SET007 Axioms: SET007+112.ax


%------------------------------------------------------------------------------
% File     : SET007+112 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Limit 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    : limfunc3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   57 (   4 unit)
%            Number of atoms       :  924 (  51 equality)
%            Maximal formula depth :   31 (  18 average)
%            Number of connectives : 1195 ( 328 ~  ;  31  |; 465  &)
%                                         (  11 <=>; 360 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   30 (   1 propositional; 0-3 arity)
%            Number of functors    :   37 (   5 constant; 0-4 arity)
%            Number of variables   :  307 (   0 singleton; 301 !;   6 ?)
%            Maximal term depth    :    5 (   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_limfunc3,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) )
             => ( ( r1_tarski(k1_rfunct_2(B),k3_xboole_0(k1_relat_1(C),k12_prob_1(A)))
                  | r1_tarski(k1_rfunct_2(B),k3_xboole_0(k1_relat_1(C),k4_limfunc1(A))) )
               => r1_tarski(k1_rfunct_2(B),k4_xboole_0(k1_relat_1(C),k1_tarski(A))) ) ) ) ) )).

fof(t2_limfunc3,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(k18_complex1(k5_real_1(A,k2_seq_1(k5_numbers,k1_numbers,B,D))),np__0)
                      & ~ r1_xreal_0(k6_real_1(np__1,k1_nat_1(D,np__1)),k18_complex1(k5_real_1(A,k2_seq_1(k5_numbers,k1_numbers,B,D))))
                      & r2_hidden(k2_seq_1(k5_numbers,k1_numbers,B,D),k1_relat_1(C)) ) )
               => ( v4_seq_2(B)
                  & k2_seq_2(B) = A
                  & r1_tarski(k1_rfunct_2(B),k1_relat_1(C))
                  & r1_tarski(k1_rfunct_2(B),k4_xboole_0(k1_relat_1(C),k1_tarski(A))) ) ) ) ) ) )).

fof(t3_limfunc3,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) )
             => ( ( v4_seq_2(B)
                  & k2_seq_2(B) = A
                  & r1_tarski(k1_rfunct_2(B),k4_xboole_0(k1_relat_1(C),k1_tarski(A))) )
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ~ ( ~ r1_xreal_0(D,np__0)
                        & ! [E] :
                            ( m2_subset_1(E,k1_numbers,k5_numbers)
                           => ? [F] :
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                                & r1_xreal_0(E,F)
                                & ~ ( ~ r1_xreal_0(k18_complex1(k5_real_1(A,k2_seq_1(k5_numbers,k1_numbers,B,F))),np__0)
                                    & ~ r1_xreal_0(D,k18_complex1(k5_real_1(A,k2_seq_1(k5_numbers,k1_numbers,B,F))))
                                    & r2_hidden(k2_seq_1(k5_numbers,k1_numbers,B,F),k1_relat_1(C)) ) ) ) ) ) ) ) ) ) )).

fof(t4_limfunc3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ~ r1_xreal_0(A,np__0)
           => k4_xboole_0(k2_rcomp_1(k5_real_1(B,A),k3_real_1(B,A)),k1_tarski(B)) = k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(B,A),B),k2_rcomp_1(B,k3_real_1(B,A))) ) ) ) )).

fof(t5_limfunc3,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(k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(B,A),B),k2_rcomp_1(B,k3_real_1(B,A))),k1_relat_1(C))
               => ( r1_xreal_0(A,np__0)
                  | ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(B,D)
                              & ~ r1_xreal_0(E,B)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ! [G] :
                                      ( m1_subset_1(G,k1_numbers)
                                     => ~ ( ~ r1_xreal_0(F,D)
                                          & ~ r1_xreal_0(B,F)
                                          & r2_hidden(F,k1_relat_1(C))
                                          & ~ r1_xreal_0(E,G)
                                          & ~ r1_xreal_0(G,B)
                                          & r2_hidden(G,k1_relat_1(C)) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t6_limfunc3,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),k1_relat_1(C)) ) )
               => ( v4_seq_2(B)
                  & k2_seq_2(B) = A
                  & r1_tarski(k1_rfunct_2(B),k4_xboole_0(k1_relat_1(C),k1_tarski(A))) ) ) ) ) ) )).

fof(t7_limfunc3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [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) = A
                  & ~ r1_xreal_0(B,np__0)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ? [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                          & r1_xreal_0(D,E)
                          & ~ ( ~ r1_xreal_0(k2_seq_1(k5_numbers,k1_numbers,C,E),k5_real_1(A,B))
                              & ~ r1_xreal_0(k3_real_1(A,B),k2_seq_1(k5_numbers,k1_numbers,C,E)) ) ) ) ) ) ) ) )).

fof(t8_limfunc3,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)
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ~ ( ~ r1_xreal_0(A,C)
                        & ~ r1_xreal_0(D,A)
                        & ! [E] :
                            ( m1_subset_1(E,k1_numbers)
                           => ! [F] :
                                ( m1_subset_1(F,k1_numbers)
                               => ~ ( ~ r1_xreal_0(E,C)
                                    & ~ r1_xreal_0(A,E)
                                    & r2_hidden(E,k1_relat_1(B))
                                    & ~ r1_xreal_0(D,F)
                                    & ~ r1_xreal_0(F,A)
                                    & r2_hidden(F,k1_relat_1(B)) ) ) ) ) ) )
          <=> ( ! [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,k1_relat_1(B)) ) ) ) )
              & ! [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,k1_relat_1(B)) ) ) ) ) ) ) ) ) )).

fof(d1_limfunc3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r1_limfunc3(A,B)
          <=> ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(B,C)
                          & ~ r1_xreal_0(D,B)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ~ ( ~ r1_xreal_0(E,C)
                                      & ~ r1_xreal_0(B,E)
                                      & r2_hidden(E,k1_relat_1(A))
                                      & ~ r1_xreal_0(D,F)
                                      & ~ r1_xreal_0(F,B)
                                      & r2_hidden(F,k1_relat_1(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),k4_xboole_0(k1_relat_1(A),k1_tarski(B))) )
                       => ( v4_seq_2(k2_rfunct_2(A,D))
                          & k2_seq_2(k2_rfunct_2(A,D)) = C ) ) ) ) ) ) ) ) )).

fof(d2_limfunc3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r2_limfunc3(A,B)
          <=> ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(B,C)
                          & ~ r1_xreal_0(D,B)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ~ ( ~ r1_xreal_0(E,C)
                                      & ~ r1_xreal_0(B,E)
                                      & r2_hidden(E,k1_relat_1(A))
                                      & ~ r1_xreal_0(D,F)
                                      & ~ r1_xreal_0(F,B)
                                      & r2_hidden(F,k1_relat_1(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),k4_xboole_0(k1_relat_1(A),k1_tarski(B))) )
                   => v1_limfunc1(k2_rfunct_2(A,C)) ) ) ) ) ) ) )).

fof(d3_limfunc3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r3_limfunc3(A,B)
          <=> ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(B,C)
                          & ~ r1_xreal_0(D,B)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ~ ( ~ r1_xreal_0(E,C)
                                      & ~ r1_xreal_0(B,E)
                                      & r2_hidden(E,k1_relat_1(A))
                                      & ~ r1_xreal_0(D,F)
                                      & ~ r1_xreal_0(F,B)
                                      & r2_hidden(F,k1_relat_1(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),k4_xboole_0(k1_relat_1(A),k1_tarski(B))) )
                   => v2_limfunc1(k2_rfunct_2(A,C)) ) ) ) ) ) ) )).

fof(t9_limfunc3,axiom,(
    $true )).

fof(t10_limfunc3,axiom,(
    $true )).

fof(t11_limfunc3,axiom,(
    $true )).

fof(t12_limfunc3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r1_limfunc3(B,A)
          <=> ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(A,C)
                          & ~ r1_xreal_0(D,A)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ~ ( ~ r1_xreal_0(E,C)
                                      & ~ r1_xreal_0(A,E)
                                      & r2_hidden(E,k1_relat_1(B))
                                      & ~ r1_xreal_0(D,F)
                                      & ~ r1_xreal_0(F,A)
                                      & r2_hidden(F,k1_relat_1(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,np__0)
                                  & ! [F] :
                                      ( m1_subset_1(F,k1_numbers)
                                     => ~ ( ~ r1_xreal_0(k18_complex1(k5_real_1(A,F)),np__0)
                                          & ~ r1_xreal_0(E,k18_complex1(k5_real_1(A,F)))
                                          & r2_hidden(F,k1_relat_1(B))
                                          & r1_xreal_0(D,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,B,F),C))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t13_limfunc3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r2_limfunc3(B,A)
          <=> ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(A,C)
                          & ~ r1_xreal_0(D,A)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ~ ( ~ r1_xreal_0(E,C)
                                      & ~ r1_xreal_0(A,E)
                                      & r2_hidden(E,k1_relat_1(B))
                                      & ~ r1_xreal_0(D,F)
                                      & ~ r1_xreal_0(F,A)
                                      & r2_hidden(F,k1_relat_1(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(k18_complex1(k5_real_1(A,E)),np__0)
                              & ~ r1_xreal_0(D,k18_complex1(k5_real_1(A,E)))
                              & r2_hidden(E,k1_relat_1(B))
                              & r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,E),C) ) ) ) ) ) ) ) ) )).

fof(t14_limfunc3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r3_limfunc3(B,A)
          <=> ( ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(A,C)
                          & ~ r1_xreal_0(D,A)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ~ ( ~ r1_xreal_0(E,C)
                                      & ~ r1_xreal_0(A,E)
                                      & r2_hidden(E,k1_relat_1(B))
                                      & ~ r1_xreal_0(D,F)
                                      & ~ r1_xreal_0(F,A)
                                      & r2_hidden(F,k1_relat_1(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(k18_complex1(k5_real_1(A,E)),np__0)
                              & ~ r1_xreal_0(D,k18_complex1(k5_real_1(A,E)))
                              & r2_hidden(E,k1_relat_1(B))
                              & r1_xreal_0(C,k2_seq_1(k1_numbers,k1_numbers,B,E)) ) ) ) ) ) ) ) ) )).

fof(t15_limfunc3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r2_limfunc3(B,A)
          <=> ( r2_limfunc2(B,A)
              & r5_limfunc2(B,A) ) ) ) ) )).

fof(t16_limfunc3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r3_limfunc3(B,A)
          <=> ( r3_limfunc2(B,A)
              & r6_limfunc2(B,A) ) ) ) ) )).

fof(t17_limfunc3,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_limfunc3(B,A)
                  & r2_limfunc3(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(A,D)
                              & ~ r1_xreal_0(E,A)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ! [G] :
                                      ( m1_subset_1(G,k1_numbers)
                                     => ~ ( ~ r1_xreal_0(F,D)
                                          & ~ r1_xreal_0(A,F)
                                          & r2_hidden(F,k3_xboole_0(k1_relat_1(B),k1_relat_1(C)))
                                          & ~ r1_xreal_0(E,G)
                                          & ~ r1_xreal_0(G,A)
                                          & r2_hidden(G,k3_xboole_0(k1_relat_1(B),k1_relat_1(C))) ) ) ) ) ) ) )
               => ( r2_limfunc3(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
                  & r2_limfunc3(k8_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) )).

fof(t18_limfunc3,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_limfunc3(B,A)
                  & r3_limfunc3(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(A,D)
                              & ~ r1_xreal_0(E,A)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ! [G] :
                                      ( m1_subset_1(G,k1_numbers)
                                     => ~ ( ~ r1_xreal_0(F,D)
                                          & ~ r1_xreal_0(A,F)
                                          & r2_hidden(F,k3_xboole_0(k1_relat_1(B),k1_relat_1(C)))
                                          & ~ r1_xreal_0(E,G)
                                          & ~ r1_xreal_0(G,A)
                                          & r2_hidden(G,k3_xboole_0(k1_relat_1(B),k1_relat_1(C))) ) ) ) ) ) ) )
               => ( r3_limfunc3(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
                  & r2_limfunc3(k8_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) )).

fof(t19_limfunc3,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_limfunc3(B,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(A,D)
                              & ~ r1_xreal_0(E,A)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ! [G] :
                                      ( m1_subset_1(G,k1_numbers)
                                     => ~ ( ~ r1_xreal_0(F,D)
                                          & ~ r1_xreal_0(A,F)
                                          & r2_hidden(F,k1_relat_1(k6_seq_1(k1_numbers,k1_numbers,B,C)))
                                          & ~ r1_xreal_0(E,G)
                                          & ~ r1_xreal_0(G,A)
                                          & r2_hidden(G,k1_relat_1(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,k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))) ) )
                  | r2_limfunc3(k6_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) )).

fof(t20_limfunc3,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_limfunc3(B,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(A,D)
                              & ~ r1_xreal_0(E,A)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ! [G] :
                                      ( m1_subset_1(G,k1_numbers)
                                     => ~ ( ~ r1_xreal_0(F,D)
                                          & ~ r1_xreal_0(A,F)
                                          & r2_hidden(F,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,B,C)))
                                          & ~ r1_xreal_0(E,G)
                                          & ~ r1_xreal_0(G,A)
                                          & r2_hidden(G,k1_relat_1(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,k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))))
                                   => r1_xreal_0(E,k2_seq_1(k1_numbers,k1_numbers,C,F)) ) ) ) ) )
                  | r2_limfunc3(k8_seq_1(k1_numbers,k1_numbers,B,C),A) ) ) ) ) ) )).

fof(t21_limfunc3,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_limfunc3(C,A)
                 => ( r1_xreal_0(B,np__0)
                    | r2_limfunc3(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) )
                & ( r2_limfunc3(C,A)
                 => ( r1_xreal_0(np__0,B)
                    | r3_limfunc3(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) )
                & ( r3_limfunc3(C,A)
                 => ( r1_xreal_0(B,np__0)
                    | r3_limfunc3(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) )
                & ( r3_limfunc3(C,A)
                 => ( r1_xreal_0(np__0,B)
                    | r2_limfunc3(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) ) ) ) ) ) )).

fof(t22_limfunc3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r2_limfunc3(B,A)
              | r3_limfunc3(B,A) )
           => r2_limfunc3(k21_seq_1(k1_numbers,k1_numbers,B),A) ) ) ) )).

fof(t23_limfunc3,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)
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ~ ( ~ r1_xreal_0(A,C)
                        & ~ r1_xreal_0(D,A)
                        & ! [E] :
                            ( m1_subset_1(E,k1_numbers)
                           => ! [F] :
                                ( m1_subset_1(F,k1_numbers)
                               => ~ ( ~ r1_xreal_0(E,C)
                                    & ~ r1_xreal_0(A,E)
                                    & r2_hidden(E,k1_relat_1(B))
                                    & ~ r1_xreal_0(D,F)
                                    & ~ r1_xreal_0(F,A)
                                    & r2_hidden(F,k1_relat_1(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))
                      & r4_rfunct_2(B,k2_rcomp_1(A,k3_real_1(A,C)))
                      & ~ r1_rfunct_1(B,k2_rcomp_1(k5_real_1(A,C),A))
                      & ~ r1_rfunct_1(B,k2_rcomp_1(A,k3_real_1(A,C))) ) )
              | r2_limfunc3(B,A) ) ) ) ) )).

fof(t24_limfunc3,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)
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ~ ( ~ r1_xreal_0(A,C)
                        & ~ r1_xreal_0(D,A)
                        & ! [E] :
                            ( m1_subset_1(E,k1_numbers)
                           => ! [F] :
                                ( m1_subset_1(F,k1_numbers)
                               => ~ ( ~ r1_xreal_0(E,C)
                                    & ~ r1_xreal_0(A,E)
                                    & r2_hidden(E,k1_relat_1(B))
                                    & ~ r1_xreal_0(D,F)
                                    & ~ r1_xreal_0(F,A)
                                    & r2_hidden(F,k1_relat_1(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))
                      & r2_rfunct_2(B,k2_rcomp_1(A,k3_real_1(A,C)))
                      & ~ r1_rfunct_1(B,k2_rcomp_1(k5_real_1(A,C),A))
                      & ~ r1_rfunct_1(B,k2_rcomp_1(A,k3_real_1(A,C))) ) )
              | r2_limfunc3(B,A) ) ) ) ) )).

fof(t25_limfunc3,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)
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ~ ( ~ r1_xreal_0(A,C)
                        & ~ r1_xreal_0(D,A)
                        & ! [E] :
                            ( m1_subset_1(E,k1_numbers)
                           => ! [F] :
                                ( m1_subset_1(F,k1_numbers)
                               => ~ ( ~ r1_xreal_0(E,C)
                                    & ~ r1_xreal_0(A,E)
                                    & r2_hidden(E,k1_relat_1(B))
                                    & ~ r1_xreal_0(D,F)
                                    & ~ r1_xreal_0(F,A)
                                    & r2_hidden(F,k1_relat_1(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))
                      & r3_rfunct_2(B,k2_rcomp_1(A,k3_real_1(A,C)))
                      & ~ r2_rfunct_1(B,k2_rcomp_1(k5_real_1(A,C),A))
                      & ~ r2_rfunct_1(B,k2_rcomp_1(A,k3_real_1(A,C))) ) )
              | r3_limfunc3(B,A) ) ) ) ) )).

fof(t26_limfunc3,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)
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ~ ( ~ r1_xreal_0(A,C)
                        & ~ r1_xreal_0(D,A)
                        & ! [E] :
                            ( m1_subset_1(E,k1_numbers)
                           => ! [F] :
                                ( m1_subset_1(F,k1_numbers)
                               => ~ ( ~ r1_xreal_0(E,C)
                                    & ~ r1_xreal_0(A,E)
                                    & r2_hidden(E,k1_relat_1(B))
                                    & ~ r1_xreal_0(D,F)
                                    & ~ r1_xreal_0(F,A)
                                    & r2_hidden(F,k1_relat_1(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))
                      & r1_rfunct_2(B,k2_rcomp_1(A,k3_real_1(A,C)))
                      & ~ r2_rfunct_1(B,k2_rcomp_1(k5_real_1(A,C),A))
                      & ~ r2_rfunct_1(B,k2_rcomp_1(A,k3_real_1(A,C))) ) )
              | r3_limfunc3(B,A) ) ) ) ) )).

fof(t27_limfunc3,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_limfunc3(B,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(A,D)
                              & ~ r1_xreal_0(E,A)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ! [G] :
                                      ( m1_subset_1(G,k1_numbers)
                                     => ~ ( ~ r1_xreal_0(F,D)
                                          & ~ r1_xreal_0(A,F)
                                          & r2_hidden(F,k1_relat_1(C))
                                          & ~ r1_xreal_0(E,G)
                                          & ~ r1_xreal_0(G,A)
                                          & r2_hidden(G,k1_relat_1(C)) ) ) ) ) ) ) )
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & r1_tarski(k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))),k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))))
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ( r2_hidden(E,k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),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)) ) ) ) )
                  | r2_limfunc3(C,A) ) ) ) ) ) )).

fof(t28_limfunc3,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_limfunc3(B,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(A,D)
                              & ~ r1_xreal_0(E,A)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ! [G] :
                                      ( m1_subset_1(G,k1_numbers)
                                     => ~ ( ~ r1_xreal_0(F,D)
                                          & ~ r1_xreal_0(A,F)
                                          & r2_hidden(F,k1_relat_1(C))
                                          & ~ r1_xreal_0(E,G)
                                          & ~ r1_xreal_0(G,A)
                                          & r2_hidden(G,k1_relat_1(C)) ) ) ) ) ) ) )
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & r1_tarski(k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))),k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))))
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ( r2_hidden(E,k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),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)) ) ) ) )
                  | r3_limfunc3(C,A) ) ) ) ) ) )).

fof(t29_limfunc3,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_limfunc3(B,A)
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & r1_tarski(k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D))),k3_xboole_0(k1_relat_1(C),k1_relat_1(B)))
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ( r2_hidden(E,k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),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)) ) ) ) )
                  | r2_limfunc3(C,A) ) ) ) ) ) )).

fof(t30_limfunc3,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_limfunc3(B,A)
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & r1_tarski(k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D))),k3_xboole_0(k1_relat_1(C),k1_relat_1(B)))
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ( r2_hidden(E,k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),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)) ) ) ) )
                  | r3_limfunc3(C,A) ) ) ) ) ) )).

fof(d4_limfunc3,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( r1_limfunc3(A,B)
           => ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ( C = k1_limfunc3(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),k4_xboole_0(k1_relat_1(A),k1_tarski(B))) )
                       => ( v4_seq_2(k2_rfunct_2(A,D))
                          & k2_seq_2(k2_rfunct_2(A,D)) = C ) ) ) ) ) ) ) ) )).

fof(t31_limfunc3,axiom,(
    $true )).

fof(t32_limfunc3,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_limfunc3(C,A)
               => ( k1_limfunc3(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,np__0)
                                  & ! [F] :
                                      ( m1_subset_1(F,k1_numbers)
                                     => ~ ( ~ r1_xreal_0(k18_complex1(k5_real_1(A,F)),np__0)
                                          & ~ r1_xreal_0(E,k18_complex1(k5_real_1(A,F)))
                                          & r2_hidden(F,k1_relat_1(C))
                                          & r1_xreal_0(D,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,C,F),B))) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t33_limfunc3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r1_limfunc3(B,A)
           => ( r1_limfunc2(B,A)
              & r4_limfunc2(B,A)
              & k1_limfunc2(B,A) = k2_limfunc2(B,A)
              & k1_limfunc3(B,A) = k1_limfunc2(B,A)
              & k1_limfunc3(B,A) = k2_limfunc2(B,A) ) ) ) ) )).

fof(t34_limfunc3,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)
              & r4_limfunc2(B,A)
              & k1_limfunc2(B,A) = k2_limfunc2(B,A) )
           => ( r1_limfunc3(B,A)
              & k1_limfunc3(B,A) = k1_limfunc2(B,A)
              & k1_limfunc3(B,A) = k2_limfunc2(B,A) ) ) ) ) )).

fof(t35_limfunc3,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_limfunc3(C,A)
               => ( r1_limfunc3(k13_seq_1(k1_numbers,k1_numbers,C,B),A)
                  & k1_limfunc3(k13_seq_1(k1_numbers,k1_numbers,C,B),A) = k4_real_1(B,k1_limfunc3(C,A)) ) ) ) ) ) )).

fof(t36_limfunc3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r1_limfunc3(B,A)
           => ( r1_limfunc3(k16_seq_1(k1_numbers,k1_numbers,B),A)
              & k1_limfunc3(k16_seq_1(k1_numbers,k1_numbers,B),A) = k1_real_1(k1_limfunc3(B,A)) ) ) ) ) )).

fof(t37_limfunc3,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_limfunc3(B,A)
                  & r1_limfunc3(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(A,D)
                              & ~ r1_xreal_0(E,A)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ! [G] :
                                      ( m1_subset_1(G,k1_numbers)
                                     => ~ ( ~ r1_xreal_0(F,D)
                                          & ~ r1_xreal_0(A,F)
                                          & r2_hidden(F,k1_relat_1(k6_seq_1(k1_numbers,k1_numbers,B,C)))
                                          & ~ r1_xreal_0(E,G)
                                          & ~ r1_xreal_0(G,A)
                                          & r2_hidden(G,k1_relat_1(k6_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) ) ) )
               => ( r1_limfunc3(k6_seq_1(k1_numbers,k1_numbers,B,C),A)
                  & k1_limfunc3(k6_seq_1(k1_numbers,k1_numbers,B,C),A) = k3_real_1(k1_limfunc3(B,A),k1_limfunc3(C,A)) ) ) ) ) ) )).

fof(t38_limfunc3,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_limfunc3(B,A)
                  & r1_limfunc3(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(A,D)
                              & ~ r1_xreal_0(E,A)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ! [G] :
                                      ( m1_subset_1(G,k1_numbers)
                                     => ~ ( ~ r1_xreal_0(F,D)
                                          & ~ r1_xreal_0(A,F)
                                          & r2_hidden(F,k1_relat_1(k7_seq_1(k1_numbers,k1_numbers,B,C)))
                                          & ~ r1_xreal_0(E,G)
                                          & ~ r1_xreal_0(G,A)
                                          & r2_hidden(G,k1_relat_1(k7_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) ) ) )
               => ( r1_limfunc3(k7_seq_1(k1_numbers,k1_numbers,B,C),A)
                  & k1_limfunc3(k7_seq_1(k1_numbers,k1_numbers,B,C),A) = k5_real_1(k1_limfunc3(B,A),k1_limfunc3(C,A)) ) ) ) ) ) )).

fof(t39_limfunc3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r1_limfunc3(B,A)
              & k3_funct_2(k1_numbers,k1_numbers,B,k1_tarski(np__0)) = k1_xboole_0 )
           => ( k1_limfunc3(B,A) = np__0
              | ( r1_limfunc3(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
                & k1_limfunc3(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = k2_real_1(k1_limfunc3(B,A)) ) ) ) ) ) )).

fof(t40_limfunc3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( r1_limfunc3(B,A)
           => ( r1_limfunc3(k21_seq_1(k1_numbers,k1_numbers,B),A)
              & k1_limfunc3(k21_seq_1(k1_numbers,k1_numbers,B),A) = k18_complex1(k1_limfunc3(B,A)) ) ) ) ) )).

fof(t41_limfunc3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r1_limfunc3(B,A)
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(A,C)
                          & ~ r1_xreal_0(D,A)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ~ ( ~ r1_xreal_0(E,C)
                                      & ~ r1_xreal_0(A,E)
                                      & r2_hidden(E,k1_relat_1(B))
                                      & ~ r1_xreal_0(D,F)
                                      & ~ r1_xreal_0(F,A)
                                      & r2_hidden(F,k1_relat_1(B))
                                      & k2_seq_1(k1_numbers,k1_numbers,B,E) != np__0
                                      & k2_seq_1(k1_numbers,k1_numbers,B,F) != np__0 ) ) ) ) ) ) )
           => ( k1_limfunc3(B,A) = np__0
              | ( r1_limfunc3(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
                & k1_limfunc3(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = k2_real_1(k1_limfunc3(B,A)) ) ) ) ) ) )).

fof(t42_limfunc3,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_limfunc3(B,A)
                  & r1_limfunc3(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(A,D)
                              & ~ r1_xreal_0(E,A)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ! [G] :
                                      ( m1_subset_1(G,k1_numbers)
                                     => ~ ( ~ r1_xreal_0(F,D)
                                          & ~ r1_xreal_0(A,F)
                                          & r2_hidden(F,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,B,C)))
                                          & ~ r1_xreal_0(E,G)
                                          & ~ r1_xreal_0(G,A)
                                          & r2_hidden(G,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) ) ) )
               => ( r1_limfunc3(k8_seq_1(k1_numbers,k1_numbers,B,C),A)
                  & k1_limfunc3(k8_seq_1(k1_numbers,k1_numbers,B,C),A) = k4_real_1(k1_limfunc3(B,A),k1_limfunc3(C,A)) ) ) ) ) ) )).

fof(t43_limfunc3,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_limfunc3(B,A)
                  & r1_limfunc3(C,A)
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(A,D)
                              & ~ r1_xreal_0(E,A)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ! [G] :
                                      ( m1_subset_1(G,k1_numbers)
                                     => ~ ( ~ r1_xreal_0(F,D)
                                          & ~ r1_xreal_0(A,F)
                                          & r2_hidden(F,k1_relat_1(k2_rfunct_1(k1_numbers,k1_numbers,B,C)))
                                          & ~ r1_xreal_0(E,G)
                                          & ~ r1_xreal_0(G,A)
                                          & r2_hidden(G,k1_relat_1(k2_rfunct_1(k1_numbers,k1_numbers,B,C))) ) ) ) ) ) ) )
               => ( k1_limfunc3(C,A) = np__0
                  | ( r1_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,B,C),A)
                    & k1_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,B,C),A) = k6_real_1(k1_limfunc3(B,A),k1_limfunc3(C,A)) ) ) ) ) ) ) )).

fof(t44_limfunc3,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_limfunc3(B,A)
                  & k1_limfunc3(B,A) = np__0
                  & ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(A,D)
                              & ~ r1_xreal_0(E,A)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ! [G] :
                                      ( m1_subset_1(G,k1_numbers)
                                     => ~ ( ~ r1_xreal_0(F,D)
                                          & ~ r1_xreal_0(A,F)
                                          & r2_hidden(F,k1_relat_1(k8_seq_1(k1_numbers,k1_numbers,B,C)))
                                          & ~ r1_xreal_0(E,G)
                                          & ~ r1_xreal_0(G,A)
                                          & r2_hidden(G,k1_relat_1(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,k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))) ) )
                  | ( r1_limfunc3(k8_seq_1(k1_numbers,k1_numbers,B,C),A)
                    & k1_limfunc3(k8_seq_1(k1_numbers,k1_numbers,B,C),A) = np__0 ) ) ) ) ) ) )).

fof(t45_limfunc3,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_limfunc3(B,A)
                      & r1_limfunc3(C,A)
                      & k1_limfunc3(B,A) = k1_limfunc3(C,A)
                      & ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ! [F] :
                              ( m1_subset_1(F,k1_numbers)
                             => ~ ( ~ r1_xreal_0(A,E)
                                  & ~ r1_xreal_0(F,A)
                                  & ! [G] :
                                      ( m1_subset_1(G,k1_numbers)
                                     => ! [H] :
                                          ( m1_subset_1(H,k1_numbers)
                                         => ~ ( ~ r1_xreal_0(G,E)
                                              & ~ r1_xreal_0(A,G)
                                              & r2_hidden(G,k1_relat_1(D))
                                              & ~ r1_xreal_0(F,H)
                                              & ~ r1_xreal_0(H,A)
                                              & r2_hidden(H,k1_relat_1(D)) ) ) ) ) ) ) )
                   => ( ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(E,np__0)
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ( r2_hidden(F,k3_xboole_0(k1_relat_1(D),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),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(k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),k2_rcomp_1(A,k3_real_1(A,E)))),k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),k2_rcomp_1(A,k3_real_1(A,E)))))
                                  & r1_tarski(k3_xboole_0(k1_relat_1(D),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),k2_rcomp_1(A,k3_real_1(A,E)))),k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),k2_rcomp_1(A,k3_real_1(A,E))))) )
                                | ( r1_tarski(k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),k2_rcomp_1(A,k3_real_1(A,E)))),k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),k2_rcomp_1(A,k3_real_1(A,E)))))
                                  & r1_tarski(k3_xboole_0(k1_relat_1(D),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),k2_rcomp_1(A,k3_real_1(A,E)))),k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),k2_rcomp_1(A,k3_real_1(A,E))))) ) ) ) )
                      | ( r1_limfunc3(D,A)
                        & k1_limfunc3(D,A) = k1_limfunc3(B,A) ) ) ) ) ) ) ) )).

fof(t46_limfunc3,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_limfunc3(B,A)
                      & r1_limfunc3(C,A)
                      & k1_limfunc3(B,A) = k1_limfunc3(C,A) )
                   => ( ! [E] :
                          ( m1_subset_1(E,k1_numbers)
                         => ~ ( ~ r1_xreal_0(E,np__0)
                              & r1_tarski(k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),k2_rcomp_1(A,k3_real_1(A,E))),k3_xboole_0(k3_xboole_0(k1_relat_1(B),k1_relat_1(C)),k1_relat_1(D)))
                              & ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ( r2_hidden(F,k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,E),A),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_limfunc3(D,A)
                        & k1_limfunc3(D,A) = k1_limfunc3(B,A) ) ) ) ) ) ) ) )).

fof(t47_limfunc3,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_limfunc3(B,A)
                  & r1_limfunc3(C,A) )
               => ( ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(D,np__0)
                          & ( ( r1_tarski(k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))),k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))))
                              & ! [E] :
                                  ( m1_subset_1(E,k1_numbers)
                                 => ( r2_hidden(E,k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),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(k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))),k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),k2_rcomp_1(A,k3_real_1(A,D)))))
                              & ! [E] :
                                  ( m1_subset_1(E,k1_numbers)
                                 => ( r2_hidden(E,k3_xboole_0(k1_relat_1(C),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,D),A),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(k1_limfunc3(B,A),k1_limfunc3(C,A)) ) ) ) ) ) )).

fof(t48_limfunc3,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)
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ~ ( ~ r1_xreal_0(A,C)
                        & ~ r1_xreal_0(D,A)
                        & ! [E] :
                            ( m1_subset_1(E,k1_numbers)
                           => ! [F] :
                                ( m1_subset_1(F,k1_numbers)
                               => ~ ( ~ r1_xreal_0(E,C)
                                    & ~ r1_xreal_0(A,E)
                                    & r2_hidden(E,k1_relat_1(B))
                                    & ~ r1_xreal_0(D,F)
                                    & ~ r1_xreal_0(F,A)
                                    & r2_hidden(F,k1_relat_1(B))
                                    & k2_seq_1(k1_numbers,k1_numbers,B,E) != np__0
                                    & k2_seq_1(k1_numbers,k1_numbers,B,F) != np__0 ) ) ) ) ) )
           => ( ( ~ r2_limfunc3(B,A)
                & ~ r3_limfunc3(B,A) )
              | ( r1_limfunc3(k4_rfunct_1(k1_numbers,k1_numbers,B),A)
                & k1_limfunc3(k4_rfunct_1(k1_numbers,k1_numbers,B),A) = np__0 ) ) ) ) ) )).

fof(t49_limfunc3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r1_limfunc3(B,A)
              & k1_limfunc3(B,A) = np__0
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(A,C)
                          & ~ r1_xreal_0(D,A)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ~ ( ~ r1_xreal_0(E,C)
                                      & ~ r1_xreal_0(A,E)
                                      & r2_hidden(E,k1_relat_1(B))
                                      & ~ r1_xreal_0(D,F)
                                      & ~ r1_xreal_0(F,A)
                                      & r2_hidden(F,k1_relat_1(B))
                                      & k2_seq_1(k1_numbers,k1_numbers,B,E) != np__0
                                      & k2_seq_1(k1_numbers,k1_numbers,B,F) != 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,k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,C),A),k2_rcomp_1(A,k3_real_1(A,C)))))
                           => r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) ) )
              | r2_limfunc3(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) )).

fof(t50_limfunc3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r1_limfunc3(B,A)
              & k1_limfunc3(B,A) = np__0
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(A,C)
                          & ~ r1_xreal_0(D,A)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ~ ( ~ r1_xreal_0(E,C)
                                      & ~ r1_xreal_0(A,E)
                                      & r2_hidden(E,k1_relat_1(B))
                                      & ~ r1_xreal_0(D,F)
                                      & ~ r1_xreal_0(F,A)
                                      & r2_hidden(F,k1_relat_1(B))
                                      & k2_seq_1(k1_numbers,k1_numbers,B,E) != np__0
                                      & k2_seq_1(k1_numbers,k1_numbers,B,F) != 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,k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,C),A),k2_rcomp_1(A,k3_real_1(A,C)))))
                           => r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,D),np__0) ) ) ) )
              | r3_limfunc3(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) )).

fof(t51_limfunc3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r1_limfunc3(B,A)
              & k1_limfunc3(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,k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,C),A),k2_rcomp_1(A,k3_real_1(A,C)))))
                              & r1_xreal_0(k2_seq_1(k1_numbers,k1_numbers,B,D),np__0) ) ) ) )
              | r2_limfunc3(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) )).

fof(t52_limfunc3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r1_limfunc3(B,A)
              & k1_limfunc3(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,k3_xboole_0(k1_relat_1(B),k4_subset_1(k1_numbers,k2_rcomp_1(k5_real_1(A,C),A),k2_rcomp_1(A,k3_real_1(A,C)))))
                              & r1_xreal_0(np__0,k2_seq_1(k1_numbers,k1_numbers,B,D)) ) ) ) )
              | r3_limfunc3(k4_rfunct_1(k1_numbers,k1_numbers,B),A) ) ) ) ) )).

fof(dt_k1_limfunc3,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_limfunc3(A,B),k1_numbers) ) )).
%------------------------------------------------------------------------------