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

theorem
  X c= Z implies X \/ Y /\ Z = (X \/ Y) /\ Z
proof
  assume
A1: X c= Z;
  thus X \/ Y /\ Z c= (X \/ Y) /\ Z
  proof
    let x be object;
    assume x in X \/ Y /\ Z;
    then
A2: x in X or x in Y /\ Z by XBOOLE_0:def 3;
    then x in X or x in Y & x in Z by XBOOLE_0:def 4;
    then
A3: x in (X \/ Y) by XBOOLE_0:def 3;
    x in Z by A1,A2,XBOOLE_0:def 4;
    hence thesis by A3,XBOOLE_0:def 4;
  end;
  let x be object;
  assume
A4: x in (X \/ Y) /\ Z;
  then x in X \/ Y by XBOOLE_0:def 4;
  then
A5: x in X or x in Y by XBOOLE_0:def 3;
  x in Z by A4,XBOOLE_0:def 4;
  then x in X & x in Z or x in Y /\ Z by A5,XBOOLE_0:def 4;
  hence thesis by XBOOLE_0:def 3;
end;
