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.