Subset Relation is Compatible with Subset Product

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

Let $\struct {\powerset S, \circ_\PP}$ be the power structure of $\struct {S, \circ}$.

Then the subset relation on $S$ is compatible with $\circ_\PP$.

That is:

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.