S$, then $U_1, U_2$ are not collinear in $\M$.

\[cor:starcaptop\] Let $S$ be a star and $T$ be a top in $\M$. If $U_1\in S$, $U_2\in T$, and $U_1\adjac U_2$, then either $S\cap T = \emptyset$, or $U_1\in S\cap T$, or $U_2\in S\cap T$.

We say that a subspace of $\M$ is a *plane* if it is (up to an isomorphism) a projective plane. It is known that every strong subspace of $\M$ carries the structure of a projective space. For a star $S = [H)_k$ a plane contained in $S$ is of the form $[H, Y]_k$, where $Y\in\Sub_{k+2}(V)$, while for a top $T = (B]_k$ a plane contained in $T$ is of the form $[Z, B]_k$, where $Z\in\Sub_{k-2}(V)$.

\[lem:top-star-top\] Let $U$ be a point and $T_i = (B_i]_k$, $i=1,2$ be distinct tops of $\M$ with $U\in T_1\cap T_2$.

If $\Pi = [Z, B_1]_k$ is a plane through $U$ contained in $T_1$, then $$\{ \starof(p)\cap T_2\colon p\in\peki_k(V),\ U\in p\subset \Pi\}$$ is the set of lines through $U$ on the plane $[Z, B_2]_k$ contained in $T_2$. Consequently, we have a bijection $f$ that maps planes through $U$ contained in $T_1$ to planes through $U$ contained in $T_2$.

Let $\Pi_i$ for $i=1,2,3$ 