
theorem
  for X,Y,Z being set st Z c= X holds X \/ (Y \ Z) = X \/ Y
proof
  let X,Y,Z be set;
  assume A1: Z c= X;
  A2: X \/ (Y \ Z) c= X \/ Y by XBOOLE_1:9;
  now
    let x be object;
    assume x in X \/ Y;
    then per cases by XBOOLE_0:def 3;
    suppose x in X;
      hence x in X \/ (Y \ Z) by XBOOLE_0:def 3;
    end;
    suppose x in Y & not x in Z;
      then x in Y \ Z by XBOOLE_0:def 5;
      hence x in X \/ (Y \ Z) by XBOOLE_0:def 3;
    end;
    suppose x in Y & x in Z;
      hence x in X \/ (Y \ Z) by A1, XBOOLE_0:def 3;
    end;
  end;
  then X \/ Y c= X \/ (Y \ Z) by TARSKI:def 3;
  hence thesis by A2, XBOOLE_0:def 10;
end;
