Two-sided inverse is unique if it exists in monoid

Statement
Suppose $$S$$ is a monoid with binary operation $$*$$ and identity element (neutral element) $$e$$. Suppose $$a \in S$$ has a two-sided inverse $$b$$, i.e., $$b * a = a * b = e$$. Then, $$b$$ is the only two-sided inverse for $$a$$, i.e., if $$c$$ is an element such that $$c * a = a * c = e$$, then $$b = c$$.

Facts used

 * 1) uses::Equality of left and right inverses in monoid

Proof
Fact (1) says that every left inverse must equal every right inverse. Thus, if we have two two-sided inverses, we can treat one of them as a left inverse and the other as a right inverse, forcing both of them to be equal.