SET007 Axioms: SET007+431.ax


%------------------------------------------------------------------------------
% File     : SET007+431 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Vertex Sequences Induced by Chains
% 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    : graph_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   90 (   5 unit)
%            Number of atoms       :  743 ( 124 equality)
%            Maximal formula depth :   25 (  10 average)
%            Number of connectives :  701 (  48 ~  ;  14  |; 346  &)
%                                         (  18 <=>; 275 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   29 (   1 propositional; 0-4 arity)
%            Number of functors    :   45 (   9 constant; 0-4 arity)
%            Number of variables   :  280 (   0 singleton; 265 !;  15 ?)
%            Maximal term depth    :    6 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(rc1_graph_2,axiom,(
    ? [A] :
      ( v1_relat_1(A)
      & v1_funct_1(A)
      & v1_finset_1(A)
      & v1_finseq_1(A)
      & v1_graph_2(A)
      & v2_graph_2(A) ) )).

fof(fc1_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ~ v1_xboole_0(u1_graph_1(A)) ) )).

fof(rc2_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ? [B] :
          ( m1_graph_1(B,A)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v2_funct_1(B)
          & v1_xboole_0(B)
          & v1_finset_1(B)
          & v1_finseq_1(B) ) ) )).

fof(rc3_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ? [B] :
          ( m1_graph_1(B,A)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v1_finset_1(B)
          & v1_finseq_1(B)
          & v3_graph_2(B,A) ) ) )).

fof(cc1_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m1_graph_1(B,A)
         => ( v1_xboole_0(B)
           => v2_funct_1(B) ) ) ) )).

fof(rc4_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ? [B] :
          ( m1_graph_1(B,A)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v2_funct_1(B)
          & v1_finset_1(B)
          & v1_finseq_1(B)
          & v3_graph_2(B,A) ) ) )).

fof(cc2_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m1_graph_1(B,A)
         => ( v1_xboole_0(B)
           => v8_graph_1(B,A) ) ) ) )).

fof(t1_graph_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(k1_nat_1(A,np__1),B)
                  & r1_xreal_0(B,C) )
              <=> ? [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                    & r1_xreal_0(A,D)
                    & ~ r1_xreal_0(C,D)
                    & B = k1_nat_1(D,np__1) ) ) ) ) ) )).

fof(t2_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( A = k7_relat_1(B,k2_finseq_1(C))
               => ( r1_xreal_0(k3_finseq_1(A),k3_finseq_1(B))
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( ( r1_xreal_0(np__1,D)
                          & r1_xreal_0(D,k3_finseq_1(A)) )
                       => k1_funct_1(B,D) = k1_funct_1(A,D) ) ) ) ) ) ) ) )).

fof(t3_graph_2,axiom,(
    ! [A,B,C] :
      ( m2_subset_1(C,k1_numbers,k5_numbers)
     => ( ( r1_tarski(A,k2_finseq_1(C))
          & r1_tarski(B,k4_finseq_1(k14_finseq_1(A))) )
       => k5_relat_1(k14_finseq_1(B),k14_finseq_1(A)) = k14_finseq_1(k2_relat_1(k7_relat_1(k14_finseq_1(A),B))) ) ) )).

fof(d1_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( ( v1_relat_1(D)
                    & v1_funct_1(D)
                    & v1_finseq_1(D) )
                 => ( ( ( r1_xreal_0(np__1,B)
                        & r1_xreal_0(B,k1_nat_1(C,np__1))
                        & r1_xreal_0(C,k3_finseq_1(A)) )
                     => ( D = k1_graph_2(A,B,C)
                      <=> ( k1_nat_1(k3_finseq_1(D),B) = k1_nat_1(C,np__1)
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ( ~ r1_xreal_0(k3_finseq_1(D),E)
                               => k1_funct_1(D,k1_nat_1(E,np__1)) = k1_funct_1(A,k1_nat_1(B,E)) ) ) ) ) )
                    & ( ~ ( r1_xreal_0(np__1,B)
                          & r1_xreal_0(B,k1_nat_1(C,np__1))
                          & r1_xreal_0(C,k3_finseq_1(A)) )
                     => ( D = k1_graph_2(A,B,C)
                      <=> D = k1_xboole_0 ) ) ) ) ) ) ) )).

fof(t6_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( r1_xreal_0(np__1,B)
              & r1_xreal_0(B,k3_finseq_1(A)) )
           => k1_graph_2(A,B,B) = k9_finseq_1(k1_funct_1(A,B)) ) ) ) )).

fof(t7_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => k1_graph_2(A,np__1,k3_finseq_1(A)) = A ) )).

fof(t8_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( ( r1_xreal_0(B,C)
                      & r1_xreal_0(C,D)
                      & r1_xreal_0(D,k3_finseq_1(A)) )
                   => k7_finseq_1(k1_graph_2(A,k1_nat_1(B,np__1),C),k1_graph_2(A,k1_nat_1(C,np__1),D)) = k1_graph_2(A,k1_nat_1(B,np__1),D) ) ) ) ) ) )).

fof(t9_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(B,k3_finseq_1(A))
           => k7_finseq_1(k1_graph_2(A,np__1,B),k1_graph_2(A,k1_nat_1(B,np__1),k3_finseq_1(A))) = A ) ) ) )).

fof(t10_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(B,C)
                  & r1_xreal_0(C,k3_finseq_1(A)) )
               => k7_finseq_1(k7_finseq_1(k1_graph_2(A,np__1,B),k1_graph_2(A,k1_nat_1(B,np__1),C)),k1_graph_2(A,k1_nat_1(C,np__1),k3_finseq_1(A))) = A ) ) ) ) )).

fof(t11_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => r1_tarski(k2_relat_1(k1_graph_2(A,B,C)),k2_relat_1(A)) ) ) ) )).

fof(t12_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(np__1,B)
                  & r1_xreal_0(B,C)
                  & r1_xreal_0(C,k3_finseq_1(A)) )
               => ( k1_funct_1(k1_graph_2(A,B,C),np__1) = k1_funct_1(A,B)
                  & k1_funct_1(k1_graph_2(A,B,C),k3_finseq_1(k1_graph_2(A,B,C))) = k1_funct_1(A,C) ) ) ) ) ) )).

fof(d2_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => k3_graph_2(A,B) = k7_finseq_1(A,k1_graph_2(B,np__2,k3_finseq_1(B))) ) ) )).

fof(t13_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( A != k1_xboole_0
           => k1_nat_1(k3_finseq_1(k3_graph_2(B,A)),np__1) = k1_nat_1(k3_finseq_1(B),k3_finseq_1(A)) ) ) ) )).

fof(t14_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(np__1,C)
                  & r1_xreal_0(C,k3_finseq_1(A)) )
               => k1_funct_1(k3_graph_2(A,B),C) = k1_funct_1(A,C) ) ) ) ) )).

fof(t15_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_xreal_0(np__1,C)
               => ( r1_xreal_0(k3_finseq_1(A),C)
                  | k1_funct_1(k3_graph_2(B,A),k1_nat_1(k3_finseq_1(B),C)) = k1_funct_1(A,k1_nat_1(C,np__1)) ) ) ) ) ) )).

fof(t16_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( ~ r1_xreal_0(k3_finseq_1(A),np__1)
           => k1_funct_1(k3_graph_2(B,A),k3_finseq_1(k3_graph_2(B,A))) = k1_funct_1(A,k3_finseq_1(A)) ) ) ) )).

fof(t17_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => r1_tarski(k2_relat_1(k3_graph_2(A,B)),k2_xboole_0(k2_relat_1(A),k2_relat_1(B))) ) ) )).

fof(t18_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( k1_funct_1(A,k3_finseq_1(A)) = k1_funct_1(B,np__1)
           => ( A = k1_xboole_0
              | B = k1_xboole_0
              | k2_relat_1(k3_graph_2(A,B)) = k2_xboole_0(k2_relat_1(A),k2_relat_1(B)) ) ) ) ) )).

fof(d3_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( v1_graph_2(A)
      <=> k4_card_1(k2_relat_1(A)) = np__2 ) ) )).

fof(t19_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( v1_graph_2(A)
      <=> ( ~ r1_xreal_0(k3_finseq_1(A),np__1)
          & ? [B,C] :
              ( B != C
              & k2_relat_1(A) = k2_tarski(B,C) ) ) ) ) )).

fof(d4_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ( v2_graph_2(A)
      <=> ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ~ ( r1_xreal_0(np__1,B)
                & r1_xreal_0(k1_nat_1(B,np__1),k3_finseq_1(A))
                & k1_funct_1(A,B) = k1_funct_1(A,k1_nat_1(B,np__1)) ) ) ) ) )).

fof(t20_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & v1_graph_2(A)
        & v2_graph_2(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B)
            & v1_graph_2(B)
            & v2_graph_2(B) )
         => ( ( k3_finseq_1(A) = k3_finseq_1(B)
              & k2_relat_1(A) = k2_relat_1(B)
              & k1_funct_1(A,np__1) = k1_funct_1(B,np__1) )
           => A = B ) ) ) )).

fof(t21_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & v1_graph_2(A)
        & v2_graph_2(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B)
            & v1_graph_2(B)
            & v2_graph_2(B) )
         => ( ( k3_finseq_1(A) = k3_finseq_1(B)
              & k2_relat_1(A) = k2_relat_1(B) )
           => ( A = B
              | ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ ( r1_xreal_0(np__1,C)
                      & r1_xreal_0(C,k3_finseq_1(A))
                      & k1_funct_1(A,C) = k1_funct_1(B,C) ) ) ) ) ) ) )).

fof(t22_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & v1_graph_2(A)
        & v2_graph_2(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B)
            & v1_graph_2(B)
            & v2_graph_2(B) )
         => ( ( k3_finseq_1(A) = k3_finseq_1(B)
              & k2_relat_1(A) = k2_relat_1(B) )
           => ( A = B
              | ! [C] :
                  ( ( v1_relat_1(C)
                    & v1_funct_1(C)
                    & v1_finseq_1(C)
                    & v1_graph_2(C)
                    & v2_graph_2(C) )
                 => ~ ( k3_finseq_1(C) = k3_finseq_1(A)
                      & k2_relat_1(C) = k2_relat_1(A)
                      & C != A
                      & C != B ) ) ) ) ) ) )).

fof(t23_graph_2,axiom,(
    ! [A,B,C] :
      ( m2_subset_1(C,k1_numbers,k5_numbers)
     => ~ ( A != B
          & ~ r1_xreal_0(C,np__1)
          & ! [D] :
              ( ( v1_relat_1(D)
                & v1_funct_1(D)
                & v1_finseq_1(D)
                & v1_graph_2(D)
                & v2_graph_2(D) )
             => ~ ( k2_relat_1(D) = k2_tarski(A,B)
                  & k3_finseq_1(D) = C
                  & k1_funct_1(D,np__1) = A ) ) ) ) )).

fof(d5_graph_2,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,A)
     => ! [C] :
          ( ( v1_relat_1(C)
            & v1_funct_1(C)
            & v2_finseq_1(C) )
         => ( m1_graph_2(C,A,B)
          <=> r1_tarski(C,B) ) ) ) )).

fof(t24_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v2_finseq_1(A) )
     => ( ( v1_relat_1(A)
          & v1_funct_1(A)
          & v1_finseq_1(A) )
       => k15_finseq_1(A) = A ) ) )).

fof(t25_graph_2,axiom,(
    $true )).

fof(t26_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v2_finseq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ! [D] :
                  ( ( v1_relat_1(D)
                    & v1_funct_1(D)
                    & v1_finseq_1(D) )
                 => ! [E] :
                      ( ( v1_relat_1(E)
                        & v1_funct_1(E)
                        & v1_finseq_1(E) )
                     => ! [F] :
                          ( ( v1_relat_1(F)
                            & v1_funct_1(F)
                            & v1_finseq_1(F) )
                         => ( ( r1_tarski(k2_relat_1(B),k1_relat_1(A))
                              & r1_tarski(k2_relat_1(C),k1_relat_1(A))
                              & D = k5_relat_1(B,A)
                              & E = k5_relat_1(C,A)
                              & F = k5_relat_1(k7_finseq_1(B,C),A) )
                           => F = k7_finseq_1(D,E) ) ) ) ) ) ) ) )).

fof(t27_graph_2,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,A)
     => ! [C] :
          ( m1_graph_2(C,A,B)
         => ( r1_tarski(k1_relat_1(C),k4_finseq_1(B))
            & r1_tarski(k2_relat_1(C),k2_relat_1(B)) ) ) ) )).

fof(t28_graph_2,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,A)
     => m1_graph_2(B,A,B) ) )).

fof(t29_graph_2,axiom,(
    ! [A,B,C] :
      ( m2_finseq_1(C,A)
     => ! [D] :
          ( m1_graph_2(D,A,C)
         => m1_graph_2(k7_relat_1(D,B),A,C) ) ) )).

fof(t30_graph_2,axiom,(
    ! [A,B] :
      ( m2_finseq_1(B,A)
     => ! [C] :
          ( m2_finseq_1(C,A)
         => ! [D] :
              ( m2_finseq_1(D,A)
             => ! [E] :
                  ( m1_graph_2(E,A,B)
                 => ! [F] :
                      ( m1_graph_2(F,A,B)
                     => ! [G] :
                          ( m1_graph_2(G,A,C)
                         => ( ( k15_finseq_1(E) = C
                              & k15_finseq_1(G) = D
                              & F = k7_relat_1(E,k2_relat_1(k7_relat_1(k14_finseq_1(k1_relat_1(E)),k1_relat_1(G)))) )
                           => k15_finseq_1(F) = D ) ) ) ) ) ) ) )).

fof(t31_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_graph_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_graph_1(A))
             => ! [D] :
                  ( r2_graph_1(A,B,C,D)
                 => r2_graph_1(A,C,B,D) ) ) ) ) )).

fof(t32_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_graph_1(A))
         => ! [C] :
              ( m1_subset_1(C,u1_graph_1(A))
             => ! [D] :
                  ( m1_subset_1(D,u1_graph_1(A))
                 => ! [E] :
                      ( m1_subset_1(E,u1_graph_1(A))
                     => ! [F] :
                          ~ ( r2_graph_1(A,B,C,F)
                            & r2_graph_1(A,D,E,F)
                            & ~ ( B = D
                                & C = E )
                            & ~ ( B = E
                                & C = D ) ) ) ) ) ) ) )).

fof(d7_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_graph_1(A))
         => ! [C] :
              ( ( v1_relat_1(C)
                & v1_funct_1(C)
                & v1_finseq_1(C) )
             => ( r1_graph_2(A,B,C)
              <=> ( k3_finseq_1(B) = k1_nat_1(k3_finseq_1(C),np__1)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( ( r1_xreal_0(np__1,D)
                          & r1_xreal_0(D,k3_finseq_1(C)) )
                       => r2_graph_1(A,k4_finseq_4(k5_numbers,u1_graph_1(A),B,D),k4_finseq_4(k5_numbers,u1_graph_1(A),B,k1_nat_1(D,np__1)),k1_funct_1(C,D)) ) ) ) ) ) ) ) )).

fof(t33_graph_2,axiom,(
    $true )).

fof(t34_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_graph_1(A))
         => ! [C] :
              ( m2_graph_1(C,A)
             => ( r1_graph_2(A,B,C)
               => ( C = k1_xboole_0
                  | k5_graph_2(A,k2_relat_1(C)) = k2_relat_1(B) ) ) ) ) ) )).

fof(t35_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_graph_1(A))
         => r1_graph_2(A,k13_binarith(u1_graph_1(A),B),k1_xboole_0) ) ) )).

fof(t36_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_graph_1(B,A)
         => ? [C] :
              ( m2_finseq_1(C,u1_graph_1(A))
              & r1_graph_2(A,C,B) ) ) ) )).

fof(t37_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_graph_1(A))
         => ! [C] :
              ( m2_finseq_1(C,u1_graph_1(A))
             => ! [D] :
                  ( m2_graph_1(D,A)
                 => ( ( r1_graph_2(A,B,D)
                      & r1_graph_2(A,C,D) )
                   => ( D = k1_xboole_0
                      | B = C
                      | ( k1_funct_1(B,np__1) != k1_funct_1(C,np__1)
                        & ! [E] :
                            ( m2_finseq_1(E,u1_graph_1(A))
                           => ~ ( r1_graph_2(A,E,D)
                                & E != B
                                & E != C ) ) ) ) ) ) ) ) ) )).

fof(d8_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( r2_graph_2(A,B)
          <=> ( r1_xreal_0(np__1,k3_finseq_1(B))
              & k1_card_1(k5_graph_2(A,k2_relat_1(B))) = np__2
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ ( r2_hidden(C,k4_finseq_1(B))
                      & k1_funct_1(u3_graph_1(A),k1_funct_1(B,C)) = k1_funct_1(u4_graph_1(A),k1_funct_1(B,C)) ) ) ) ) ) ) )).

fof(t38_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_graph_1(A))
         => ! [C] :
              ( m2_graph_1(C,A)
             => ( ( r2_graph_2(A,C)
                  & r1_graph_2(A,B,C) )
               => ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ~ ( r2_hidden(D,k4_finseq_1(C))
                        & k1_funct_1(B,D) = k1_funct_1(B,k1_nat_1(D,np__1)) ) ) ) ) ) ) )).

fof(t39_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_graph_1(A))
         => ! [C] :
              ( m2_graph_1(C,A)
             => ( ( r2_graph_2(A,C)
                  & r1_graph_2(A,B,C) )
               => k2_relat_1(B) = k2_tarski(k1_funct_1(u3_graph_1(A),k1_funct_1(C,np__1)),k1_funct_1(u4_graph_1(A),k1_funct_1(C,np__1))) ) ) ) ) )).

fof(t40_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_graph_1(A))
         => ! [C] :
              ( m2_graph_1(C,A)
             => ( ( r2_graph_2(A,C)
                  & r1_graph_2(A,B,C) )
               => ( v1_relat_1(B)
                  & v1_funct_1(B)
                  & v1_finseq_1(B)
                  & v1_graph_2(B)
                  & v2_graph_2(B) ) ) ) ) ) )).

fof(t41_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_graph_1(B,A)
         => ~ ( r2_graph_2(A,B)
              & ! [C] :
                  ( m2_finseq_1(C,u1_graph_1(A))
                 => ! [D] :
                      ( m2_finseq_1(D,u1_graph_1(A))
                     => ~ ( C != D
                          & r1_graph_2(A,C,B)
                          & r1_graph_2(A,D,B)
                          & ! [E] :
                              ( m2_finseq_1(E,u1_graph_1(A))
                             => ~ ( r1_graph_2(A,E,B)
                                  & E != C
                                  & E != D ) ) ) ) ) ) ) ) )).

fof(t42_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_graph_1(A))
         => ! [C] :
              ( m2_graph_1(C,A)
             => ( r1_graph_2(A,B,C)
               => ( ( k1_card_1(u1_graph_1(A)) = np__1
                    | ( C != k1_xboole_0
                      & ~ r2_graph_2(A,C) ) )
                <=> ! [D] :
                      ( m2_finseq_1(D,u1_graph_1(A))
                     => ( r1_graph_2(A,D,C)
                       => D = B ) ) ) ) ) ) ) )).

fof(d9_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_graph_1(B,A)
         => ( ( k1_card_1(u1_graph_1(A)) = np__1
              | ( B != k1_xboole_0
                & ~ r2_graph_2(A,B) ) )
           => ! [C] :
                ( m2_finseq_1(C,u1_graph_1(A))
               => ( C = k6_graph_2(A,B)
                <=> r1_graph_2(A,C,B) ) ) ) ) ) )).

fof(t43_graph_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ! [C] :
              ( m2_finseq_1(C,u1_graph_1(B))
             => ! [D] :
                  ( m2_finseq_1(D,u1_graph_1(B))
                 => ! [E] :
                      ( m2_graph_1(E,B)
                     => ! [F] :
                          ( m2_graph_1(F,B)
                         => ( ( r1_graph_2(B,C,E)
                              & F = k7_relat_1(E,k2_finseq_1(A))
                              & D = k7_relat_1(C,k2_finseq_1(k1_nat_1(A,np__1))) )
                           => r1_graph_2(B,D,F) ) ) ) ) ) ) ) )).

fof(t44_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( ( v2_graph_1(D)
                    & l1_graph_1(D) )
                 => ! [E] :
                      ( m2_graph_1(E,D)
                     => ( ( r1_xreal_0(np__1,B)
                          & r1_xreal_0(B,C)
                          & r1_xreal_0(C,k3_finseq_1(E))
                          & A = k2_graph_2(u2_graph_1(D),E,B,C) )
                       => m2_graph_1(A,D) ) ) ) ) ) ) )).

fof(t45_graph_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v2_graph_1(C)
                & l1_graph_1(C) )
             => ! [D] :
                  ( m2_finseq_1(D,u1_graph_1(C))
                 => ! [E] :
                      ( m2_finseq_1(E,u1_graph_1(C))
                     => ! [F] :
                          ( m2_graph_1(F,C)
                         => ! [G] :
                              ( m2_graph_1(G,C)
                             => ( ( r1_xreal_0(np__1,A)
                                  & r1_xreal_0(A,B)
                                  & r1_xreal_0(B,k3_finseq_1(F))
                                  & G = k2_graph_2(u2_graph_1(C),F,A,B)
                                  & r1_graph_2(C,D,F)
                                  & E = k2_graph_2(u1_graph_1(C),D,A,k1_nat_1(B,np__1)) )
                               => r1_graph_2(C,E,G) ) ) ) ) ) ) ) ) )).

fof(t46_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_graph_1(A))
         => ! [C] :
              ( m2_finseq_1(C,u1_graph_1(A))
             => ! [D] :
                  ( m2_graph_1(D,A)
                 => ! [E] :
                      ( m2_graph_1(E,A)
                     => ( ( r1_graph_2(A,B,D)
                          & r1_graph_2(A,C,E)
                          & k1_funct_1(B,k3_finseq_1(B)) = k1_funct_1(C,np__1) )
                       => m2_graph_1(k8_finseq_1(u2_graph_1(A),D,E),A) ) ) ) ) ) ) )).

fof(t47_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_graph_1(A))
         => ! [C] :
              ( m2_finseq_1(C,u1_graph_1(A))
             => ! [D] :
                  ( m2_finseq_1(D,u1_graph_1(A))
                 => ! [E] :
                      ( m2_graph_1(E,A)
                     => ! [F] :
                          ( m2_graph_1(F,A)
                         => ! [G] :
                              ( m2_graph_1(G,A)
                             => ( ( r1_graph_2(A,B,E)
                                  & r1_graph_2(A,C,F)
                                  & k1_funct_1(B,k3_finseq_1(B)) = k1_funct_1(C,np__1)
                                  & G = k8_finseq_1(u2_graph_1(A),E,F)
                                  & D = k4_graph_2(u1_graph_1(A),B,C) )
                               => r1_graph_2(A,D,G) ) ) ) ) ) ) ) ) )).

fof(d10_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_graph_1(B,A)
         => ( v3_graph_2(B,A)
          <=> ? [C] :
                ( m2_finseq_1(C,u1_graph_1(A))
                & r1_graph_2(A,C,B)
                & ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( ( r1_xreal_0(np__1,D)
                            & r1_xreal_0(E,k3_finseq_1(C))
                            & k1_funct_1(C,D) = k1_funct_1(C,E) )
                         => ( r1_xreal_0(E,D)
                            | ( D = np__1
                              & E = k3_finseq_1(C) ) ) ) ) ) ) ) ) ) )).

fof(t48_graph_2,axiom,(
    $true )).

fof(t49_graph_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ! [C] :
              ( ( v3_graph_2(C,B)
                & m2_graph_1(C,B) )
             => ( v3_graph_2(k7_relat_1(C,k2_finseq_1(A)),B)
                & m2_graph_1(k7_relat_1(C,k2_finseq_1(A)),B) ) ) ) ) )).

fof(t50_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_graph_1(A))
         => ! [C] :
              ( m2_finseq_1(C,u1_graph_1(A))
             => ! [D] :
                  ( ( v3_graph_2(D,A)
                    & m2_graph_1(D,A) )
                 => ( ( r1_graph_2(A,B,D)
                      & r1_graph_2(A,C,D) )
                   => ( r1_xreal_0(k3_finseq_1(D),np__2)
                      | B = C ) ) ) ) ) ) )).

fof(t51_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_graph_1(A))
         => ! [C] :
              ( ( v3_graph_2(C,A)
                & m2_graph_1(C,A) )
             => ( r1_graph_2(A,B,C)
               => ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( ( r1_xreal_0(np__1,D)
                            & r1_xreal_0(E,k3_finseq_1(B))
                            & k1_funct_1(B,D) = k1_funct_1(B,E) )
                         => ( r1_xreal_0(E,D)
                            | ( D = np__1
                              & E = k3_finseq_1(B) ) ) ) ) ) ) ) ) ) )).

fof(t52_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_graph_1(A))
         => ! [C] :
              ( m2_graph_1(C,A)
             => ~ ( ~ ( v3_graph_2(C,A)
                      & m2_graph_1(C,A) )
                  & r1_graph_2(A,B,C)
                  & ! [D] :
                      ( m1_graph_2(D,u2_graph_1(A),C)
                     => ! [E] :
                          ( m1_graph_2(E,u1_graph_1(A),B)
                         => ! [F] :
                              ( m2_graph_1(F,A)
                             => ! [G] :
                                  ( m2_finseq_1(G,u1_graph_1(A))
                                 => ~ ( ~ r1_xreal_0(k3_finseq_1(C),k3_finseq_1(F))
                                      & r1_graph_2(A,G,F)
                                      & ~ r1_xreal_0(k3_finseq_1(B),k3_finseq_1(G))
                                      & k1_funct_1(B,np__1) = k1_funct_1(G,np__1)
                                      & k1_funct_1(B,k3_finseq_1(B)) = k1_funct_1(G,k3_finseq_1(G))
                                      & k15_finseq_1(D) = F
                                      & k15_finseq_1(E) = G ) ) ) ) ) ) ) ) ) )).

fof(t53_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_graph_1(A))
         => ! [C] :
              ( m2_graph_1(C,A)
             => ~ ( r1_graph_2(A,B,C)
                  & ! [D] :
                      ( m1_graph_2(D,u2_graph_1(A),C)
                     => ! [E] :
                          ( m1_graph_2(E,u1_graph_1(A),B)
                         => ! [F] :
                              ( ( v3_graph_2(F,A)
                                & m2_graph_1(F,A) )
                             => ! [G] :
                                  ( m2_finseq_1(G,u1_graph_1(A))
                                 => ~ ( k15_finseq_1(D) = F
                                      & k15_finseq_1(E) = G
                                      & r1_graph_2(A,G,F)
                                      & k1_funct_1(B,np__1) = k1_funct_1(G,np__1)
                                      & k1_funct_1(B,k3_finseq_1(B)) = k1_funct_1(G,k3_finseq_1(G)) ) ) ) ) ) ) ) ) ) )).

fof(t54_graph_2,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( v2_graph_1(C)
                & l1_graph_1(C) )
             => ( ( v2_funct_1(A)
                  & m2_graph_1(A,C) )
               => ( v2_funct_1(k7_relat_1(A,k2_finseq_1(B)))
                  & m2_graph_1(k7_relat_1(A,k2_finseq_1(B)),C) ) ) ) ) ) )).

fof(t55_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v3_graph_2(B,A)
            & m2_graph_1(B,A) )
         => ( ~ r1_xreal_0(k3_finseq_1(B),np__2)
           => ( v2_funct_1(B)
              & m2_graph_1(B,A) ) ) ) ) )).

fof(t56_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v3_graph_2(B,A)
            & m2_graph_1(B,A) )
         => ( ( v2_funct_1(B)
              & m2_graph_1(B,A) )
          <=> ~ ( k3_finseq_1(B) != np__0
                & k3_finseq_1(B) != np__1
                & k1_funct_1(B,np__1) = k1_funct_1(B,np__2) ) ) ) ) )).

fof(d11_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v8_graph_1(B,A)
            & m2_graph_1(B,A) )
         => ( B != k1_xboole_0
           => ! [C] :
                ( m2_finseq_1(C,u1_graph_1(A))
               => ( C = k7_graph_2(A,B)
                <=> ( r1_graph_2(A,C,B)
                    & k1_funct_1(C,np__1) = k1_funct_1(u3_graph_1(A),k1_funct_1(B,np__1)) ) ) ) ) ) ) )).

fof(dt_m1_graph_2,axiom,(
    ! [A,B] :
      ( m1_finseq_1(B,A)
     => ! [C] :
          ( m1_graph_2(C,A,B)
         => ( v1_relat_1(C)
            & v1_funct_1(C)
            & v2_finseq_1(C) ) ) ) )).

fof(existence_m1_graph_2,axiom,(
    ! [A,B] :
      ( m1_finseq_1(B,A)
     => ? [C] : m1_graph_2(C,A,B) ) )).

fof(dt_k1_graph_2,axiom,(
    ! [A,B,C] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k5_numbers) )
     => ( v1_relat_1(k1_graph_2(A,B,C))
        & v1_funct_1(k1_graph_2(A,B,C))
        & v1_finseq_1(k1_graph_2(A,B,C)) ) ) )).

fof(dt_k2_graph_2,axiom,(
    ! [A,B,C,D] :
      ( ( m1_finseq_1(B,A)
        & m1_subset_1(C,k5_numbers)
        & m1_subset_1(D,k5_numbers) )
     => m2_finseq_1(k2_graph_2(A,B,C,D),A) ) )).

fof(redefinition_k2_graph_2,axiom,(
    ! [A,B,C,D] :
      ( ( m1_finseq_1(B,A)
        & m1_subset_1(C,k5_numbers)
        & m1_subset_1(D,k5_numbers) )
     => k2_graph_2(A,B,C,D) = k1_graph_2(B,C,D) ) )).

fof(dt_k3_graph_2,axiom,(
    ! [A,B] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A)
        & v1_relat_1(B)
        & v1_funct_1(B)
        & v1_finseq_1(B) )
     => ( v1_relat_1(k3_graph_2(A,B))
        & v1_funct_1(k3_graph_2(A,B))
        & v1_finseq_1(k3_graph_2(A,B)) ) ) )).

fof(dt_k4_graph_2,axiom,(
    ! [A,B,C] :
      ( ( m1_finseq_1(B,A)
        & m1_finseq_1(C,A) )
     => m2_finseq_1(k4_graph_2(A,B,C),A) ) )).

fof(redefinition_k4_graph_2,axiom,(
    ! [A,B,C] :
      ( ( m1_finseq_1(B,A)
        & m1_finseq_1(C,A) )
     => k4_graph_2(A,B,C) = k3_graph_2(B,C) ) )).

fof(dt_k5_graph_2,axiom,(
    $true )).

fof(dt_k6_graph_2,axiom,(
    ! [A,B] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A)
        & m1_graph_1(B,A) )
     => m2_finseq_1(k6_graph_2(A,B),u1_graph_1(A)) ) )).

fof(dt_k7_graph_2,axiom,(
    ! [A,B] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A)
        & v8_graph_1(B,A)
        & m1_graph_1(B,A) )
     => m2_finseq_1(k7_graph_2(A,B),u1_graph_1(A)) ) )).

fof(t4_graph_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k1_card_1(a_2_0_graph_2(A,B)) = k1_nat_1(B,np__1) ) ) )).

fof(t5_graph_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(np__1,C)
                  & r1_xreal_0(C,A) )
               => k1_recdef_1(k5_numbers,k14_finseq_1(a_2_1_graph_2(A,B)),C) = k1_nat_1(B,C) ) ) ) ) )).

fof(d6_graph_2,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] : k5_graph_2(A,B) = a_2_2_graph_2(A,B) ) )).

fof(s1_graph_2,axiom,(
    v1_finset_1(a_0_0_graph_2) )).

fof(fraenkel_a_2_0_graph_2,axiom,(
    ! [A,B,C] :
      ( ( m2_subset_1(B,k1_numbers,k5_numbers)
        & m2_subset_1(C,k1_numbers,k5_numbers) )
     => ( r2_hidden(A,a_2_0_graph_2(B,C))
      <=> ? [D] :
            ( m2_subset_1(D,k1_numbers,k5_numbers)
            & A = D
            & r1_xreal_0(B,D)
            & r1_xreal_0(D,k1_nat_1(B,C)) ) ) ) )).

fof(fraenkel_a_2_1_graph_2,axiom,(
    ! [A,B,C] :
      ( ( m2_subset_1(B,k1_numbers,k5_numbers)
        & m2_subset_1(C,k1_numbers,k5_numbers) )
     => ( r2_hidden(A,a_2_1_graph_2(B,C))
      <=> ? [D] :
            ( m2_subset_1(D,k1_numbers,k5_numbers)
            & A = D
            & r1_xreal_0(k1_nat_1(C,np__1),D)
            & r1_xreal_0(D,k1_nat_1(C,B)) ) ) ) )).

fof(fraenkel_a_2_2_graph_2,axiom,(
    ! [A,B,C] :
      ( ( v2_graph_1(B)
        & l1_graph_1(B) )
     => ( r2_hidden(A,a_2_2_graph_2(B,C))
      <=> ? [D] :
            ( m1_subset_1(D,u1_graph_1(B))
            & A = D
            & ? [E] :
                ( m1_subset_1(E,u2_graph_1(B))
                & r2_hidden(E,C)
                & ( D = k1_funct_1(u3_graph_1(B),E)
                  | D = k1_funct_1(u4_graph_1(B),E) ) ) ) ) ) )).

fof(fraenkel_a_0_0_graph_2,axiom,(
    ! [A] :
      ( r2_hidden(A,a_0_0_graph_2)
    <=> ? [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
          & A = f3_s1_graph_2(B)
          & r1_xreal_0(f1_s1_graph_2,B)
          & r1_xreal_0(B,f2_s1_graph_2)
          & p1_s1_graph_2(B) ) ) )).
%------------------------------------------------------------------------------