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: