Corollary

Let $\Sigma$ be a set of reflections generating W. Then every reflection belonging to W is conjugate to an element of $\Sigma$.

Let $\mathfrak{H}'$ be the set of hyperplanes of the form $w(H)$ with $w \in W$ and $H \in \mathfrak{H}$ such that $s_H \in \Sigma$. Since W is generated by the family $(s_H)_{H \in \mathfrak{H}'}$ and since $\mathfrak{H}'$ is stable under W, we can apply all the results of this no. to $\mathfrak{H}'$ instead of $\mathfrak{H}$; but Th. 1, (iv) shows that every reflection in W is of the form $s_H$ with $H \in \mathfrak{H}'$, hence the corollary.