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. 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.