If ## p\geq q\geq 5 ## and ## p ## and ## q ## are both primes ....

[PeroK]

That's far too long. You only need $p = q + 2k$. You don't need the rest of those variables. To do something rigorously you don't need to write out every detail.

You seem so focused on the details that you forget the ideas involved and what is important.