Equivalence of Definitions of Self-Inverse

Theorem
Let $\left({S, \circ}\right)$ be a monoid whose identity is $e_S$.

Let $x \in \left({S, \circ}\right)$ be such that $x \circ x = e_S$.

Then $x = x^{-1}$.

Such an $x$ is said to be self-inverse.