Proposition 3

Suppose that $\mathscr{C}$ is hereditary. Let E be a module of type $\mathscr{C}$ and $(E_i)_{0\leqslant i\leqslant n}$ a composition series of E (I, §4, No. 7, p. 41). For $1\leqslant i\leqslant n$, the module $E_{i-1}/E_i$ is of type $\mathscr{C}$, and we have

Since $\mathscr{C}$ is hereditary, the modules $E_0/E_1$ and $E_1$ are of type $\mathscr{C}$, and we have $\varphi (E) =\varphi (E_0/E_1) +\varphi (E_1)$. Since the sequence $(E_{i+1})_{0\leqslant i\leqslant n-1}$ is a composition series of $E_1$, Proposition 3 follows by induction on $n$.