Intersection Distributes over Symmetric Difference

Theorem
Intersection is distributive over symmetric difference:


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

Proof
From Set Intersection Distributes over Set Difference, we have $$\left({R \setminus S}\right) \cap T = \left({R \cap T}\right) \setminus \left({S \cap T}\right)$$. So:

$$ $$ $$ $$

The second part of the proof is a direct consequence of the fact that both Intersection is Commutative and Symmetric Difference is Commutative.