$3$ dimensional, take a point $p$ not on $\Pi$. Regardless of whether $p$ is proper or not, there is a proper point $a_3$ on the line $\LineOn(p, a_1)$ with $a_3\neq a_1$ (possibly $a_3 = p$) by \[fact:basic\]. Lines $k_1$, $k_2$, $k_3 := \LineOn(a, a_3)$ are the required lines.

A straightforward outcome of \[lem:bundle\] is as follows.

\[lem:cosik\]

\[lem:cosik:a\] There is a proper line through every point of $\fixproj$.

\[lem:cosik:b\] There is a proper plane through every point of $\fixproj$.

\[lem:affine-lines\] There are at least $\fixind+2$ affine lines through every improper point on a proper plane.

Consider an improper point $a$ and a plane $\Pi$ of $\fixout$ through $a$. By there are at least $2\fixind+2$ lines through $a$ on $\Pi$. Assume that there are $\lambda$ improper lines $k_1,\dots,k_\lambda$ contained in $\Pi$ through $a$. Take a proper point $b$ of $\Pi$ and a point $a_i\in k_i$ for some $i=1,\dots,\lambda$ distinct from $a$. The line $l:=\LineOn(a_i, b)$ is proper by \[fact:basic\], and intersects every of $k_1,\dots,k_\lambda$. There are up to $\fixind$ improper points on $l$, thus $\lambda\leq \fixind$. The remaining lines through $a$ are affine.

\[impintersec\] For every improper line $k$ there are at least $\fixind+2$ proper planes containing $k$.

Consider the bundle 