Inverse of Product/Monoid

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

Let $a, b \in S$ be invertible for $\circ$, with inverses $a^{-1}, b^{-1}$.

Then $a \circ b$ is invertible for $\circ$, and:
 * $\left({a \circ b}\right)^{-1} = b^{-1} \circ a^{-1}$

Proof
Similarly for $\left({b^{-1} \circ a^{-1}}\right) \circ \left({a \circ b}\right)$.