User blog comment:LittlePeng9/FOOT is not as strong as I thought/@comment-5529393-20161225200516/@comment-1605058-20170127194355

Refering again to the post I have referenced above for definitions, we do have to prove *. Since * is an axiom of T+*, then the part "[\(T\) proves] \(i(\phi)\) for each axiom \(\phi\) of \(S\)" (note \(T\) and \(S\) in the post are the other way around that T, S we consider), means that if S is to interpret T+*, then S has to prove \(\phi\) (or, more precisely, \(i(\phi)\).

Also, we do have to manage theorems of the form "not * implies ...", it's just that they are easy to deal with, once we prove * in S.