Hence, by \[prop:veblen\] there is a required line $l\in\linesout$ and points $c_1, c_2, c_3\in\pointsout$.

Note that the lines $k_1,k_2$ intersect in an improper point, i.e. $\emptyset\neq k_1\cap k_2\in\W$. So, $k_1\parallel k_2$ by .

Thanks to \[prop:parallel\] we can distinguish affine lines using the internal language of $\fixout$.

\[fact:affinelines\] A line $k\in\linesout$ is affine iff there exists a line $l\in\linesout$ such that $l\neq k\parallel l$.

Planes
------

By analogy to proper line we introduce the term *proper plane*. Namely we say that a plane $\Pi$ of $\fixproj$ is proper (or is a *plane of $\fixout$*) if $\Pi\not\subset\W$. In the following lemma we try to justify this terminology.

\[lem:triangle-plane\] There is a proper triangle on every plane of $\fixout$.

Let $\Pi$ be a plane of $\fixout$. There is a proper point $a$ on $\Pi$, since $\Pi\not\subset\W$. Every line through $a$ is proper by \[fact:basic\]. There are at least $2\fixind +2$ lines through $a$ on $\Pi$ by . So, take two distinct lines $l_1,l_2$ with $a\in l_1,l_2\subset\Pi$, and consider two points $b,c\neq a$ such that $a\in l_1$, $b\in l_2$. Then $a,b,c$ and $l_1, l_2, \LineOn(b,c)$ form the required triangle.

By \[lem:triangle-plane\] we have a proper triangle on a plane $\Pi$ of $\fixout$. Now 