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

theorem Th52:
  X \ (Y \ Z) = (X \ Y) \/ X /\ Z
proof
  thus for x being object
    holds x in X \ (Y \ Z) implies x in (X \ Y) \/ X /\ Z
  proof let x be object;
    assume
A1: x in X \ (Y \ Z);
    then not x in (Y \ Z) by XBOOLE_0:def 5;
    then x in X & not x in Y or x in X & x in Z by A1,XBOOLE_0:def 5;
    then x in (X \ Y) or x in X /\ Z by XBOOLE_0:def 4,def 5;
    hence thesis by XBOOLE_0:def 3;
  end;
  let x be object;
    assume x in (X \ Y) \/ X /\ Z;
    then x in (X \ Y) or x in X /\ Z by XBOOLE_0:def 3;
    then
A2: x in X & not x in Y or x in X & x in Z by XBOOLE_0:def 4,def 5;
    then not x in (Y \ Z) by XBOOLE_0:def 5;
    hence thesis by A2,XBOOLE_0:def 5;
end;
