Power Set is Closed under Symmetric Difference

Theorem
Let $$S$$ be a set.

Let $$\mathcal P \left({S}\right)$$ be the power set of $$S$$.

Then:
 * $$\forall A, B \in \mathcal P \left({S}\right): A * B \in \mathcal P \left({S}\right)$$

where $$A * B$$ is the symmetric difference between $$A$$ and $$B$$.

Proof
Let $$A, B \subseteq S$$, i.e. $$A, B \in \mathcal P \left({S}\right)$$.

$$ $$ $$ $$

Thus $$A * B \in \mathcal P \left({S}\right)$$.