Please have a look at sentence below:

Before one votes on the propositions, it's truly vital that he or she becomes familiar with the reasons for voting both for the proposition and against the proposition.

Does it have any grammar issue? My book says, "become familiar", "proposal". That is, becomes should be become. Agree?

Please have a look at the sentence below:

Before one votes on the propositions, it's truly vital that he or she becomes familiar with the reasons for voting both for the proposition and against the proposition.*

My book says, become familiar… proposal. That is, becomes should be become. Is there a grammatical difference? Is one more correct than the other?