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

theorem
  X \ (X \ Y) = X /\ Y
proof
  thus for x being object holds x in X \ (X \ Y) implies x in X /\ Y
  proof let x be object;
    assume
A1: x in X \ (X \ Y);
    then not x in (X \ Y) by XBOOLE_0:def 5;
    then
A2: not x in X or x in Y by XBOOLE_0:def 5;
    x in X by A1,XBOOLE_0:def 5;
    hence thesis by A2,XBOOLE_0:def 4;
  end;
  thus for x being object holds x in X /\ Y implies x in X \ (X \ Y)
  proof let x be object;
    assume
A3: x in X /\ Y;
    then not x in X or x in Y by XBOOLE_0:def 4;
    then
A4: not x in (X \ Y) by XBOOLE_0:def 5;
    x in X by A3,XBOOLE_0:def 4;
    hence thesis by A4,XBOOLE_0:def 5;
  end;
end;
