The question I'm about to ask might sound weird, I hope i can deliver the idea.
I have noticed that in some mathematics books (especially English ones) the proofs are written in words and symbols are used only when necessary, example: for all, implies, there exists, if and only if... etc, but in other books symbols are used more, example: $\implies$, $\forall$, $\exists$, $\iff$... etc
** So is there a rule that should be followed here? Or is it a choice that the writer makes? Will it be familiar if the proofs are mostly written with symbols?
I am asking this question because, right now, I am translating a book to English, and the writer of the original book barely used words in proofs, he only used connectors like: Therefore, Hence, Thus... etc. I do not want to translate over 300 pages for nothing.
** So should I keep the original proofs? Or should I reformulate them?
Sorry for the long question, but I am really confused here, I would be grateful if you help me. Thank you all in advance.
Keep the original style. You will respect the author's intent, and, more importantly, you will avoid introducing errors due to possible misunderstandings or clumsy rewrites. As a bonus, you don't have to translate the symbols, they are international.