are definable in $\fixout$ by \[cor:planes\]. As for top directions they need not to exist on every plane. If for example from a plane $\Pi$ we remove a single point, two points, a single line, a line and a point, or two lines then there is no top direction on $\Pi$.

The main reasoning {#sec:main}
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We are going to reconstruct the horizon $\W$ in $\fixout$.

In view of \[cor:star-directions\], as it has been already stated, to every improper point $a$ of $\fixproj$ we can uniquely associate the star direction $\stardir(a)$. Hence we formally have $$S = \pointsout \cup S^\infty,$$ where $S^\infty = \bset{ \pclique\colon \pclique \text{ is a maximal $\parallel$-clique that fails $(\ast)$} }$. The incidence relation of $\fixout$ can be extended now to the set of improper points as follows. If $a$ is an improper point and $k\in\linesout$, then $$\label{eq:incinfty}
  a\in k :\iff k\in\stardir(a).$$ Let us summarize what we have so far.

The structure $\struct{S, \linesout}$ is definable in $\fixout$.

To get the entire projective space $\fixproj$ we need to recover improper lines and tell what does it mean that a point, proper or improper, lies on such an improper line, everything in terms of $\fixout$.

Thanks to 