Reference for the lemma in Nakagawa's article

Question

In K.Nakagawa's article "On the orders of automorphisms of a closed Riemann surface" there is Lemma 3, that doesn't have any proof or reference. Specifically, let $S$ be a closed Riemann surface, $G$ be the cyclic group of order $N$, acting on $S$ by automorphisms and $\tilde{S} = S / G$ be the closed Riemann surface obtained by identifying those point on $S$ which are equivalent under the action of $G$. Then, the lemma claims that if the number of branch points of covering map $\varphi : S \rightarrow \tilde{S}$ is $0$, then the Riemann surface $S$ is conformally equivalent to $y^N = f(x)$, where $f(x)$ is a polynomial in $x$. Where can I find the reference for it or is there an obvious proof that I'm missing?