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

Then for all $$T \in \mathcal P \left({S}\right)$$, both $$\leftarrow$$ and $$\rightarrow$$ are closed on $$T$$.

Thus, for all $$T \in \mathcal P \left({S}\right)$$:
 * $$\left({T, \leftarrow}\right)$$ is a subsemigroup of $$\left({S, \leftarrow}\right)$$;
 * $$\left({T, \rightarrow}\right)$$ is a subsemigroup of $$\left({S, \rightarrow}\right)$$.

Proof
From Right Operation All Elements Left Identities we have that $$\left({S, \rightarrow}\right)$$ is a semigroup, whatever the nature of $$S$$.

From Left Operation All Elements Right Identities we have that $$\left({S, \leftarrow}\right)$$ is a semigroup, whatever the nature of $$S$$.

Let $$T \in \mathcal P \left({S}\right)$$.

Then:
 * From Right Operation All Elements Left Identities, $$\left({T, \rightarrow}\right)$$ is a semigroup, and therefore a subsemigroup of $$\left({S, \rightarrow}\right)$$.
 * From Left Operation All Elements Right Identities, $$\left({T, \leftarrow}\right)$$ is a semigroup, and therefore a subsemigroup of $$\left({S, \leftarrow}\right)$$.

This applies whatever $$S$$ is and whatever the subset $$T$$ is.