Inverse of Product

Theorem
Let $$\left({S, \circ}\right)$$ be a monoid whose identity is $$e$$.

Let $$x, y \in S$$ be invertible for $$\circ$$, with inverses $$x^{-1}, y^{-1}$$.

Then $$x \circ y$$ is invertible for $$\circ$$, and $$\left({x \circ y}\right)^{-1} = y^{-1} \circ x^{-1}$$.

Proof
Similarly for $$\left({y^{-1} \circ x^{-1}}\right) \circ \left({x \circ y}\right)$$.