Section 8.5 due 12/6

I got lost in the proof of Theorem 8.30.

As I was reading this section, I kept thinking about how difficult it would be to come up with these theorems and then to go about proving them. Which comes first: the pattern to see the result of the theorem or the steps that are gone through in the theorem and then check if the idea actually works? Whatever the process is, it seems like it would be really tedious and difficult.