Let $W$ be a fixed matrix. Define $$\operatorname{pos}W \triangleq \{t \mid Wy =t , y≥ 0\}.$$ It is called the positive hull of $W$. It represents
the set of right-hand sides that can be obtained by a non-negative combination
of the columns of $W$. The positive hull is easily seen to be a convex cone.

Let $p$ be a point not in the set $\operatorname{pos}W$. Then, there exists a hyperplane
$H \triangleq \{x\mid\sigma^Tx =0\}$ that separates $p$ and $\operatorname{pos}W$.

How can we prove that the number of possible separating hyperplanes (separating $p$ and $\operatorname{pos}W$) is finite based on the fact that $\operatorname{pos}W$ is finitely generated?