I would like to have the (simplest) proofs for the following theorem (there may be more than one interesting approach):

Let $f=\frac{p}{q}:\mathbb{C}\longrightarrow\mathbb{C}$ be a quotient of polynomials (of course, at some points it may be undefined) and $\bar{f}:\mathbb{S}^2\longrightarrow\mathbb{S}^2$ the natural extension to the Riemann sphere. Then $deg(\bar{f})=deg(p)-deg(q)$.

Any (original) proof is welcome.