De Morgan's Laws (Set Theory)/Relative Complement/General Case

Theorem
Let $S$ be a set.

Let $T$ be a subset of $S$.

Let $\mathcal P \left({T}\right)$ be the power set of $T$.

Let $\mathbb T \subseteq \mathcal P \left({T}\right)$.

Then:


 * $(1): \quad \displaystyle \complement_S \left({\bigcap \mathbb T}\right) = \bigcup_{H \mathop \in \mathbb T} \complement_S \left({H}\right)$


 * $(2): \quad \displaystyle \complement_S \left({\bigcup \mathbb T}\right) = \bigcap_{H \mathop \in \mathbb T} \complement_S \left({H}\right)$