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

theorem Th23:
  X /\ (Y \/ Z) = X /\ Y \/ X /\ Z
proof
  thus X /\ (Y \/ Z) c= X /\ Y \/ X /\ Z
  proof
    let x be object;
    assume
A1: x in X /\ (Y \/ Z);
    then x in Y \/ Z by XBOOLE_0:def 4;
    then
A2: x in Y or x in Z by XBOOLE_0:def 3;
    x in X by A1,XBOOLE_0:def 4;
    then x in X /\ Y or x in X /\ Z by A2,XBOOLE_0:def 4;
    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
A3: x in X & x in Y or x in X & x in Z by XBOOLE_0:def 4;
  then x in Y \/ Z by XBOOLE_0:def 3;
  hence thesis by A3,XBOOLE_0:def 4;
end;
