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

theorem Th16:
  (X /\ Y) /\ Z = X /\ (Y /\ Z)
proof
  thus (X /\ Y) /\ Z c= X /\ (Y /\ Z)
  proof
    let x be object;
    assume
A1: x in (X /\ Y) /\ Z;
    then
A2: x in Z by XBOOLE_0:def 4;
A3: x in X /\ Y by A1,XBOOLE_0:def 4;
    then
A4: x in X by XBOOLE_0:def 4;
    x in Y by A3,XBOOLE_0:def 4;
    then x in Y /\ Z by A2,XBOOLE_0:def 4;
    hence thesis by A4,XBOOLE_0:def 4;
  end;
  let x be object;
  assume
A5: x in X /\ (Y /\ Z);
  then
A6: x in Y /\ Z by XBOOLE_0:def 4;
  then
A7: x in Y by XBOOLE_0:def 4;
A8: x in Z by A6,XBOOLE_0:def 4;
  x in X by A5,XBOOLE_0:def 4;
  then x in X /\ Y by A7,XBOOLE_0:def 4;
  hence thesis by A8,XBOOLE_0:def 4;
end;
