that determines $\pclique$. Let $a_1\in k_1$, $a_2\in k_2$ be proper points and let $l_1 := \LineOn(a_1, a_2)$. Now, in view of \[fact:basic\], take a proper point $b_2\in k_2$ with $b_2\neq a_2$. Then, take a proper point $b_3\in k_3$ and $l_2:=\LineOn(b_2, b_3)$. Lines $l_1, l_2$ are proper and skew, a contradiction.

Let $\pclique$ be a top direction and let $l_1, l_2$ be distinct proper lines sharing two distinct points with lines of $\pclique$ each. Every two lines intersect each other on a projective plane. Hence, the lines $l_1, l_2$ share a proper or an improper point, and accordingly, $l_1$ intersects $l_2$ or $l_1\parallel l_2$.

Consequences of \[prop:cliques\] and \[prop:stars\] together with \[prop:parallel\] are quite essential for our future construction and will be put down as follows.

\[cor:star-directions\] Star directions are definable in terms of $\fixout$.

Moreover, every star direction $\pclique$ can be identified with the improper point shared by all the lines of $\pclique$. We will write $\stardir(a)$ for the star direction determined by an improper point $a$.

On a side note, in $(\ast)$ we can claim, equivalently, that every two distinct proper lines, each of which shares two distinct points with lines of a top direction, are coplanar as planes 