Grassmannians. In this paper we propose a solution involving parallelism, imitating the affine parallelism, valid under certain assumptions on the size of improper parts of lines. Our solution is just [*a*]{} solution: there are fragments of projective spaces (e.g. ruled quadrics) which do not satisfy our requirements and from which the surrounding spaces are recoverable, but with completely different methods.

Motivations and references
--------------------------

In this paper we deal with a point-line space $\A$ where a subset $\W$ of points, a horizon, and those lines that are entirely contained in $\W$ are removed. We call such a reduct of $\A$ the complement of $\W$ in $\A$ and denote it by $\fixoutf(\A,\W)$. As for which lines to remove from $\A$ our approach is not unique and there are different complements considered in the literature, e.g. in [@sulima], where all the lines that meet $\W$ are deleted.

The most classic and vivid example of a complement $\fixoutf(\A,\W)$, aforementioned at the beginning, is an affine space i.e. the complement of a hyperplane $\W$ in a projective space $\A$. There are more examples where removing a hyperplane is also successful and where in result we get some affine geometries. In a polar space deleting 