Intersection Distributes over Symmetric Difference

Theorem
Intersection is distributive over symmetric difference:


 * $\paren {R * S} \cap T = \paren {R \cap T} * \paren {S \cap T}$
 * $T \cap \paren {R * S} = \paren {T \cap R} * \paren {T \cap S}$

Proof
From Set Intersection Distributes over Set Difference, we have:
 * $\paren {R \setminus S} \cap T = \paren {R \cap T} \setminus \paren {S \cap T}$

So:

The second part of the proof is a direct consequence of the fact that Intersection is Commutative.