Fibres of a Holomorphic Function Let $f$ : $U\rightarrow V$ be a proper holomorphic map where $U$ and $V$ are open subsets of $\mathbb{C}$ with $V$ connected. Show that the cardinality of the fibres of $f$, i.e. $f^{-1}(\{z\})$ counted with the multiplicities are the same for each $z \in$ $V$. This looks like the property of covering maps and so I was trying to prove if $f$ is a local homeomorphism or a covering map, but to no avail. Thanks for any help.

Here's a useful lemma:

Lemma. Let $X,Y$ be locally compact Hausdorff topological spaces, $f : X \to Y$ continuous, open, proper, surjective and with discrete fibers. Let $K$ be a neighborhood of $y \in Y$, $y_1, \ldots, y_n$ its preimages (a finite number, by properness and discreteness) and $K_i$ a neighborhood of each $y_i$. Then there exist disjoint open neighborhoods $V_i \subseteq K_i$ of the $y_i$ and an open neighborhood $V \subseteq K$ of $y$ such that $f^{-1}(V)$ is the disjoint union of the $V_i$.

It is nothing deep (you can prove it). It is similar to the proof that a proper surjective local homeomorphism is a covering map. The only difference is that we don't have local injectivity here.
Note that proper implies closed, so proper+open+ $Y$ connected implies surjective.
Now let $X=U$, $Y=V$ as in the question. By the lemma, outside of the set $B$ of branch points (images of points where the derivative $f'$ vanishes) $f$ is a covering map (by the lemma + we now have local injectivity at the preimages). In particular, the size of fibers is constant on connected components of $Y - B$, say equal to $n$.
If $f$ proper, the set of branch points is closed (by closedness) and discrete (by properness) in $V$. In particular, $Y-B$ is connected.
It remains to check what happens in a neighborhood of a branch point $y$. Take $V$ and $V_i$ as in the lemma. It suffices to check that the number of preimages is constant for each of the restrictions $f : V_i \to V$. On such $V_i$, $f$ has the form $y + a_{m_i}(z-y_i)^{m_i} + a_{m_i+1} (z-y_i)^{m_i+1} + \cdots$ by Taylor-expansion, for some $m_i$ ($a_{m_i} \neq 0$). We want $\sum m_i=n$. W.l.o.g. we may assume $y_i=y=0$. We have that 
$$a_{m_i} z^{m_i} + a_{m_i+1} z^{m_i+1} + \cdots = g(z)^{m_i}$$
for some holomorphic $g$ which is a homeomorphism between open neighborhoods of $0$. Replacing $f(z)$ by $(f \circ g^{-1}) (z) = z^{m_i}$ does not change the number of preimages. So the fibers have cardinality $\sum m_i$ in a neighborhood of $y$, including at $y$. Comparing this to any point different from $y$ in that neighborhood, we conclude that $\sum m_i=n$.
