$\cal F$ of all the planes in $\fixproj$ containing the line $k$ and let $a\in k$. By \[lem:bundle\] we have three proper lines $k_1, k_2, k_3$ through $a$ that span three pairwise distinct proper planes. Note that the line $k$, as an improper one, can be contained in at most one of those planes. Take $\Pi$ to be one of the other two, thus $\Pi$ is a proper plane with $a\in\Pi$ and $k\not\subset\Pi$.

Every plane from the family $\cal F$ intersects $\Pi$ in a line of $\fixproj$. All these lines form a pencil through $a$ on $\Pi$. There are at least $\fixind+2$ proper lines in that pencil by \[lem:affine-lines\]. Each of these proper lines together with $k$ span a proper plane from $\cal F$ which completes the proof.

Cliques of parallelism {#sec:cliques}
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Let $\pclique$ be a maximal clique of $\parallel$ in $\fixout$. There are two possibilities:

there is an improper point $a$ such that $a\in k$ for all $k\in\pclique$, or

there is a plane $\Pi$ in $\fixout$ such that $k\subset\Pi$ for all $k\in\pclique$.

A maximal $\parallel$-clique in the first case will be called a *star direction*, and a *top direction* in the second.

\[prop:cliques\] There are two disjoint classes 