# Intersection with Set Difference is Set Difference with Intersection/Proof 2

$\left({R \setminus S}\right) \cap T = \left({R \cap T}\right) \setminus S$
 $\ds$  $\ds x \in \paren {R \setminus S} \cap T$ $\ds \leadstoandfrom \ \$ $\ds$  $\ds \paren {x \in R \land x \notin S} \land x \in T$ Definition of Set Intersection and Definition of Set Difference $\ds \leadstoandfrom \ \$ $\ds$  $\ds \paren {x \in R \land x \in T} \land x \notin S$ Rule of Commutation and Rule of Association $\ds \leadstoandfrom \ \$ $\ds$  $\ds x \in \paren {R \cap T} \setminus S$ Definition of Set Intersection and Definition of Set Difference
$\blacksquare$