Refinement of a Refinement is Refinement of Cover

Theorem
Let $S$ be a set.

Let $\UU = \set {U_\alpha}$, $\VV = \set {V_\beta}$ and $\WW = \set {W_\gamma}$ be covers of $S$.

Let $\VV$ be a refinement of $\UU$.

Let $\WW$ be a refinement of $\VV$.

Then:
 * $\WW$ is a refinement of $\UU$

Proof
Let $W \in \WW$.

By definition of refinement:
 * $\exists V \in \VV : W \subseteq V$

Similarly:
 * $\exists U \in \UU : V \subseteq U$

From Subset Relation is Transitive:
 * $W \subseteq U$

Since $W$ was arbitrary:
 * $\forall W \in \WW : \exists U \in \UU : W \subseteq U$

It follows that $\WW$ is a refinement of $\UU$ by definition.