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

theorem Th24:
  X \/ Y /\ Z = (X \/ Y) /\ (X \/ Z)
proof
  thus X \/ Y /\ Z c= (X \/ Y) /\ (X \/ Z)
  proof
    let x be object;
    assume x in X \/ Y /\ Z;
    then 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 x in X \/ Y & x in X \/ Z by XBOOLE_0:def 3;
    hence thesis by XBOOLE_0:def 4;
  end;
  let x be object;
  assume
A1: x in (X \/ Y) /\ (X \/ Z);
  then x in X \/ Z by XBOOLE_0:def 4;
  then
A2: x in X or x in Z by XBOOLE_0:def 3;
  x in X \/ Y by A1,XBOOLE_0:def 4;
  then x in X or x in Y by XBOOLE_0:def 3;
  then x in X or x in Y /\ Z by A2,XBOOLE_0:def 4;
  hence thesis by XBOOLE_0:def 3;
end;
