Corollary 2

Let X be a closed irreducible subset of V and H a hypersurface in V. Only three cases are possible:

1) one has $X \subset H$; 2) the set $X \cap H$ is nonempty and each of its irreducible components Z satisfies $\operatorname{codim}(Z, X) = 1$; 3) the set $X \cap H$ is empty.

Suppose $X' = X \cap H$ nonempty and distinct from X; every irreducible component Z of $X'$ is distinct from X and satisfies $\operatorname{codim}(Z, X) \leq 1$ by Prop. 4, b). Corollary 2 follows at once from this.