I think there's something wrong with this proof for uniqueness of inverses in a set with an associative binary operation, but I can't properly word it.
We take for granted that if $a \in S$ has a left inverse $\ell$ and a right inverse $r$, then $\ell = r$. It follows immediately that the inverse is unique. If $b_1, b_2$ are inverses of $a$, then $b_1 = b_2$.
The assumptions in uniqueness seem much stronger than in the lemma. The assumption is $a b_1 = b_2 a = e$ and $ab_2 = b_2 a = e$. But if we regard, say, $b_1$ as a left inverse and $b_2$ as a right inverse, then $b_1 = b_2$ by the same argument. The way I proved it is by a similar set of steps.
Is the proof ok or am I missing something?