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

theorem Th42:
  (X \/ Y) \ Z = (X \ Z) \/ (Y \ Z)
proof
  thus (X \/ Y) \ Z c= (X \ Z) \/ (Y \ Z)
  proof
    let x be object;
    assume
A1: x in (X \/ Y) \ Z;
    then x in (X \/ Y) by XBOOLE_0:def 5;
    then x in X & not x in Z or x in Y & not x in Z by A1,XBOOLE_0:def 3,def 5;
    then x in (X \ Z) or x in (Y \ Z) by XBOOLE_0:def 5;
    hence thesis by XBOOLE_0:def 3;
  end;
  let x be object;
  assume x in (( X \ Z) \/ (Y \ Z));
  then x in (X \ Z) or x in (Y \ Z) by XBOOLE_0:def 3;
  then
A2: x in X & not x in Z or x in Y & not x in Z by XBOOLE_0:def 5;
  then x in (X \/ Y) by XBOOLE_0:def 3;
  hence thesis by A2,XBOOLE_0:def 5;
end;
