Relative Complement of Relative Complement

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

$$\mathcal{C}_S \left({\mathcal{C}_S \left({T}\right)}\right) = T$$

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

But we have $$T \subseteq S \iff T \cap S = T$$ from Subset Equivalences, thus $$\mathcal{C}_S \left({\mathcal{C}_S \left({T}\right)}\right) = T$$ follows directly.