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

theorem Th51:
  X /\ Y \/ (X \ Y) = X
proof
  thus X /\ Y \/ (X \ Y) c= X
  proof
    let x be object;
    assume x in X /\ Y \/ (X \ Y);
    then x in X /\ Y or x in (X \ Y) by XBOOLE_0:def 3;
    hence thesis by XBOOLE_0:def 4,def 5;
  end;
  let x be object;
  assume x in X;
  then x in X & x in Y or x in (X\Y) by XBOOLE_0:def 5;
  then x in X /\ Y or x in (X \ Y) by XBOOLE_0:def 4;
  hence thesis by XBOOLE_0:def 3;
end;
