There are major problems with definitions in this area of probability.
Here is the standard quote:
"If are mutually independent events 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 .
Now you should be able to finish.