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.