Set Intersection Preserves Subsets/Corollary/Proof 2

Theorem
Let $A, B, S$ be sets.

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

Proof
From the Factor Principles, themselves a corollary of the Praeclarum Theorema:
 * $\left({p \implies q}\right) \vdash \left({p \land r}\right) \implies \left ({q \land r}\right)$

applying it as:
 * $\left({x \in A \implies x \in B}\right) \implies \left({x \in A \land x \in S \implies x \in B \land x \in S}\right)$

and so on.