Set Intersection Distributes over Set Difference

Theorem
Set intersection is distributive over set difference.

Let $R, S, T$ be sets.

Then:
 * $\paren {R \setminus S} \cap T = \paren {R \cap T} \setminus \paren {S \cap T}$
 * $R \cap \paren {S \setminus T} = \paren {R \cap S} \setminus \paren {R \cap T}$

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

Proof
Then: