Lemma 13.3.4.label Let $E$ be a vector space over $\real$, $K \subset E$ be convex, $A \subset K$ be extreme, and $x \in A$ be an extreme point of $A$, then $x$ is an extreme point of $K$.
Proof. Let $y, z \in K$ such that $x \in (y, z)$. Since $A$ is extreme, $y, z \in A$. As $x$ is extremal in $A$, $x = y = z$, so $x$ is extremal in $K$.$\square$
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