Suppose $X$, $Y$, $Z$ are projective varieties in $\mathbb{P}^n\_K$ of dimension $n-1$, where $K$ is a field. $X$, $Y$, $Z$ intersect properly, and $P$ is one of their intersection irreducible components of dimension $n-3$. Suppose their intersection multiplicity at $P$ is $i(P; X,Y,Z; \mathbb{P}^n)$. Do we always have $i(P; X,Y,Z; \mathbb{P}^n)\geq e\_P(X)e\_P(Y)e\_P(Z)$? Where $e\_P(X)$ is the multiplicity of $P$ in $X$. 


If $P$ is an isolated point, this result is true from Fulton's intersection theory book (Thm 12.4 and Cor 12.4), and it is also true for two varieties intersecting case. I want to know whether the general case is right and a reference. Thank you.



Relation between intersection multiplicity and Hilbert-Samuel multiplicities
It's a result in "méthodes d'algèbre abstraite en géométrie algébrique" written by Samuel.


