Proposition 43.19.3. Let $X$ be a nonsingular variety. Let $V \subset X$ and $W \subset Y$ be closed subvarieties which intersect properly. Let $Z \subset V \cap W$ be an irreducible component. Then $e(X, V \cdot W, Z) > 0$.
This is one of the main results of [Serre_algebre_locale].
By Lemma 43.19.2 we have
Proof. By Lemma 43.19.2 we have
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