# Two-sided inverse is unique if it exists in monoid

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 $b = c$.