Identity is Unique/Proof 1

Theorem
Let $\left({S, \circ}\right)$ be an algebraic structure that has an identity element $e \in S$.

Then $e$ is unique.

Proof
Suppose $e_1$ and $e_2$ are both identities of $\left({S, \circ}\right)$.

Then by the definition of identity element:
 * $\forall s \in S: s \circ e_1 = s = e_2 \circ s$

Then:
 * $e_1 = e_2 \circ e_1 = e_2$

So $e_1 = e_2$ and there is only one identity after all.