Set Intersection Preserves Subsets/Corollary/Proof 2

Corollary to Set Intersection Preserves Subsets
Let $A, B, S$ be sets.

Then:
 * $A \subseteq B \implies A \cap S \subseteq B \cap S$

Proof
Recall the Factor Principles, themselves a corollary of the Praeclarum Theorema:
 * $\paren {p \implies q} \vdash \paren {p \land r} \implies \paren {q \land r}$

This is applied as: