Show that if events A and B are independent, then events A' and B are independent.
If events are independent, then we know that
by definition. That's our key.
For the proof, we will use the following lemma:
Proof of the lemma:
So now we know that lemma is true, we can use it to conclude that:
Looking at the right hand side, we have:
(using our key assumption)
So we conclude that
as desired.