Subcover is Refinement of Cover

Theorem
Let $S$ be a set.

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

Let $\VV$ be a subcover of $\CC$.

Then $\VV$ is a refinement of $\CC$.

Proof
From definition of subcover:
 * $\VV \subseteq \CC$

That is, every element of $\VV$ is an element of $\CC$.

From definition of subset, every element of $\VV$ is the subset of some element of $\CC$.

This is precisely the definition of refinement.