Set Difference with Union

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

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

where:
 * $R \setminus S$ denotes set difference
 * $R \cup T$ denotes set union.

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

Then:

Then: