Corollary 4

Let E be a complete metrisable vector space over a non-discrete valued division ring. If M and N are two closed vector subspaces, that are (algebraic) complements in E, then E is the direct topological sum of M and N.

For $M \times N$ is a complete metrisable vector space and the mapping $(y, z) \mapsto y + z$ of $M \times N$ on E is continuous and bijective, therefore an isomorphism (cor. 1).