Uniqueness of identity

Proposition (Uniqueness of identity). Let $G$ be a group with binary operation written multiplicatively. An element $e\in G$ is called an identity element if for every $g\in G$ one has $eg=g$ and $ge=g$. If $e,e'\in G$ are both identity elements, then $e=e'$.

Context. This shows that “the” identity element of a group is well-defined (not dependent on choices).

Proof sketch. Since $e$ is an identity, $ee'=e'$. Since $e'$ is an identity, $ee'=e$. Hence $e=e'$.