Corollary

Let $G$ be a group with operators in which there exists a Jordan-Hölder series. If $\Sigma$ is any strictly decreasing composition series of $G$, there exists a Jordan-Hölder series finer than $\Sigma$.

Let $\Sigma_0$ be a Jordan-Hölder series of $G$; by Theorem 5, there exist two equivalent composition series, $\Sigma'$ and $\Sigma'_0$ respectively finer than $\Sigma$ and $\Sigma_0$; the argument of Theorem 6 shows that, by suppressing from $\Sigma'$ the repetitions, a sequence $\Sigma''$ is obtained which is equivalent to $\Sigma_0$ and hence a Jordan-Hölder series, since all its quotients are simple (Proposition 9). As $\Sigma$ is strictly decreasing, $\Sigma''$ is finer than $\Sigma$, whence the corollary.