reserve a,b,x,y,z,z1,z2,z3,y1,y3,y4,A,B,C,D,G,M,N,X,Y,Z,W0,W00 for set,
  R,S,T, W,W1,W2 for Relation,
  F,H,H1 for Function;

theorem Th2:
  X <> {} & Y <> {} & W = [: X,Y :] implies field W = X \/ Y
proof
  set a = the Element of X,b = the Element of Y;
  assume that
A1: X <> {} and
A2: Y <> {} and
A3: W = [: X,Y :];
A4: for x being object holds x in field W implies x in X \/ Y
  proof let x be object;
    assume x in field W;
    then consider y being object such that
A5: [x,y] in W or [y,x] in W by Th1;
A6: [y,x] in W implies x in X \/ Y
    proof
      assume [y,x] in W;
      then x in Y by A3,ZFMISC_1:87;
      hence thesis by XBOOLE_0:def 3;
    end;
    [x,y] in W implies x in X \/ Y
    proof
      assume [x,y] in W;
      then x in X by A3,ZFMISC_1:87;
      hence thesis by XBOOLE_0:def 3;
    end;
    hence thesis by A5,A6;
  end;
  for x being object holds x in X \/ Y implies x in field W
  proof let x be object;
A7: x in X implies x in field W
    proof
      assume x in X;
      then [x,b] in W by A2,A3,ZFMISC_1:87;
      hence thesis by Th1;
    end;
A8: x in Y implies x in field W
    proof
      assume x in Y;
      then [a,x] in W by A1,A3,ZFMISC_1:87;
      hence thesis by Th1;
    end;
    assume x in X \/ Y;
    hence thesis by A7,A8,XBOOLE_0:def 3;
  end;
  hence thesis by A4,TARSKI:2;
end;
