# Set Difference is Right Distributive over Set Intersection/Proof 2

$\paren {R \cap S} \setminus T = \paren {R \setminus T} \cap \paren {S \setminus T}$
 $\displaystyle x$ $\in$ $\displaystyle \paren {R \cap S} \setminus T$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x$ $\in$ $\displaystyle R \land x \in S$ Definition of Set Intersection $\, \displaystyle \land \,$ $\displaystyle x$ $\notin$ $\displaystyle T$ Definition of Set Difference $\displaystyle \leadstoandfrom \ \$ $\displaystyle x$ $\in$ $\displaystyle R \land x \in S$ Rule of Idempotence $\, \displaystyle \land \,$ $\displaystyle x$ $\notin$ $\displaystyle T \land x \notin T$ Rule of Idempotence $\displaystyle \leadstoandfrom \ \$ $\displaystyle x$ $\in$ $\displaystyle R$ Rule of Association $\, \displaystyle \land \,$ $\displaystyle x$ $\in$ $\displaystyle x \in S \land x \notin T$ $\, \displaystyle \land \,$ $\displaystyle x$ $\notin$ $\displaystyle T$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x$ $\in$ $\displaystyle R$ $\, \displaystyle \land \,$ $\displaystyle x$ $\notin$ $\displaystyle T \land x \in S$ Rule of Commutation $\, \displaystyle \land \,$ $\displaystyle x$ $\notin$ $\displaystyle T$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x$ $\in$ $\displaystyle R \land x \notin T$ Rule of Association $\, \displaystyle \land \,$ $\displaystyle x$ $\in$ $\displaystyle S \land x \notin T$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x$ $\in$ $\displaystyle \paren {R \setminus T}$ Definition of Set Difference $\, \displaystyle \land \,$ $\displaystyle x$ $\in$ $\displaystyle \paren {S \setminus T}$ Definition of Set Difference $\displaystyle \leadstoandfrom \ \$ $\displaystyle x$ $\in$ $\displaystyle \paren {R \setminus T} \cap \paren {S \setminus T}$ Definition of Set Intersection
$\blacksquare$