Suppose we have a line $L$ in the projective space $ \mathbb{CP}^2$ and we choose a point $R \in \mathbb{CP}^2$, such that $R \notin L$. For any other point $P \in \mathbb{CP}^2$, we define $L_{PR}$ to be the line through $P$ and $R$. Now we define the following map: $$ \phi: \mathbb{CP}^2 \setminus {\{R\}} \to L, \ \ P \mapsto \{L \cap L_{PR}\}. $$
Here $\{L \cap L_{PR}\}$ is the unique intersection point of those two lines. The map $\phi$ is apparently holomorphic, but how would one go about proving that it is holomorphic?