Powerset of Subset is Closed under Intersection

Theorem
Let $S$ be a set.

Let $T \subseteq S$ be a subset of $S$.

Let $\powerset S$ denote the power set of $S$.

Then $\powerset T$ is a closed subset of $\powerset S$ under set intersection:


 * $\forall A, B \in \powerset T: A \cap B \in \powerset T$

Proof
A direct application of Power Set is Closed under Intersection.