line of\/ $\A$ has size at least\/ $2\indf(\A,\W)+2$},$$ which is obviously too strong for this specific purpose but will become essential later. Now, the very first observation.

\[fact:basic\]

\[fact:basic:proper\] Every line through a proper point is proper.

\[fact:basic:min\] There are at least $\fixind +2$ proper points on a proper line.

\[fact:basic:notall\] $\W \neq S$.

The consequence of \[fact:basic\] is that there are at least 3 proper points on every line of the complement $\fixoutf(\A,\W)$, thus it is a partial linear space.

Grassmann spaces
----------------

Let $V$ be a vector space of dimension $n$ with $3\le n<\infty$. The set of all subspaces of $V$ will be written as $\Sub(V)$ and the set of all $k$-dimensional subspaces (or $k$-subspaces in short) as $\Sub_k(V)$. The most basic concept for us here is a *$k$-pencil* that is the set of the form $$\penc(H, B) := \{ U\in\Sub_k(V)\colon H\subset U\subset B\},$$ where $H\in\Sub_{k-1}(V)$, $B\in\Sub_{k+1}(V)$, and $H\subset B$. The family of all such $k$-pencils will be denoted by $\peki_k(V)$. We consider a *Grassmann space* (also known as a *space of pencils*) $$\M = \PencSpace(V, k) = \struct{\Sub_k(V), \peki_k(V)},$$ a point-line space with $k$-subspaces of $V$ as points and $k$-pencils as lines (see [@polargras], [@slitgras] for a 