Subset Relation is Compatible with Subset Product/Corollary 1

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

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

Let $\circ_\mathcal P$ be the operation induced on $\mathcal P \left({S}\right)$ by $\circ$. 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 $\circ_\mathcal P$.

By Subset Relation is Transitive, $\subseteq$ is transitive.

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