Product of Commuting Elements with Inverses

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 $$x \circ y \circ x^{-1} \circ y^{-1} = e_S$$ iff $$x$$ and $$y$$ commute.

Proof
As $$\left({S, \circ}\right)$$ is a monoid, it is taken for granted that $$\circ$$ is associative, so we can dispense with parentheses.

We also take for granted the fact that $$x, y, x^{-1}, y^{-1}$$ are cancellable from Invertible also Cancellable.

So:

$$ $$ $$ $$