Uniqueness of inverses

Proposition (Uniqueness of inverses). Let $G$ be a group . For $g\in G$, an element $x\in G$ is an inverse of $g$ if $xg=e$ and $gx=e$, where $e$ is the identity element of $G$. If $x$ and $y$ are both inverses of $g$, then $x=y$.

Context. This justifies writing $g^{-1}$ for the inverse of $g$.

Proof sketch. Using associativity and the defining equations,

(Here $e$ denotes the identity in $G$.)