Inverse of Inverse of Subset of Group

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

Let $X \subseteq G$.

Then $\left({X^{-1}}\right)^{-1} = X$.

That is, the inverse of the inverse of $X$ is $X$ itself.

Proof
By the definition of inverse of subset:


 * $X^{-1} = \left\{{x^{-1}: x \in X}\right\}$

Thus: