Subset Relation is Compatible with Subset Product/Corollary 1

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

Let $\circ_{\mathcal P}$ be the subset product on $\mathcal P \left({S}\right)$, the power set of $S$. Let $A, B, C, D \in \mathcal P \left({S}\right)$.

Let $A \subseteq B$ and $C \subseteq D$.

Then $A \circ_{\mathcal P} C \subseteq B \circ_{\mathcal P} D$

Proof
By Subset Relation is Compatible with Subset Product, $\subseteq$ is compatible with $\circ_{\mathcal P}$.

Subset Relation is Transitive.

Thus the theorem holds by Operating on Transitive Relationships Compatible with Operation.