Subset Relation is Compatible with Subset Product

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

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

Let $\circ_\PP$ be the operation induced on $\powerset S$ by $\circ$.

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

Proof
Let $A, B, C \in \powerset S$.

Let $A \subseteq B$.

Let $x \in A \circ_\PP C$.

Then for some $a \in A$ and some $c \in C$:
 * $x = a \circ c$

Since $A \subseteq B$, $a \in B$.

Thus $x \in B \circ_\PP C$.

Since this holds for all $x \in A \circ_\PP C$:
 * $A \circ_\PP C \subseteq B \circ_\PP C$

The same argument shows that:
 * $C \circ_\PP A \subseteq C \circ_\PP B$