Proposition 4

Let $(E_\alpha, f_{\beta \alpha})$ be a direct system of $A_\alpha$*-modules, $E = \lim \to E_\alpha$ its direct limit and $\phi_\alpha : A_\alpha \to A$ and $f_\alpha : E_\alpha \to E$ the canonical mappings for all $\alpha \in I$. If, for all $\alpha \in I$, $S_\alpha$ is a generating system of $E_\alpha$, then $S = \bigcup_{\alpha \in I} f_\alpha(S_\alpha)$ is a generating system of $E$.

Every $x \in E$ is of the form $f_\alpha(x_\alpha)$ for some $\alpha \in I$ and some $x_\alpha \in E_\alpha$ and by hypothesis $x = \sum_i \lambda^{(i)}_\alpha y^{(i)}_\alpha$, where $\lambda^{(i)}_\alpha \in A_\alpha$ and $y^{(i)}_\alpha \in S_\alpha$; writing $\lambda^{(i)} = \phi_\alpha(\lambda^{(i)}_\alpha)$, $y^{(i)} = f_\alpha(y^{(i)}_\alpha)$, we obtain $x = \sum_i \lambda^{(i)} y^{(i)}$.