Sigma-Locally Finite Cover and Countable Locally Finite Cover have Common Locally Finite Refinement

Theorem
Let $T = \struct{X, \tau}$ be a topological space.

Let $\SS = \ds \bigcup_{n = 0}^\infty \SS_n$ be a $\sigma$-locally finite cover of $X$, where each $\SS_n$ is locally finite for all $n \in \N$.

Let $\CC = \set{A_n : n \in \N}$ be a countable locally finite cover of $X$.

Then:
 * there exists a common locally finite refinement $\AA$ of both $\SS$ and $\CC$