Proposition 8

Let $G$ be a topological group which has a Hausdorff completion $\hat{G}'$. Then every continuous homomorphism $u$ of $G$ into a complete Hausdorff group $H$ can be uniquely factorized into $u = v \circ \varphi$, where $v$ is a continuous homomorphism of $\hat{G}'$ into $H$ and $\varphi$ is the canonical mapping of $G$ into $\hat{G}'$ (the composition of the canonical injection of $G'$ into $\hat{G}'$ and the canonical homomorphism $\psi$ of $G$ onto $G/N = G'$).

Since the kernel of $u$ is closed and contains $e$, it contains $N$, and hence $u$ can be written as $u = w \circ \psi$, where $w$ is a continuous homomorphism of $G'$ into $H$; now apply Proposition 5 of no. 3 to $w$.