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

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Theorem

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


Proof

\(\ds \) \(\) \(\ds x \in \paren {R \setminus S} \cap T\)
\(\ds \leadstoandfrom \ \ \) \(\ds \) \(\) \(\ds \paren {x \in R \land x \notin S} \land x \in T\) 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

$\blacksquare$