The definition for intersection multiplicity used in Fulton's Algebraic Curves is given by:
$$I(P,F\cap G):=\dim_{\mathbb{K}}(\mathscr{O}_{P,\mathbb{A}^2}/(F,G)),$$
(see section 3.3) where $\mathscr{O}_{P,\mathbb{A}^2}$ denotes the local ring of $\mathbb{A}^2$ at $P$.
Later, Fulton proves a property (Property (9)) about intersection multiplicity:
$$\sum_{P}I(P,F\cap G)=\dim_{\mathbb{K}}(\mathbb{K}[X,Y]/(F,G)).$$
However, the proof of this result refers to an earlier result, which Fulton gave a proof which is a bit complicated (see section 2.9; it refers to several earlier results and exercises, as well as using some tools like "higher" tools Radical).
I want to know if there's a simpler way of showing that $\sum_{P}I(P,F\cap G)=\dim_{\mathbb{K}}(\mathbb{K}[X,Y]/(F,G))$ without using tools like Radicals, etc, or does the nature of this result demands a not-simple proof?
Or, since there's "only one intersection multiplicity", is there another way of defining intersection multiplicity which makes this result easier to proof (but without referring to resultant)?
Edit: I am actually preparing a short talk for a seminar course, where most of us have only taken a one-year undergrad algebra course, which, as far as I know, does not cover topics like local rings, radicals, etc. However, basics of Noetherian and Artinian rings are introduced (so at least they know what Noetherian and Artinian rings are, and I believe I can use this as a starting point).