Subset Relation is Compatible with Subset Product

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

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

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

Then:
 * $X \subseteq Y \implies \left({X \circ Z}\right) \subseteq \left({Y \circ Z}\right)$
 * $X \subseteq Y \implies \left({Z \circ X}\right) \subseteq \left({Z \circ Y}\right)$

where $X \circ Z$ etc. denotes subset product.

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

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

Thus:
 * $x \circ z \in Y \circ Z$ and $z \circ x \in Z \circ Y$

and the result follows.