\[cor:planes\] we are allowed to use the term *proper plane* in the language of $\fixout$. Taking into account we can express what does it mean that a proper line is contained in such a plane. Moreover, by \[lem:affine-lines\] there is a proper line through an improper point on every proper plane. Therefore, for an improper point $a$ and a proper plane $\Pi$ we can make the following definition $$a\in\Pi :\iff (\exists k\in\linesout)\bigl[\, a\in k \Land k\subseteq\Pi \,\bigr],$$ where the formula on the right hand side is a sentence in the language of $\fixout$.

Now, for improper points $a, b$ we define *improper adjacency* which means that $a, b$ lie on an improper line, formally: $$\label{eq:impadjac}
  a\impadjac b :\iff (\nexists\; k\in\linesout)\bigl[\,
    k\in\stardir(a) \Land k\in\stardir(b)\,\bigr].$$ Let $\pplanes$ be the class of proper planes in $\fixproj$. Consider three points $a, b, c$ of $\fixproj$. In view of \[impintersec\], with the formula $$\begin{gathered}
\label{collinearity}
  \operatorname{L^\infty}(a, b, c) :\iff \impadjac(a, b, c) \Land (\exists \Pi_1, \Pi_2\in\pplanes)
    \bigl[\; \Pi_1\neq\Pi_2 \Land
      a, b, c\in \Pi_1, \Pi_2 \,\bigr]\end{gathered}$$ we define *improper collinearity* relation. An improper line in $\fixproj$ is an equivalence class of that relation $\operatorname{L^\infty}$ with two distinct points fixed. That is, if $a, b$ are two 