State and prove the Valency Theorem and define the degree of a non-constant holomorphic map between compact Riemann surfaces.

Let $X$ be a compact Riemann surface of genus $g$ and $\pi: X \rightarrow \mathbb{C}_{\infty}$ a holomorphic map of degree two. Find the cardinality of the set $R$ of ramification points of $\pi$. Find also the cardinality of the set of branch points of $\pi$. [You may use standard results from lectures provided they are clearly stated.]

Define $\sigma: X \rightarrow X$ as follows: if $p \in R$, then $\sigma(p)=p$; otherwise, $\sigma(p)=q$ where $q$ is the unique point such that $\pi(q)=\pi(p)$ and $p \neq q$. Show that $\sigma$ is a conformal equivalence with $\pi \sigma=\pi$ and $\sigma \sigma=$ id.