Conjugate of Commuting Elements

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 $$x \circ y \circ x^{-1} = y$$ 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 $$x$$ and $$y$$ are cancellable from Invertible also Cancellable.

So:

$$ $$ $$