Set Intersection Distributes over Set Difference

Theorem
Set intersection is distributive over set difference.

Let $$R, S, T$$ be sets.

Then: where:
 * $$\left({R \setminus S}\right) \cap T = \left({R \cap T}\right) \setminus \left({S \cap T}\right)$$
 * $$R \cap \left({S \setminus T}\right) = \left({R \cap S}\right) \setminus \left({R \cap T}\right)$$
 * $$R \setminus S$$ denotes set difference;
 * $$R \cap T$$ denotes set intersection.

Proof
$$ $$ $$ $$

Then:

$$ $$ $$