Define the terms Riemann surface, holomorphic map between Riemann surfaces, and biholomorphic map.

(a) Prove that if two holomorphic maps $f, g$ coincide on a non-empty open subset of a connected Riemann surface $R$ then $f=g$ everywhere on $R$.

(b) Prove that if $f: R \rightarrow S$ is a non-constant holomorphic map between Riemann surfaces and $p \in R$ then there is a choice of co-ordinate charts $\phi$ near $p$ and $\psi$ near $f(p)$, such that $\left(\psi \circ f \circ \phi^{-1}\right)(z)=z^{n}$, for some non-negative integer $n$. Deduce that a holomorphic bijective map between Riemann surfaces is biholomorphic.

[The inverse function theorem for holomorphic functions on open domains in $\mathbb{C}$ may be used without proof if accurately stated.]