Intersection number for projective plane curves

Question

Context: In our lecture about algebraic geometry we defined the intersection number of two algebraic curves $F,G$ to be $dim_k (\mathcal{O}_P (\mathbb{A}^2)/(F,G))$. Then we proved that it satisfies the 7 Properties for intersection numbers (I am not sure if these are well known or not, but I mean the ones from William Fultons book about algebraic curves).
My question is regarding the intersection number of two projective curves $F,G$, which we defined by $dim_k (\mathcal{O}_P (\mathbb{P}^2)/(F_*,G_*))$, where $F_*$ and $G_*$ are the corresponding dehomogenized forms. Our professor only told us that this satisfies the 7 properties again (however you have to change 2 of them a little so that they make sense in the projective case). Now I wanted to know if you actually have to check every single one of the properties again, or if there is a short argument tracing back to the affine case.
Thx in advance!