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

theorem Th41:
  (X \ Y) \ Z = X \ (Y \/ Z)
proof
  thus for x being object holds x in (X \ Y) \ Z implies x in X \ (Y \/ Z)
  proof let x be object;
    assume
A1: x in (X \ Y) \ Z;
    then
A2: not x in Z by XBOOLE_0:def 5;
A3: x in (X \ Y) by A1,XBOOLE_0:def 5;
    then
A4: x in X by XBOOLE_0:def 5;
    not x in Y by A3,XBOOLE_0:def 5;
    then not x in (Y \/ Z) by A2,XBOOLE_0:def 3;
    hence thesis by A4,XBOOLE_0:def 5;
  end;
  thus for x being object holds x in X \ (Y \/ Z) implies x in (X \ Y) \ Z
  proof let x be object;
    assume
A5: x in X \ (Y \/ Z);
    then
A6: not x in (Y \/ Z) by XBOOLE_0:def 5;
    then
A7: not x in Y by XBOOLE_0:def 3;
A8: not x in Z by A6,XBOOLE_0:def 3;
    x in X by A5,XBOOLE_0:def 5;
    then x in (X \ Y) by A7,XBOOLE_0:def 5;
    hence thesis by A8,XBOOLE_0:def 5;
  end;
end;
