Severi Formula for Intersection Multiplicities

I say in advance that I am a novice in Intersection Theory, so forgive me if my question is trivial.

Let $ X\subseteq\mathbb{P}^N$ be a smooth irreducible projective variety of dimension $ n$ and $ V, W\subseteq X$ be two irreducible subvarieties meeting properly in $ X$ . Choose a generic projection $ f\colon X\longrightarrow \mathbb{P}^n$ and denote by $ i(Z,V\cdot W;X)$ the intersection multiplicity of $ V$ and $ W$ at $ Z$ , in the ambient space $ X$ . As Fulton writes in his Intersection Theory (Example 8.2.6), it holds true the following formula $ $ i(Z, V\cdot W;X)=i(f(Z),f(V)\cdot f(W);\mathbb{P}^n).$ $

This was one of Severi’s methods for reducing the intersections on general varieties to intersections on the projective space. I would like to prove the formula above by using the Serre intersection formula.

I am pretty sure that there are results from Homological/Commutative Algebra (that I don’t know) that would make the proof of the formula an easy exercise.
Any help is well accepted.