Proposition 8

In a vector space E over a field K, every vector subspace F is the intersection of the hyperplanes which contain it.

It suffices to show that for all $x \notin F$ there exists a hyperplane H containing F and not containing x. By hypothesis $F \cap Kx = \{0\}$ and hence the sum M of F and $Kx$ is direct. Let N be supplementary to M in E; E is then the direct sum of $H = F + N$ and $Kx$ and H is therefore a hyperplane with the desired property.