Intersection with Set Difference is Set Difference with Intersection

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

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

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

Proof 1
Consider $$R, S, T \subseteq \mathbb U$$, where $$\mathbb U$$ is considered as the universe.

Then:

$$ $$ $$

Proof 2
$$ $$ $$ $$