In Milne's notes on Modular Forms, he sketches the proof of Prop 1.16 (p. 20): "Let $f$ be a nonconstant meromorphic function with valence $n$ on a compact Riemann surface $X$. Then every meromorphic function $g$ on $X$ is a root of a polynomial of degree $n$ with coefficients in $\mathbb{C}(f)$".
The proof uses the following result: If $f:X\rightarrow S$ is a meromorphic function from a compact Riemann surface $X$ to the Riemann sphere $S$ with valence $n$, $g:X\rightarrow S$ another meromporphic function, and $c_0\in S$ has $n$ distinct preimages under $f$, then $r(c)=\sum g(x)$, where the sum ranges over the preimages of $c$, is holomorphic at $c_0$ (assuming $g$ itself is holomorphic at all preimages). (Milne uses this result for $r(c)$ defined by other symmetric polynomials in the $g(x)$ values, not only their sum as I wrote).
Why is this true? Even if $X=S$ and $g$ is the identity function, I'm not sure how to prove this.