How to prove that : $ d.e = \displaystyle \sum_{ p \in C \cap D } \mathrm{dim} \ \mathcal{O}_{ \mathbb{P}^{2} } / (f,g) $?

Question

Let $ C \subset \mathbb{P}^2 $ be a smooth curve defined by a homogeneous polynomial $ f $ of degree $ d $.

Let $ D \subset \mathbb{P}^2 $ be a second smooth curve defined by a homogeneous polynomial $ g $ of degree $ e $.

How to show that : $$ d ~ .e = \displaystyle \sum_{ p \in C \cap D } \mathrm{dim} \ \mathcal{O}_{ \mathbb{P}^{2} } / (f,g) $$

Thank you in advance for help.