Subset Relation is Compatible with Subset Product/Corollary 1

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 $\circ$. Let $A, B, C, D \in \powerset S$.

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

Then:


 * $A \circ_\PP C \subseteq B \circ_\PP D$

Proof
By Subset Relation is Compatible with Subset Product, $\subseteq$ is compatible with $\circ_\PP$.

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

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