Set Difference of Complements

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Theorem

$\map \complement S \setminus \map \complement T = T \setminus S$


Proof

\(\ds \map \complement S \setminus \map \complement T\) \(=\) \(\ds \set {x: x \in \map \complement S \land x \notin \map \complement T}\) Definition of Set Difference
\(\ds \) \(=\) \(\ds \set {x: x \notin S \land x \in T}\) Definition of Set Complement
\(\ds \) \(=\) \(\ds \set {x: x \in T \land x \notin S}\) Rule of Commutation
\(\ds \) \(=\) \(\ds T \setminus S\) Definition of Set Difference

$\blacksquare$