Relative Complement of Relative Complement

Theorem
The relative complement of the relative complement of a set is itself:


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

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

But we have $T \subseteq S \iff T \cap S = T$ from Intersection with Subset is Subset‎.

Thus $\complement_S \left({\complement_S \left({T}\right)}\right) = T$ follows directly.