SET007 Axioms: SET007+562.ax


%------------------------------------------------------------------------------
% File     : SET007+562 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Oriented 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_4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   42 (   4 unit)
%            Number of atoms       :  376 (  39 equality)
%            Maximal formula depth :   25 (  10 average)
%            Number of connectives :  342 (   8 ~  ;   3  |; 185  &)
%                                         (   7 <=>; 139 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   32 (   1 propositional; 0-4 arity)
%            Number of functors    :   26 (   4 constant; 0-4 arity)
%            Number of variables   :  138 (   0 singleton; 128 !;  10 ?)
%            Maximal term depth    :    4 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(rc1_graph_4,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ? [B] :
          ( m1_graph_1(B,A)
          & v1_xboole_0(B)
          & v1_relat_1(B)
          & v1_funct_1(B)
          & v2_funct_1(B)
          & v1_finset_1(B)
          & v1_finseq_1(B)
          & v8_graph_1(B,A)
          & v1_membered(B)
          & v2_membered(B)
          & v3_membered(B)
          & v4_membered(B)
          & v5_membered(B) ) ) )).

fof(rc2_graph_4,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)
          & v8_graph_1(B,A)
          & v1_graph_4(B,A) ) ) )).

fof(rc3_graph_4,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)
          & v8_graph_1(B,A)
          & v3_graph_2(B,A) ) ) )).

fof(cc1_graph_4,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) )
           => ( v2_funct_1(B)
              & v8_graph_1(B,A) ) ) ) ) )).

fof(d1_graph_4,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] :
                  ( r1_graph_4(A,B,C,D)
                <=> ( k1_funct_1(u3_graph_1(A),D) = B
                    & k1_funct_1(u4_graph_1(A),D) = C ) ) ) ) ) )).

fof(t1_graph_4,axiom,(
    ! [A,B] :
      ( ( v2_graph_1(B)
        & l1_graph_1(B) )
     => ! [C] :
          ( m1_subset_1(C,u1_graph_1(B))
         => ! [D] :
              ( m1_subset_1(D,u1_graph_1(B))
             => ( r1_graph_4(B,C,D,A)
               => r2_graph_1(B,C,D,A) ) ) ) ) )).

fof(d2_graph_4,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))
             => ( r2_graph_4(A,B,C)
              <=> ? [D] :
                    ( r2_hidden(D,u2_graph_1(A))
                    & r1_graph_4(A,B,C,D) ) ) ) ) ) )).

fof(t2_graph_4,axiom,(
    ! [A,B] :
      ( ( v2_graph_1(B)
        & l1_graph_1(B) )
     => ! [C] :
          ( m1_subset_1(C,u1_graph_1(B))
         => ! [D] :
              ( m1_subset_1(D,u1_graph_1(B))
             => ! [E] :
                  ( m1_subset_1(E,u1_graph_1(B))
                 => ! [F] :
                      ( m1_subset_1(F,u1_graph_1(B))
                     => ( ( r1_graph_4(B,C,D,A)
                          & r1_graph_4(B,E,F,A) )
                       => ( C = E
                          & D = F ) ) ) ) ) ) ) )).

fof(t3_graph_4,axiom,(
    $true )).

fof(t4_graph_4,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ( k1_graph_4(A,k1_xboole_0) = k1_xboole_0
        & k2_graph_4(A,k1_xboole_0) = k1_xboole_0 ) ) )).

fof(d5_graph_4,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) )
             => ( r3_graph_4(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)) )
                       => r1_graph_4(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(t5_graph_4,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_graph_1(A))
         => ! [C] :
              ( ( v8_graph_1(C,A)
                & m2_graph_1(C,A) )
             => ( r3_graph_4(A,B,C)
               => r1_graph_2(A,B,C) ) ) ) ) )).

fof(t6_graph_4,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_graph_1(A))
         => ! [C] :
              ( ( v8_graph_1(C,A)
                & m2_graph_1(C,A) )
             => ( r3_graph_4(A,B,C)
               => r1_tarski(k1_graph_4(A,k2_relat_1(C)),k2_relat_1(B)) ) ) ) ) )).

fof(t7_graph_4,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_graph_1(A))
         => ! [C] :
              ( ( v8_graph_1(C,A)
                & m2_graph_1(C,A) )
             => ( r3_graph_4(A,B,C)
               => r1_tarski(k2_graph_4(A,k2_relat_1(C)),k2_relat_1(B)) ) ) ) ) )).

fof(t8_graph_4,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_graph_1(A))
         => ! [C] :
              ( ( v8_graph_1(C,A)
                & m2_graph_1(C,A) )
             => ( r3_graph_4(A,B,C)
               => ( C = k1_xboole_0
                  | r1_tarski(k2_relat_1(B),k2_xboole_0(k1_graph_4(A,k2_relat_1(C)),k2_graph_4(A,k2_relat_1(C)))) ) ) ) ) ) )).

fof(t9_graph_4,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m1_subset_1(B,u1_graph_1(A))
         => r3_graph_4(A,k13_binarith(u1_graph_1(A),B),k1_xboole_0) ) ) )).

fof(t10_graph_4,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v8_graph_1(B,A)
            & m2_graph_1(B,A) )
         => ? [C] :
              ( m2_finseq_1(C,u1_graph_1(A))
              & r3_graph_4(A,C,B) ) ) ) )).

fof(t11_graph_4,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] :
                  ( ( v8_graph_1(D,A)
                    & m2_graph_1(D,A) )
                 => ( ( r3_graph_4(A,B,D)
                      & r3_graph_4(A,C,D) )
                   => ( D = k1_xboole_0
                      | B = C ) ) ) ) ) ) )).

fof(d6_graph_4,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 = k3_graph_4(A,B)
                <=> r3_graph_4(A,C,B) ) ) ) ) ) )).

fof(t12_graph_4,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] :
                      ( ( v8_graph_1(E,B)
                        & m2_graph_1(E,B) )
                     => ! [F] :
                          ( ( v8_graph_1(F,B)
                            & m2_graph_1(F,B) )
                         => ( ( r3_graph_4(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))) )
                           => r3_graph_4(B,D,F) ) ) ) ) ) ) ) )).

fof(t13_graph_4,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] :
                      ( ( v8_graph_1(E,D)
                        & 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) )
                       => ( v8_graph_1(A,D)
                          & m2_graph_1(A,D) ) ) ) ) ) ) ) )).

fof(t14_graph_4,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] :
                          ( ( v8_graph_1(F,C)
                            & m2_graph_1(F,C) )
                         => ! [G] :
                              ( ( v8_graph_1(G,C)
                                & 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)
                                  & r3_graph_4(C,D,F)
                                  & E = k2_graph_2(u1_graph_1(C),D,A,k1_nat_1(B,np__1)) )
                               => r3_graph_4(C,E,G) ) ) ) ) ) ) ) ) )).

fof(t15_graph_4,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] :
                  ( ( v8_graph_1(D,A)
                    & m2_graph_1(D,A) )
                 => ! [E] :
                      ( ( v8_graph_1(E,A)
                        & m2_graph_1(E,A) )
                     => ( ( r3_graph_4(A,B,D)
                          & r3_graph_4(A,C,E)
                          & k1_funct_1(B,k3_finseq_1(B)) = k1_funct_1(C,np__1) )
                       => ( v8_graph_1(k8_finseq_1(u2_graph_1(A),D,E),A)
                          & m2_graph_1(k8_finseq_1(u2_graph_1(A),D,E),A) ) ) ) ) ) ) ) )).

fof(t16_graph_4,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] :
                      ( ( v8_graph_1(E,A)
                        & m2_graph_1(E,A) )
                     => ! [F] :
                          ( ( v8_graph_1(F,A)
                            & m2_graph_1(F,A) )
                         => ! [G] :
                              ( ( v8_graph_1(G,A)
                                & m2_graph_1(G,A) )
                             => ( ( r3_graph_4(A,B,E)
                                  & r3_graph_4(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) )
                               => r3_graph_4(A,D,G) ) ) ) ) ) ) ) ) )).

fof(d7_graph_4,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v8_graph_1(B,A)
            & m2_graph_1(B,A) )
         => ( v1_graph_4(B,A)
          <=> ? [C] :
                ( m2_finseq_1(C,u1_graph_1(A))
                & r3_graph_4(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(t17_graph_4,axiom,(
    $true )).

fof(t18_graph_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ! [C] :
              ( ( v8_graph_1(C,B)
                & m2_graph_1(C,B) )
             => ( v8_graph_1(k7_relat_1(C,k2_finseq_1(A)),B)
                & m2_graph_1(k7_relat_1(C,k2_finseq_1(A)),B) ) ) ) ) )).

fof(t19_graph_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v2_graph_1(B)
            & l1_graph_1(B) )
         => ! [C] :
              ( ( v8_graph_1(C,B)
                & v3_graph_2(C,B)
                & m2_graph_1(C,B) )
             => ( v8_graph_1(k7_relat_1(C,k2_finseq_1(A)),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(t20_graph_4,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v8_graph_1(B,A)
            & v3_graph_2(B,A)
            & m2_graph_1(B,A) )
         => ! [C] :
              ( ( v8_graph_1(C,A)
                & m2_graph_1(C,A) )
             => ( C = B
               => v1_graph_4(C,A) ) ) ) ) )).

fof(t21_graph_4,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v8_graph_1(B,A)
            & v1_graph_4(B,A)
            & m2_graph_1(B,A) )
         => ( v8_graph_1(B,A)
            & v3_graph_2(B,A)
            & m2_graph_1(B,A) ) ) ) )).

fof(t22_graph_4,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_graph_1(A))
         => ! [C] :
              ( ( v8_graph_1(C,A)
                & m2_graph_1(C,A) )
             => ~ ( ~ v1_graph_4(C,A)
                  & r3_graph_4(A,B,C)
                  & ! [D] :
                      ( m1_graph_2(D,u2_graph_1(A),C)
                     => ! [E] :
                          ( m1_graph_2(E,u1_graph_1(A),B)
                         => ! [F] :
                              ( ( v8_graph_1(F,A)
                                & 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))
                                      & r3_graph_4(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(t23_graph_4,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( m2_finseq_1(B,u1_graph_1(A))
         => ! [C] :
              ( ( v8_graph_1(C,A)
                & m2_graph_1(C,A) )
             => ~ ( r3_graph_4(A,B,C)
                  & ! [D] :
                      ( m1_graph_2(D,u2_graph_1(A),C)
                     => ! [E] :
                          ( m1_graph_2(E,u1_graph_1(A),B)
                         => ! [F] :
                              ( ( v8_graph_1(F,A)
                                & 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
                                      & r3_graph_4(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(t24_graph_4,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)
                  & v8_graph_1(A,C)
                  & m2_graph_1(A,C) )
               => ( v2_funct_1(k7_relat_1(A,k2_finseq_1(B)))
                  & v8_graph_1(k7_relat_1(A,k2_finseq_1(B)),C)
                  & m2_graph_1(k7_relat_1(A,k2_finseq_1(B)),C) ) ) ) ) ) )).

fof(t25_graph_4,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v8_graph_1(B,A)
            & v3_graph_2(B,A)
            & m2_graph_1(B,A) )
         => ( v2_funct_1(B)
            & v8_graph_1(B,A)
            & m2_graph_1(B,A) ) ) ) )).

fof(t26_graph_4,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( ( ( v8_graph_1(B,A)
                & v1_graph_4(B,A)
                & m2_graph_1(B,A) )
             => ( v8_graph_1(B,A)
                & v3_graph_2(B,A)
                & m2_graph_1(B,A) ) )
            & ( ( v8_graph_1(B,A)
                & v3_graph_2(B,A)
                & m2_graph_1(B,A) )
             => ( v8_graph_1(B,A)
                & v1_graph_4(B,A)
                & m2_graph_1(B,A) ) )
            & ( ( v8_graph_1(B,A)
                & v3_graph_2(B,A)
                & m2_graph_1(B,A) )
             => ( v2_funct_1(B)
                & v8_graph_1(B,A)
                & m2_graph_1(B,A) ) ) ) ) ) )).

fof(dt_k1_graph_4,axiom,(
    $true )).

fof(dt_k2_graph_4,axiom,(
    $true )).

fof(dt_k3_graph_4,axiom,(
    ! [A,B] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A)
        & v8_graph_1(B,A)
        & m1_graph_1(B,A) )
     => m2_finseq_1(k3_graph_4(A,B),u1_graph_1(A)) ) )).

fof(d3_graph_4,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] : k1_graph_4(A,B) = a_2_0_graph_4(A,B) ) )).

fof(d4_graph_4,axiom,(
    ! [A] :
      ( ( v2_graph_1(A)
        & l1_graph_1(A) )
     => ! [B] : k2_graph_4(A,B) = a_2_1_graph_4(A,B) ) )).

fof(fraenkel_a_2_0_graph_4,axiom,(
    ! [A,B,C] :
      ( ( v2_graph_1(B)
        & l1_graph_1(B) )
     => ( r2_hidden(A,a_2_0_graph_4(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) ) ) ) ) )).

fof(fraenkel_a_2_1_graph_4,axiom,(
    ! [A,B,C] :
      ( ( v2_graph_1(B)
        & l1_graph_1(B) )
     => ( r2_hidden(A,a_2_1_graph_4(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(u4_graph_1(B),E) ) ) ) ) )).
%------------------------------------------------------------------------------