Subset of Cover is Cover of Subset

Theorem
Let $S$ be a set.

Let $\CC$ be a cover of $S$.

Let $T \subseteq S$.

Let:
 * $\CC_T = \set{C \in \CC : C \cap T \ne \O}$.

Then $\CC_T$ is a cover of $T$:
 * $T \subseteq \ds \bigcup \CC_T$

Proof
Let $x \in T$.

By definition of a cover:
 * $\exists C \in \CC : x \in C$

By definition of set intersection:
 * $x \in C \cap T$

Hence:
 * $C \in \CC_T$

Since $x$ was arbitrary, it follows that $\CC_T$ is a cover of $T$ by definition and:
 * $T \subseteq \ds \bigcup \CC_T$