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

Proof
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