the sense of Definition \[Def\_parallelSimplices\].

2.  They contain sub-Weyl chambers $S'\subset S$ and $T'\subset T$ such that $S'$ and $T'$ are contained in a common apartment and are translates of one another in this apartment.

3.  $S$ and $T$ are at bounded Hausdorff distance, i.e. are parallel in the sense of [@Parreau].

\[Lem\_subWeyl\]

1.  If $C$ is a sub-Weyl chamber of $D$, then $C$ is a translate of $D$.

2.  If $S$ is a translate of the Weyl chamber $T$ in an apartment $A$, then $S\cap T$ contains a common Weyl chamber of both.

3.  Given sub-Weyl chambers $S$ and $T$ of the same Weyl chamber $U$, then $S\cap T$ contains a Weyl chamber.

Since $C$ is a sub-Weyl chamber of $D$ these two are at bounded distance. By Proposition 4.23.2 in [@Diss] there exists then a Weyl chamber $U\subset C\cap D = C$ having bounded distance to both. By 1. of the same Proposition this is equivalent to the fact that one is a translate of the other. Hence (1).

To prove the second assertion observe that $S=t+T$ for some translation $T$ in the affine Weyl group. Therefore $S$ ant $T$ are parallel in the sense of Bennett by 