> Let F be a field, and let a,b∈F such that ab=0. There are two cases to consider: a=0 and a≠0. If a=0, then indeed ab=0 by Proposition 1. This part is a bit w

> Let F be a field, and let a,b∈F such that ab=0. There are two cases to consider: a=0 and a≠0. If a=0, then indeed ab=0 by Proposition 1. This part is a bit weird. If a=0, then we are already done, there's no need to prove ab=0 (which was already the assumption). The other case can also be proved in a shorter way by just multiplying both sides of ab=0 with a^(-1) from the left.