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

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Theorem

$\paren {S \setminus T} \cap S = S \setminus T$


Proof

\(\ds \paren {S \setminus T} \cap S\) \(=\) \(\ds \paren {S \cap S} \setminus T\) Intersection with Set Difference is Set Difference with Intersection
\(\ds \) \(=\) \(\ds S \setminus T\) Set Intersection is Idempotent

$\blacksquare$