of lines through $U$ on the plane $[H_2, Y]_k$ contained in $S_2$. Consequently, we have a bijection $f$ that maps planes through $U$ contained in $S_1$ to planes through $U$ contained in $S_2$.

Let $\Pi_i$ for $i=1,2,3$ be pairwise distinct planes through $U$ contained in $S_1$. If $\Pi_1\cap\Pi_2\cap\Pi_3$ is a line, then $f(\Pi_1)\cap f(\Pi_2)\cap f(\Pi_3)$ is a line.

Complements in projective spaces {#sec:proj}
================================

Let $\fixproj=\struct{S, \lines}$ be a projective space of dimension at least $3$ and let $\W\subset S$ such that is satisfied. We will investigate the complement of $\W$ in $\fixproj$, that is the structure $\fixout := \fixoutf(\fixproj,\W)$.

Note that in the case where $\W$ is a hyperplane, the complement $\fixout$ is an affine space, while if $W$ is not necessarily a hyperplane but a proper subspace of $\fixproj$, then $\fixout$ is a slit space (cf. [@KM67], [@KP70], [@slitgras]).

The goal of the first part of our paper is as follows.

\[thm:main-proj\] If $\fixproj$ is a projective space of dimension at least $3$ and $\W$ is its point subset satisfying , then both $\fixproj$ and $\W$ can be recovered in the complement $\fixoutf(\fixproj,\W)$.

More in vein of Klein’s Erlangen Program this theorem can be rephrased to the language 