Corollary 5

In a locally convex space $E$, the closure of each convex cone $C$ of vertex $0$ is the intersection of closed half-spaces containing $C$ determined by closed hyperplanes that pass through $0$.

For $\overline{C}$ is a convex cone of vertex $0$ (II, p. 13, prop. 14). For $x \notin \overline{C}$, there exists a closed hyperplane $H$ that separates $x$ strictly from $\overline{C}$(prop. 4). It is now just necessary to apply the following lemma :