Others have already given some good answers, but I'll add that a part of it is that this sort of thing is to be expected, especially when dealing with proofs. A

Others have already given some good answers, but I'll add that a part of it is that this sort of thing is to be expected, especially when dealing with proofs. All proofs have at least some ambiguity because whether or not to leave certain parts out is subjective. Don't expect to just get a nontrivial proof on the first read. You have to convince yourself that the proof is valid. It's easy to think of this as a waste of time, because the author could have just been more clear, but to a certain extent it's a good thing. It forces you to understand the context around this proof, and ultimately is more illuminating than just reading and memorizing the proof.