Lemma 1

Let E, $E'$, and $E''$ be A-modules of finite length and

0 // ${E'}^i$ // E $^p$ // $E''$ // 0

an exact sequence. We have $\ell_S(E) =\ell_S(E') +\ell_S(E'')$.

Let $\Sigma '$ and $\Sigma ''$ be Jordan–Hölder series of $i(E')$ and $E/i(E')$, respectively. There exists a Jordan–Hölder series Σ of E whose sequence of quotients can be obtained by juxtaposing the sequence of quotients of Σ and that of $\Sigma '$ (I, §4, No. 8, p. 44).