Corollary 1

If the condition (AP) is satisfied and if in addition the (Hausdorff) groups $G/H_\alpha$ are complete, then the group $G$ has a Hausdorff completion which can be identified with $\tilde{G}$; and the mapping $i : G \to \tilde{G}$ is then identified with the canonical mapping ($§ 3$, no. 3, Proposition 5).

For $\tilde{G}$ is then complete (no. 2), and Proposition 2 shows that $i(G)$ is isomorphic with the Hausdorff group associated with $G$; since it is dense in $\tilde{G}$, the corollary follows ($§ 3$, no. 3, Proposition 5). In particular: