Inverse of Product/Monoid

Corollary to Inverse of Product in Associative Structure
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
By definition, a monoid is an algebraic structure whose operation is associative.

The result follows by Inverse of Product in Associative Structure.