Lem 3: $\theta,\psi$ non-const meromorphic functions on compact riemann surface $S$ of valence $m,n$ respectively. $\exists F(X,Y)$ of degree at most $n$ in $X$ and $m$ in $Y$ s.t. $F(\theta,\phi)=0$.
Let $p_i$ be values of $\psi(p_i)=c$ s.t. $p_i$ avoids poles and zeroes of $\theta$. Then one can determine $\sum_i\theta(p_i)^r$ as a rational function of $c=\psi(p_i)$ via considering residue of form $\theta^r\frac{d\psi}{\psi-c}$ which sums to $0$ and this yields a rational function of $c$. This rational function is independent of $p_i$ as it only depends upon poles and $\theta$ and $\psi$. This yields $\theta$ satisfies a polynomial of degree at most $n$ as one consider all symmetric polynomial generated by $\theta(p_i)$.
$\textbf{Q:}$ "Now the argument applies symmetrically to $\psi$. This yields a polynomial of degree at most $n$ in $\psi$. Hence $F$ has the property as described." How do I know the argument extends symmetrically yields the same $F$? I do not see any good reason that symmetric argument applies here somehow.
Ref: Analytic Theory of Abelian Varieties by P. Swinnerton-Dyer