Inverse of Commuting Pair

Theorem
Let $\left({S, \circ}\right)$ be a monoid whose identity is $e_S$. Let $x, y \in S$ such that $x$ and $y$ are both invertible.

Then $x$ commutes with $y$ iff $\left({x \circ y}\right)^{-1} = x^{-1} \circ y^{-1}$.