Degree-sized invariant separates finite-group orbits

Let a finite group $G$ of order $d$ act analytically on a compact Riemann surface, and let an invariant analytic map $F$ have degree $d$. At a point with stabilizer of order $e$, invariance forces the local multiplicity of $F$ to be at least $e$. Its orbit has $d/e$ points, so it already contributes at least $d$ to the fibre multiplicity. The valency theorem forces equality and shows that every fibre is exactly one orbit.