There are major problems with definitions in this area of probability.

Here is the standard quote:

"If aremutually independentevents then every event determined by a subcollection of these events is independent of every event determined by a subcollection of the remaining events, Jim Pitman.

If we use that definition, it seems to solve this question.

Have you proved that: If are independent events then are independent events?

Here is the prove of that.

We know that .

So

.

Now you should be able to finish.