Subset of Subset Product

Theorem
Let $\struct {S, \circ}$ be a magma.

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

Let $X, Y, Z \in \powerset S$.

Then:
 * $X \subseteq Y \implies \paren {X \circ Z} \subseteq \paren {Y \circ Z}$
 * $X \subseteq Y \implies \paren {Z \circ X} \subseteq \paren {Z \circ Y}$

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

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.