Equal Set Differences iff Equal Intersections

Theorem

 * $R \setminus S = R \setminus T \iff R \cap S = R \cap T$

Proof
Assuming either $R \setminus S = R \setminus T$ or $R \cap S = R \cap T$ we can subtract those terms from both sides because the unions are disjoint.

Assume for instance $R \setminus S = R \setminus T$. Subtracting this set we obtain