Proposition 1

Let a be a point of E. There exists a connected open neighbourhood of a that does not meet any hyperplane H that belongs to $\mathscr{H}$ and does not pass through a. Moreover, there exist only finitely many hyperplanes that belong to $\mathscr{H}$ and pass through a.

The set $\mathscr{N}$ of hyperplanes H such that $H \in \mathscr{H}$ and $a \notin H$ is locally finite since it is contained in $\mathscr{H}$. Hence, the set U of points of E that do not belong to any of the hyperplanes of the set $\mathscr{N}$ is open. Since $a \in U$, there is a connected open neighbourhood of a contained in U. The remainder of the proposition is clear.

Given two points x and y of E, denote by R\{x, y\} the relation

" For any hyperplane $H \in \mathscr{H}$, either $x \in H$ and $y \in H$ or $x$ and $y$ are strictly on the same side of H."

Clearly, R is an equivalence relation on E.