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

\(\displaystyle \) \(\) \(\displaystyle x \in \left({R \setminus S}\right) \cap T\)
\(\displaystyle \) \(\iff\) \(\displaystyle \left({x \in R \land x \notin S}\right) \land x \in T\) Definitions of Set Intersection and Set Difference
\(\displaystyle \) \(\iff\) \(\displaystyle \left({x \in R \land x \in T}\right) \land x \notin S\) Rules of Commutation and Association
\(\displaystyle \) \(\iff\) \(\displaystyle x \in \left({R \cap T}\right) \setminus S\) Definitions of Set Intersection and Set Difference

$\blacksquare$