Left and Right Operation are Closed for All Subsets

Theorem
Let $S$ be a set.

Let:
 * $\leftarrow$ be the left operation on $S$
 * $\rightarrow$ be the right operation on $S$.

That is:
 * $\forall x, y \in S: x \leftarrow y = x$
 * $\forall x, y \in S: x \rightarrow y = y$

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

Then for all $T \in \powerset S$, both $\leftarrow$ and $\rightarrow$ are closed on $T$.

That is, for all $T \in \powerset S$: