Inverse of Identity Element is Itself

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

Let the inverse of $e$ be $e^{-1}$.

Then:
 * $e^{-1} = e$

That is, $e$ is self-inverse.

Proof
From Identities are Idempotent:
 * $e \circ e = e$

Hence the result by definition of identity element.