Hatcher 1.3. problem 16

Given maps $X\to Y\to Z$ such that both $Y\to Z$ and the composition $X\to Z$ are covering spaces, show that $X\to Y$ is a covering space if $Z$ is locally path-connected.

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Yes, that is a homework problem from Hatcher's textbook. Do you have a question? –  Ryan Budney Feb 23 '12 at 6:08
Please anyone can help? I`m totally stuck in it. –  Danny Feb 24 '12 at 3:00
I decided to do this problem and made some progress by considering the two coverings $\{U_\alpha \}$ and $\{V_\beta \}$ associated with the two covering maps $q: Y \to Z$ and $qp: X \to Z$, and looking at the cover of $Y$ by the disjoint sets in $\{ q^{-1}(U_\alpha \cap V_\beta) \}_{\alpha , \beta}$, but I don't see where local path-connectedness comes into play. I'd appreciate if someone could contribute. –  Carl Feb 29 '12 at 7:28
Seems like it looks a lot like Lemma 80.2. in Munkres. –  user123412 Aug 11 '13 at 9:24