"the" is required for nouns that are specific objects, like the one and only.
For example: I live in Indonesia. It is an archipelago country found beside "the" Pacific Ocean.
For your sentences, using "a" or "an" instead of "the" would be more preferable since:
1. There's nothing specific about "PE class".
2. There could be more than one proof, thus making "proof" not specific.
However if there is something special about the "PE class" and "proof" making it somewhat unique or so, then it is better to use "the".