Inverse of Product/Monoid

Theorem
Let $\struct {S, \circ}$ 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:
 * $\paren {a \circ b}^{-1} = b^{-1} \circ a^{-1}$

Proof
Similarly for $\paren {b^{-1} \circ a^{-1} } \circ \paren {a \circ b}$.