[@Bennett Prop. 2.7]. Proposition [@Bennett Proposition 3.4] implies that $S\cap T$ contains a sub-Weyl chamber parallel to both. This implies (2).

Using the first item we can conclude that the sub-Weyl chambers $S$ and $T$ of $U$ in the last item are both translates of $U$. Hence $S$ is a translate of $T$ and they are, by 2.23.1 in [@Diss], at bounded distance of one another. By the second assertion of the same proposition, their intersection therefore contains a sub-Weyl chamber of both.

Let $F$ and $G$ be Weyl simplices in an affine building $X$. Let $S$ and $T$ be Weyl chambers such that $F$ is a face of $S$ and $G$ one of $T$. By (A4) there exists an apartment $A$ containing sub-Weyl chambers $S'\subset S$ and $T'\subset T$. In an apartment containing $S$ the sub-Weyl chamber $S'$ is a translate of $S$ and thus there exists a face $F'$ of $S'$ which is a translate of $F$ in this apartment. We say that $F'$ *corresponds to* $F$. In the same manner there is a face $G'$ of $T'$ corresponding to $G$.

\[Def\_parallelSimplex\] Two Weyl simplices $F$ and $G$ are *parallel* if the corresponding Weyl simplices $F'$ and $G'$ 