Subset Relation is Compatible with Subset Product

Theorem
Let $$\left({S, \circ}\right)$$ be a groupoid.

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

Let $$X, Y, Z \in \mathcal P \left({S}\right)$$.

Then: where $$X \circ Z$$ etc. denotes subset product.
 * $$X \subseteq Y \implies \left({X \circ Z}\right) \subseteq \left({Y \circ Z}\right)$$
 * $$X \subseteq Y \implies \left({Z \circ X}\right) \subseteq \left({Z \circ Y}\right)$$

Proof
Let $$x \in X, z \in Z$$.

Then $$x \circ z \in X \circ Z$$ and $$z \circ x \in Z \circ X$$.

Now:
 * $$Y \circ Z = \left\{{y \circ z: y \in Y, z \in Z}\right\}$$, and
 * $$Z \circ Y = \left\{{z \circ y: y \in Y, z \in Z}\right\}$$.

But $$x \in X \implies x \in Y$$ by the definition of a subset.

Thus $$x \circ z \in Y \circ Z$$ and $$z \circ x \in Z \circ Y$$ and the result follows.