Identity is Unique/Proof 2

Theorem

Let $\struct {S, \circ}$ be an algebraic structure that has an identity element $e \in S$.

Then $e$ is unique.

Proof

Let $e_S$ be an identity of $\struct {S, \circ}$.

Then by definition, $e_S$ is both a left identity and a right identity.

By More than one Left Identity then no Right Identity, if there is more than one of either, there cannot be one of the other.

So there can be only one of each.

By Left and Right Identity are the Same, they are one and the same thing.

$\blacksquare$