[Spivak]

I don't really think you need too much to prove it yourself.

You are being governed with consent when the person who's elected is someone you are okay being governed by. And the person who wins an approval election is the person with that has the most people fine with being governed by them. Because approval voting doesn't ask people to rank candidates voting for someone you disapprove of only hurts you and voting for any subset of people you do approve of is sincere.

It's not some deep thing because it changes the target to something much easier. Finding the best candidate is hard, finding the candidate most people find acceptable is less so.

Approval voting gets more mathematically interesting when you assume people have preferences among the candidates they approve of and whether the best candidate gets elected but IRL you don't actually care about that anymore. You're fine electing someone who isn't the best.