Conjugate of Set with Inverse Closed for Inverses

Theorem
Let $G$ be a group.

Let $S \subseteq G$.

Let $\hat S = S \cup S^{-1}$, where $S^{-1}$ is the set of all the inverses of all the elements of $S$.

Let $\tilde S = \left\{{a s a^{-1}: s \in \hat S, a \in G}\right\}$.

That is, $\tilde S$ is the set containing all the conjugates of the elements of $S$ and all their inverses.

Then:
 * $\forall x \in \tilde S: x^{-1} \in \tilde S$

Proof
Let $x \in \tilde S$. That is, $\exists s \in \hat S: x = a s a^{-1}$.

Thus:

Since $s^{-1} \in \hat S$, it follows that $x^{-1} \in \tilde S$.