Theorem 6 (Jordan-Hölder)

Two Jordan-Hölder series of a group with operators are equivalent.

Let $\Sigma_1, \Sigma_2$ be two Jordan-Hölder series of a group with operators $G$; by applying Theorem 5 two equivalent composition series $\Sigma'_1, \Sigma'_2$ are obtained which are respectively finer than $\Sigma_1$ and $\Sigma_2$; since the latter are Jordan-Hölder series, $\Sigma'_1$ is identical with $\Sigma_1$ or is derived from it by repeating certain terms; the series of quotients of $\Sigma'_1$ is derived from that for $\Sigma_1$ by inserting a number of terms isomorphic to the group $\{e\}$; since $\Sigma_1$ is strictly decreasing, the series of quotients of $\Sigma_1$ is derived from that of $\Sigma'_1$ by suppressing in the latter all the terms isomorphic to $\{e\}$. Similarly for $\Sigma_2$ and $\Sigma'_2$. As the series of quotients of $\Sigma'_1$ and $\Sigma'_2$ differ (up to isomorphism) only in the order of the terms, the same is true of $\Sigma_1$ and $\Sigma_2$; the theorem is proved.