Relative Complement of Relative Complement/Proof 2

Proof
The definition of the relative complement requires that:
 * $T \subseteq S$

But from Intersection with Subset is Subset‎:
 * $T \subseteq S \iff T \cap S = T$

Thus:
 * $\complement_S \left({\complement_S \left({T}\right)}\right) = T$

follows directly.