of collineations.

Under assumptions of \[thm:main-proj\] for every collineation $f$ of the complement $\fixoutf(\fixproj,\W)$ there is a collineation $F$ of $\fixproj$, leaving $\W$ invariant, such that $f = F|\pointsout$.

Parallelism
-----------

In affine structures parallelism is a key notion for projective completion. We are going to use it here in a similar way.

Usually, two lines are said to be parallel if they meet on the horizon. Following this idea we define *parallelism* $\parallel$ so that for lines $k_1, k_2\in\lines$ $$\label{eq:parallel}
  k_1\parallel k_2 :\iff \emptyset\neq k_1\cap k_2\subseteq\W.$$ In case $\fixout$ is a slit space our requirement that lines are of size at least 3 makes good sense as it is known that in an affine space with lines of size 2 parallelism cannot be defined in terms of incidence.

Definition involves the incidence relation of the ambient projective space $\fixproj$. To give an internal definition of our parallelism, expressible purely in terms of the complement $\fixout$, we begin by proving a variant of Veblen (Pasch) axiom for $\fixout$.

\[prop:veblen\] If proper lines $k_1, k_2, k_3$ form a triangle in $\fixproj$ with at least one improper vertex, then there is a proper line that intersects $k_1, k_2, k_3$ in pairwise distinct 