ~ r1_xreal_0(A,D)
                            & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
           => ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & r1_rfunct_2(B,k2_rcomp_1(k5_real_1(A,C),A))
                      & ~ r1_rfunct_1(B,k2_rcomp_1(k5_real_1(A,C),A)) ) )
              | r2_limfunc2(B,A) ) ) ) ) )).

fof(t33_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ~ ( ~ r1_xreal_0(A,C)
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( ~ r1_xreal_0(D,C)
                            & ~ r1_xreal_0(A,D)
                            & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
           => ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & r4_rfunct_2(B,k2_rcomp_1(k5_real_1(A,C),A))
                      & ~ r2_rfunct_1(B,k2_rcomp_1(k5_real_1(A,C),A)) ) )
              | r3_limfunc2(B,A) ) ) ) ) )).

fof(t34_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ~ ( ~ r1_xreal_0(A,C)
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( ~ r1_xreal_0(D,C)
                            & ~ r1_xreal_0(A,D)
                            & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
           => ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & r2_rfunct_2(B,k2_rcomp_1(k5_real_1(A,C),A))
                      & ~ r2_rfunct_1(B,k2_rcomp_1(k5_real_1(A,C),A)) ) )
              | r3_limfunc2(B,A) ) ) ) ) )).

fof(t35_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ~ ( ~ r1_xreal_0(C,A)
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( ~ r1_xreal_0(C,D)
                            & ~ r1_xreal_0(D,A)
                            & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
           => ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & r4_rfunct_2(B,k2_rcomp_1(A,k3_real_1(A,C)))
                      & ~ r1_rfunct_1(B,k2_rcomp_1(A,k3_real_1(A,C))) ) )
              | r5_limfunc2(B,A) ) ) ) ) )).

fof(t36_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ~ ( ~ r1_xreal_0(C,A)
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( ~ r1_xreal_0(C,D)
                            & ~ r1_xreal_0(D,A)
                            & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
           => ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & r2_rfunct_2(B,k2_rcomp_1(A,k3_real_1(A,C)))
                      & ~ r1_rfunct_1(B,k2_rcomp_1(A,k3_real_1(A,C))) ) )
              | r5_limfunc2(B,A) ) ) ) ) )).

fof(t37_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ~ ( ~ r1_xreal_0(C,A)
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( ~ r1_xreal_0(C,D)
                            & ~ r1_xreal_0(D,A)
                            & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
           => ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & r3_rfunct_2(B,k2_rcomp_1(A,k3_real_1(A,C)))
                      & ~ r2_rfunct_1(B,k2_rcomp_1(A,k3_real_1(A,C))) ) )
              | r6_limfunc2(B,A) ) ) ) ) )).

fof(t38_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ~ ( ~ r1_xreal_0(C,A)
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( ~ r1_xreal_0(C,D)
                            & ~ r1_xreal_0(D,A)
                            & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B)) ) ) ) )
           => ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & r1_rfunct_2(B,k2_rcomp_1(A,k3_real_1(A,C)))
                      & ~ r2_rfunct_1(B,k2_rcomp_1(A,k3_real_1(A,C))) ) )
              | r6_limfunc2(B,A) ) ) ) ) )).

fof(t39_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r2_limfunc2(B,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(A,D)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(E,D)
                                  & ~ r1_xreal_0(A,E)
                                  & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,C)) ) ) ) ) )
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,D),A)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,D),A)))
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ( r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,D),A)))
                               => r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),k2_seq_1(k1_numbers,k1_numbers,C,E)) ) ) ) )
                  | r2_limfunc2(C,A) ) ) ) ) ) )).

fof(t40_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r3_limfunc2(B,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(A,D)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(E,D)
                                  & ~ r1_xreal_0(A,E)
                                  & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,C)) ) ) ) ) )
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,D),A)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,D),A)))
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ( r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,D),A)))
                               => r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,C,E),k2_seq_1(k1_numbers,k1_numbers,B,E)) ) ) ) )
                  | r3_limfunc2(C,A) ) ) ) ) ) )).

fof(t41_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r5_limfunc2(B,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,A)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(D,E)
                                  & ~ r1_xreal_0(E,A)
                                  & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,C)) ) ) ) ) )
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,D))),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,D))))
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ( r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,D))))
                               => r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),k2_seq_1(k1_numbers,k1_numbers,C,E)) ) ) ) )
                  | r5_limfunc2(C,A) ) ) ) ) ) )).

fof(t42_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r6_limfunc2(B,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,A)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(D,E)
                                  & ~ r1_xreal_0(E,A)
                                  & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,C)) ) ) ) ) )
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,D))),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,D))))
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ( r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,D))))
                               => r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,C,E),k2_seq_1(k1_numbers,k1_numbers,B,E)) ) ) ) )
                  | r6_limfunc2(C,A) ) ) ) ) ) )).

fof(t43_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r2_limfunc2(B,A)
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & r1_tarski(k2_rcomp_1(k5_real_1(A,D),A),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k4_relset_1(k1_numbers,k1_numbers,B)))
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ( r2_hidden(E,k2_rcomp_1(k5_real_1(A,D),A))
                               => r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),k2_seq_1(k1_numbers,k1_numbers,C,E)) ) ) ) )
                  | r2_limfunc2(C,A) ) ) ) ) ) )).

fof(t44_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r3_limfunc2(B,A)
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & r1_tarski(k2_rcomp_1(k5_real_1(A,D),A),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k4_relset_1(k1_numbers,k1_numbers,B)))
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ( r2_hidden(E,k2_rcomp_1(k5_real_1(A,D),A))
                               => r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,C,E),k2_seq_1(k1_numbers,k1_numbers,B,E)) ) ) ) )
                  | r3_limfunc2(C,A) ) ) ) ) ) )).

fof(t45_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r5_limfunc2(B,A)
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & r1_tarski(k2_rcomp_1(A,k3_real_1(A,D)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k4_relset_1(k1_numbers,k1_numbers,B)))
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ( r2_hidden(E,k2_rcomp_1(A,k3_real_1(A,D)))
                               => r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),k2_seq_1(k1_numbers,k1_numbers,C,E)) ) ) ) )
                  | r5_limfunc2(C,A) ) ) ) ) ) )).

fof(t46_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r6_limfunc2(B,A)
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & r1_tarski(k2_rcomp_1(A,k3_real_1(A,D)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k4_relset_1(k1_numbers,k1_numbers,B)))
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ( r2_hidden(E,k2_rcomp_1(A,k3_real_1(A,D)))
                               => r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,C,E),k2_seq_1(k1_numbers,k1_numbers,B,E)) ) ) ) )
                  | r6_limfunc2(C,A) ) ) ) ) ) )).

fof(d7_limfunc2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r1_limfunc2(A,B)
           => ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ( C = k1_limfunc2(A,B)
                <=> ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,k5_numbers,k1_numbers)
                        & m2_relset_1(D,k5_numbers,k1_numbers) )
                     => ( ( v4_seq_2(D)
                          & k2_seq_2(D) = B
                          & r1_tarski(k1_rfunct_2(D),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(B))) )
                       => ( v4_seq_2(k2_rfunct_2(A,D))
                          & k2_seq_2(k2_rfunct_2(A,D)) = C ) ) ) ) ) ) ) ) )).

fof(d8_limfunc2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r4_limfunc2(A,B)
           => ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ( C = k2_limfunc2(A,B)
                <=> ! [D] :
                      ( ( v1_funct_1(D)
                        & v1_funct_2(D,k5_numbers,k1_numbers)
                        & m2_relset_1(D,k5_numbers,k1_numbers) )
                     => ( ( v4_seq_2(D)
                          & k2_seq_2(D) = B
                          & r1_tarski(k1_rfunct_2(D),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(B))) )
                       => ( v4_seq_2(k2_rfunct_2(A,D))
                          & k2_seq_2(k2_rfunct_2(A,D)) = C ) ) ) ) ) ) ) ) )).

fof(t47_limfunc2,axiom,(
    $true )).

fof(t48_limfunc2,axiom,(
    $true )).

fof(t49_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r1_limfunc2(C,A)
               => ( k1_limfunc2(C,A) = B
                <=> ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(A,E)
                                  & ! [F] :
                                      ( m1_subset_1(F,k1_numbers)
                                     => ~ ( ~ r1_xreal_0(F,E)
                                          & ~ r1_xreal_0(A,F)
                                          & r2_hidden(F,k4_relset_1(k1_numbers,k1_numbers,C))
                                          & r1_xreal_0(D,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,C,F),B))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t50_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r4_limfunc2(C,A)
               => ( k2_limfunc2(C,A) = B
                <=> ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(E,A)
                                  & ! [F] :
                                      ( m1_subset_1(F,k1_numbers)
                                     => ~ ( ~ r1_xreal_0(E,F)
                                          & ~ r1_xreal_0(F,A)
                                          & r2_hidden(F,k4_relset_1(k1_numbers,k1_numbers,C))
                                          & r1_xreal_0(D,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,C,F),B))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t51_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r1_limfunc2(C,A)
               => ( r1_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A)
                  & k1_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) = k4_real_1(B,k1_limfunc2(C,A)) ) ) ) ) ) )).

fof(t52_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r1_limfunc2(B,A)
           => ( r1_limfunc2(k16_seq_1(k1_numbers,k1_numbers,B),A)
              & k1_limfunc2(k16_seq_1(k1_numbers,k1_numbers,B),A) = k1_real_1(k1_limfunc2(B,A)) ) ) ) ) )).

fof(t53_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r1_limfunc2(B,A)
                  & r1_limfunc2(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(A,D)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(E,D)
                                  & ~ r1_xreal_0(A,E)
                                  & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
               => ( r1_limfunc2(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
                  & k1_limfunc2(k6_seq_1(k1_numbers,k1_numbers,B,C),A) = k3_real_1(k1_limfunc2(B,A),k1_limfunc2(C,A)) ) ) ) ) ) )).

fof(t54_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r1_limfunc2(B,A)
                  & r1_limfunc2(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(A,D)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(E,D)
                                  & ~ r1_xreal_0(A,E)
                                  & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k7_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
               => ( r1_limfunc2(k7_seq_1(k1_numbers,k1_numbers,B,C),A)
                  & k1_limfunc2(k7_seq_1(k1_numbers,k1_numbers,B,C),A) = k5_real_1(k1_limfunc2(B,A),k1_limfunc2(C,A)) ) ) ) ) ) )).

fof(t55_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r1_limfunc2(B,A)
              & k11_relset_1(k1_numbers,k1_numbers,B,k1_tarski(np__0)) = k1_xboole_0 )
           => ( k1_limfunc2(B,A) = np__0
              | ( r1_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
                & k1_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = k2_real_1(k1_limfunc2(B,A)) ) ) ) ) ) )).

fof(t56_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r1_limfunc2(B,A)
           => ( r1_limfunc2(k21_seq_1(k1_numbers,k1_numbers,B),A)
              & k1_limfunc2(k21_seq_1(k1_numbers,k1_numbers,B),A) = k18_complex1(k1_limfunc2(B,A)) ) ) ) ) )).

fof(t57_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r1_limfunc2(B,A)
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(A,C)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( ~ r1_xreal_0(D,C)
                              & ~ r1_xreal_0(A,D)
                              & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B))
                              & k2_seq_1(k1_numbers,k1_numbers,B,D) != np__0 ) ) ) ) )
           => ( k1_limfunc2(B,A) = np__0
              | ( r1_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
                & k1_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = k2_real_1(k1_limfunc2(B,A)) ) ) ) ) ) )).

fof(t58_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r1_limfunc2(B,A)
                  & r1_limfunc2(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(A,D)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(E,D)
                                  & ~ r1_xreal_0(A,E)
                                  & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
               => ( r1_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A)
                  & k1_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) = k4_real_1(k1_limfunc2(B,A),k1_limfunc2(C,A)) ) ) ) ) ) )).

fof(t59_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r1_limfunc2(B,A)
                  & r1_limfunc2(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(A,D)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(E,D)
                                  & ~ r1_xreal_0(A,E)
                                  & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
               => ( k1_limfunc2(C,A) = np__0
                  | ( r1_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,B,C),A)
                    & k1_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,B,C),A) = k6_real_1(k1_limfunc2(B,A),k1_limfunc2(C,A)) ) ) ) ) ) ) )).

fof(t60_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( r4_limfunc2(C,A)
               => ( r4_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A)
                  & k2_limfunc2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) = k4_real_1(B,k2_limfunc2(C,A)) ) ) ) ) ) )).

fof(t61_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r4_limfunc2(B,A)
           => ( r4_limfunc2(k16_seq_1(k1_numbers,k1_numbers,B),A)
              & k2_limfunc2(k16_seq_1(k1_numbers,k1_numbers,B),A) = k1_real_1(k2_limfunc2(B,A)) ) ) ) ) )).

fof(t62_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r4_limfunc2(B,A)
                  & r4_limfunc2(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,A)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(D,E)
                                  & ~ r1_xreal_0(E,A)
                                  & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k6_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
               => ( r4_limfunc2(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
                  & k2_limfunc2(k6_seq_1(k1_numbers,k1_numbers,B,C),A) = k3_real_1(k2_limfunc2(B,A),k2_limfunc2(C,A)) ) ) ) ) ) )).

fof(t63_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r4_limfunc2(B,A)
                  & r4_limfunc2(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,A)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(D,E)
                                  & ~ r1_xreal_0(E,A)
                                  & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k7_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
               => ( r4_limfunc2(k7_seq_1(k1_numbers,k1_numbers,B,C),A)
                  & k2_limfunc2(k7_seq_1(k1_numbers,k1_numbers,B,C),A) = k5_real_1(k2_limfunc2(B,A),k2_limfunc2(C,A)) ) ) ) ) ) )).

fof(t64_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r4_limfunc2(B,A)
              & k11_relset_1(k1_numbers,k1_numbers,B,k1_tarski(np__0)) = k1_xboole_0 )
           => ( k2_limfunc2(B,A) = np__0
              | ( r4_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
                & k2_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = k2_real_1(k2_limfunc2(B,A)) ) ) ) ) ) )).

fof(t65_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r4_limfunc2(B,A)
           => ( r4_limfunc2(k21_seq_1(k1_numbers,k1_numbers,B),A)
              & k2_limfunc2(k21_seq_1(k1_numbers,k1_numbers,B),A) = k18_complex1(k2_limfunc2(B,A)) ) ) ) ) )).

fof(t66_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r4_limfunc2(B,A)
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,A)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( ~ r1_xreal_0(C,D)
                              & ~ r1_xreal_0(D,A)
                              & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B))
                              & k2_seq_1(k1_numbers,k1_numbers,B,D) != np__0 ) ) ) ) )
           => ( k2_limfunc2(B,A) = np__0
              | ( r4_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
                & k2_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = k2_real_1(k2_limfunc2(B,A)) ) ) ) ) ) )).

fof(t67_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r4_limfunc2(B,A)
                  & r4_limfunc2(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,A)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(D,E)
                                  & ~ r1_xreal_0(E,A)
                                  & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
               => ( r4_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A)
                  & k2_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) = k4_real_1(k2_limfunc2(B,A),k2_limfunc2(C,A)) ) ) ) ) ) )).

fof(t68_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r4_limfunc2(B,A)
                  & r4_limfunc2(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,A)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(D,E)
                                  & ~ r1_xreal_0(E,A)
                                  & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
               => ( k2_limfunc2(C,A) = np__0
                  | ( r4_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,B,C),A)
                    & k2_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,B,C),A) = k6_real_1(k2_limfunc2(B,A),k2_limfunc2(C,A)) ) ) ) ) ) ) )).

fof(t69_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r1_limfunc2(B,A)
                  & k1_limfunc2(B,A) = np__0
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(A,D)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(E,D)
                                  & ~ r1_xreal_0(A,E)
                                  & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & r3_rfunct_1(C,k2_rcomp_1(k5_real_1(A,D),A)) ) )
                  | ( r1_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A)
                    & k1_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) = np__0 ) ) ) ) ) ) )).

fof(t70_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r4_limfunc2(B,A)
                  & k2_limfunc2(B,A) = np__0
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,A)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ~ ( ~ r1_xreal_0(D,E)
                                  & ~ r1_xreal_0(E,A)
                                  & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,k8_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) )
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & r3_rfunct_1(C,k2_rcomp_1(A,k3_real_1(A,D))) ) )
                  | ( r4_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A)
                    & k2_limfunc2(k8_seq_1(k1_numbers,k1_numbers,B,C),A) = np__0 ) ) ) ) ) ) )).

fof(t71_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,k1_numbers,k1_numbers) )
                 => ( ( r1_limfunc2(B,A)
                      & r1_limfunc2(C,A)
                      & k1_limfunc2(B,A) = k1_limfunc2(C,A)
                      & ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(A,E)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ~ ( ~ r1_xreal_0(F,E)
                                      & ~ r1_xreal_0(A,F)
                                      & r2_hidden(F,k4_relset_1(k1_numbers,k1_numbers,D)) ) ) ) ) )
                   => ( ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(E,np__0)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ( r2_hidden(F,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,D),k2_rcomp_1(k5_real_1(A,E),A)))
                                   => ( r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,F),k2_seq_1(k1_numbers,k1_numbers,D,F))
                                      & r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,D,F),k2_seq_1(k1_numbers,k1_numbers,C,F)) ) ) )
                              & ( ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,E),A)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,E),A)))
                                  & r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,D),k2_rcomp_1(k5_real_1(A,E),A)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,E),A))) )
                                | ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,E),A)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,E),A)))
                                  & r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,D),k2_rcomp_1(k5_real_1(A,E),A)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,E),A))) ) ) ) )
                      | ( r1_limfunc2(D,A)
                        & k1_limfunc2(D,A) = k1_limfunc2(B,A) ) ) ) ) ) ) ) )).

fof(t72_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,k1_numbers,k1_numbers) )
                 => ( ( r1_limfunc2(B,A)
                      & r1_limfunc2(C,A)
                      & k1_limfunc2(B,A) = k1_limfunc2(C,A) )
                   => ( ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(E,np__0)
                              & r1_tarski(k2_rcomp_1(k5_real_1(A,E),A),k5_subset_1(k1_numbers,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_relset_1(k1_numbers,k1_numbers,C)),k4_relset_1(k1_numbers,k1_numbers,D)))
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ( r2_hidden(F,k2_rcomp_1(k5_real_1(A,E),A))
                                   => ( r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,F),k2_seq_1(k1_numbers,k1_numbers,D,F))
                                      & r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,D,F),k2_seq_1(k1_numbers,k1_numbers,C,F)) ) ) ) ) )
                      | ( r1_limfunc2(D,A)
                        & k1_limfunc2(D,A) = k1_limfunc2(B,A) ) ) ) ) ) ) ) )).

fof(t73_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,k1_numbers,k1_numbers) )
                 => ( ( r4_limfunc2(B,A)
                      & r4_limfunc2(C,A)
                      & k2_limfunc2(B,A) = k2_limfunc2(C,A)
                      & ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(E,A)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ~ ( ~ r1_xreal_0(E,F)
                                      & ~ r1_xreal_0(F,A)
                                      & r2_hidden(F,k4_relset_1(k1_numbers,k1_numbers,D)) ) ) ) ) )
                   => ( ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(E,np__0)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ( r2_hidden(F,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,D),k2_rcomp_1(A,k3_real_1(A,E))))
                                   => ( r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,F),k2_seq_1(k1_numbers,k1_numbers,D,F))
                                      & r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,D,F),k2_seq_1(k1_numbers,k1_numbers,C,F)) ) ) )
                              & ( ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,E))),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,E))))
                                  & r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,D),k2_rcomp_1(A,k3_real_1(A,E))),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,E)))) )
                                | ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,E))),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,E))))
                                  & r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,D),k2_rcomp_1(A,k3_real_1(A,E))),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,E)))) ) ) ) )
                      | ( r4_limfunc2(D,A)
                        & k2_limfunc2(D,A) = k2_limfunc2(B,A) ) ) ) ) ) ) ) )).

fof(t74_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ! [D] :
                  ( ( v1_funct_1(D)
                    & m2_relset_1(D,k1_numbers,k1_numbers) )
                 => ( ( r4_limfunc2(B,A)
                      & r4_limfunc2(C,A)
                      & k2_limfunc2(B,A) = k2_limfunc2(C,A) )
                   => ( ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(E,np__0)
                              & r1_tarski(k2_rcomp_1(A,k3_real_1(A,E)),k5_subset_1(k1_numbers,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k4_relset_1(k1_numbers,k1_numbers,C)),k4_relset_1(k1_numbers,k1_numbers,D)))
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ( r2_hidden(F,k2_rcomp_1(A,k3_real_1(A,E)))
                                   => ( r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,F),k2_seq_1(k1_numbers,k1_numbers,D,F))
                                      & r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,D,F),k2_seq_1(k1_numbers,k1_numbers,C,F)) ) ) ) ) )
                      | ( r4_limfunc2(D,A)
                        & k2_limfunc2(D,A) = k2_limfunc2(B,A) ) ) ) ) ) ) ) )).

fof(t75_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r1_limfunc2(B,A)
                  & r1_limfunc2(C,A) )
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & ( ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,D),A)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,D),A)))
                              & ! [E] :
                                  ( m1_subset_1(E,k1_numbers)
                                 => ( r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,D),A)))
                                   => r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),k2_seq_1(k1_numbers,k1_numbers,C,E)) ) ) )
                            | ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,D),A)),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,D),A)))
                              & ! [E] :
                                  ( m1_subset_1(E,k1_numbers)
                                 => ( r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(k5_real_1(A,D),A)))
                                   => r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),k2_seq_1(k1_numbers,k1_numbers,C,E)) ) ) ) ) ) )
                  | r1_xreal_0(k1_limfunc2(B,A),k1_limfunc2(C,A)) ) ) ) ) ) )).

fof(t76_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r4_limfunc2(B,A)
                  & r4_limfunc2(C,A) )
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & ( ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,D))),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,D))))
                              & ! [E] :
                                  ( m1_subset_1(E,k1_numbers)
                                 => ( r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,D))))
                                   => r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),k2_seq_1(k1_numbers,k1_numbers,C,E)) ) ) )
                            | ( r1_tarski(k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,D))),k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,D))))
                              & ! [E] :
                                  ( m1_subset_1(E,k1_numbers)
                                 => ( r2_hidden(E,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,C),k2_rcomp_1(A,k3_real_1(A,D))))
                                   => r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),k2_seq_1(k1_numbers,k1_numbers,C,E)) ) ) ) ) ) )
                  | r1_xreal_0(k2_limfunc2(B,A),k2_limfunc2(C,A)) ) ) ) ) ) )).

fof(t77_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ~ ( ~ r1_xreal_0(A,C)
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( ~ r1_xreal_0(D,C)
                            & ~ r1_xreal_0(A,D)
                            & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B))
                            & k2_seq_1(k1_numbers,k1_numbers,B,D) != np__0 ) ) ) )
           => ( ( ~ r2_limfunc2(B,A)
                & ~ r3_limfunc2(B,A) )
              | ( r1_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
                & k1_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = np__0 ) ) ) ) ) )).

fof(t78_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ~ ( ~ r1_xreal_0(C,A)
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( ~ r1_xreal_0(C,D)
                            & ~ r1_xreal_0(D,A)
                            & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B))
                            & k2_seq_1(k1_numbers,k1_numbers,B,D) != np__0 ) ) ) )
           => ( ( ~ r5_limfunc2(B,A)
                & ~ r6_limfunc2(B,A) )
              | ( r4_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
                & k2_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = np__0 ) ) ) ) ) )).

fof(t79_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r1_limfunc2(B,A)
              & k1_limfunc2(B,A) = np__0 )
           => ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,C),A)))
                              & r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,D),np__0) ) ) ) )
              | r2_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) )).

fof(t80_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r1_limfunc2(B,A)
              & k1_limfunc2(B,A) = np__0 )
           => ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,C),A)))
                              & r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) ) )
              | r3_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) )).

fof(t81_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r4_limfunc2(B,A)
              & k2_limfunc2(B,A) = np__0 )
           => ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,C))))
                              & r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,D),np__0) ) ) ) )
              | r5_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) )).

fof(t82_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r4_limfunc2(B,A)
              & k2_limfunc2(B,A) = np__0 )
           => ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,C))))
                              & r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) ) )
              | r6_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) )).

fof(t83_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r1_limfunc2(B,A)
              & k1_limfunc2(B,A) = np__0
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(A,C)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( ~ r1_xreal_0(D,C)
                              & ~ r1_xreal_0(A,D)
                              & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B))
                              & k2_seq_1(k1_numbers,k1_numbers,B,D) != np__0 ) ) ) ) )
           => ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,C),A)))
                           => r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) ) )
              | r2_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) )).

fof(t84_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r1_limfunc2(B,A)
              & k1_limfunc2(B,A) = np__0
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(A,C)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( ~ r1_xreal_0(D,C)
                              & ~ r1_xreal_0(A,D)
                              & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B))
                              & k2_seq_1(k1_numbers,k1_numbers,B,D) != np__0 ) ) ) ) )
           => ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(k5_real_1(A,C),A)))
                           => r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,D),np__0) ) ) ) )
              | r3_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) )).

fof(t85_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r4_limfunc2(B,A)
              & k2_limfunc2(B,A) = np__0
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,A)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( ~ r1_xreal_0(C,D)
                              & ~ r1_xreal_0(D,A)
                              & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B))
                              & k2_seq_1(k1_numbers,k1_numbers,B,D) != np__0 ) ) ) ) )
           => ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,C))))
                           => r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) ) )
              | r5_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) )).

fof(t86_limfunc2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r4_limfunc2(B,A)
              & k2_limfunc2(B,A) = np__0
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,A)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ~ ( ~ r1_xreal_0(C,D)
                              & ~ r1_xreal_0(D,A)
                              & r2_hidden(D,k4_relset_1(k1_numbers,k1_numbers,B))
                              & k2_seq_1(k1_numbers,k1_numbers,B,D) != np__0 ) ) ) ) )
           => ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ~ ( ~ r1_xreal_0(C,np__0)
                      & ! [D] :
                          ( m1_subset_1(D,k1_numbers)
                         => ( r2_hidden(D,k5_subset_1(k1_numbers,k4_relset_1(k1_numbers,k1_numbers,B),k2_rcomp_1(A,k3_real_1(A,C))))
                           => r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,D),np__0) ) ) ) )
              | r6_limfunc2(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) )).

fof(dt_k1_limfunc2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & m1_relset_1(A,k1_numbers,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k1_limfunc2(A,B),k1_numbers) ) )).

fof(dt_k2_limfunc2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(A)
        & m1_relset_1(A,k1_numbers,k1_numbers)
        & m1_subset_1(B,k1_numbers) )
     => m1_subset_1(k2_limfunc2(A,B),k1_numbers) ) )).
%------------------------------------------------------------------------------