reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;
reserve A,B for set;
reserve x,y,i,j,k for object;
reserve x for set;
reserve x for object;
reserve A1,A2,B1,B2 for non empty set,
  f for Function of A1,B1,
  g for Function of A2,B2,
  Y1 for non empty Subset of A1,
  Y2 for non empty Subset of A2;
reserve a,b,c,x,y,z,w,d for object;

theorem
  for a,b,c,d,e,i,j,k being object, g being Function
   st dom g = {a,b,c,d} &  g.a = e & g.b = i & g.c = j & g.d=k
  holds g = (a,b,c,d) --> (e,i,j,k)
proof
  let a,b,c,d,e,i,j,k be object;
  let h be Function such that
A1: dom h = {a,b,c,d} and
A2: h.a = e and
A3: h.b = i and
A4: h.c = j and
A5: h.d=k;
  set f = (a,b) --> (e,i);
  set g = (c,d) --> (j,k);
A6: dom f = {a,b} by Th62;
A7: dom g = {c,d} by Th62;
  then
A8: dom h = dom f \/ dom g by A1,A6,ENUMSET1:5;
  now
    let x be object such that
A9: x in dom f \/ dom g;
    thus x in dom g implies h.x = g.x
    proof
      assume
A10:  x in dom g;
      per cases by A7,A10,TARSKI:def 2;
      suppose x=c & c <> d;
        hence thesis by A4,Th63;
      end;
      suppose x=d;
        hence thesis by A5,Th63;
      end;
    end;
    thus not x in dom g implies h.x = f.x
    proof
      assume not x in dom g;
      then
A11:  x in dom f by A9,XBOOLE_0:def 3;
      per cases by A6,A11,TARSKI:def 2;
      suppose x=a & a <> b;
        hence thesis by A2,Th63;
      end;
      suppose x=b;
        hence thesis by A3,Th63;
      end;
    end;
  end;
  hence thesis by A8,Def1;
end;
