reserve x,A,B,X,X9,Y,Y9,Z,V for set;

theorem Th55:
  (X \/ Y) \ (X /\ Y) = (X \ Y) \/ (Y \ X)
proof
 for x being object holds
    x in (X \/ Y) \ (X /\ Y) iff x in (X \ Y) \/ (Y \ X)
  proof let x be object;
    thus x in (X \/ Y) \ (X /\ Y) implies x in (X \ Y) \/ (Y \ X)
    proof
      assume
A1:   x in (X \/ Y) \ (X /\ Y);
      then not x in (X /\ Y) by XBOOLE_0:def 5;
      then
A2:   not x in X or not x in Y by XBOOLE_0:def 4;
      x in (X \/ Y) by A1,XBOOLE_0:def 5;
      then x in X or x in Y by XBOOLE_0:def 3;
      then x in (X \ Y) or x in( Y \ X) by A2,XBOOLE_0:def 5;
      hence thesis by XBOOLE_0:def 3;
    end;
    assume x in (X \ Y) \/ (Y \ X);
    then x in (X \ Y) or x in (Y \ X) by XBOOLE_0:def 3;
    then x in X & not x in Y or x in Y & not x in X by XBOOLE_0:def 5;
    then ( not x in (X /\ Y))& x in (X \/ Y) by XBOOLE_0:def 3,def 4;
    hence thesis by XBOOLE_0:def 5;
  end;
  hence thesis by TARSKI:2;
end;
