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.