Identity of Group is Unique/Proof 2

Theorem
Let $\left({G, \circ}\right)$ be a group which has an identity element $e \in G$.

Then $e$ is unique.

Proof
Let $e$ and $f$ both be identity elements of a group $\left({G, \circ}\right)$.

Then:

So $e = f$ and there is only one identity after all.