Set Intersection Distributes over Set Difference

Theorem
Set intersection is distributive over set difference.

Let $R, S, T$ be sets.

Then:
 * $\left({R \setminus S}\right) \cap T = \left({R \cap T}\right) \setminus \left({S \cap T}\right)$
 * $R \cap \left({S \setminus T}\right) = \left({R \cap S}\right) \setminus \left({R \cap T}\right)$

where:
 * $R \setminus S$ denotes set difference
 * $R \cap T$ denotes set intersection.

Proof
Then: