I'm doing fine in all the other probability stuff I'm doing, but I got stuck in proving the following theorem:

Source:If , and are any three events, then

Probability and Statistical Inference, 7e, by Hogg and Tanis

Here is my work thus far:

This implies that

Now,

So, substituting this into the equation I get:

I have no clue how to break up the part in red. Instead of working it out for me, give me a hint or two to help me get started.

Thanks in advance for any help!!

--Chris