Inverse of Product of Subsets of Group

Theorem
Let $\left({G, \circ}\right)$ be a group.

Let $X, Y \subseteq G$.

Then:
 * $\left({X \circ Y}\right)^{-1} = Y^{-1} \circ X^{-1}$

where $X^{-1}$ is the inverse of $X$.

Proof
First, note that.

Now:

By a similar argument we see that $\left({X \circ Y}\right)^{-1} \subseteq Y^{-1} \circ X^{-1}$.

Hence the result.