we described above are translates of one another in an apartment containing both.

This definition is clearly independent of the choice of $A$ since every sub-Weyl chamber of a Weyl chamber $S$ is a translate of $S$ in every apartment containing $S$ (see Lemma \[Lem\_subWeyl\]). By Proposition \[Prop\_parallel\] it is equivalent to the definition used in [@Bennett] or [@Parreau].

\[Prop\_parallelEquiv\] Parallelism is an equivalence relation on Weyl simplices.

Reflexivity and symmetry is clear. Hence it remains to prove transitivity. Let $F$, $G$ and $H$ be Weyl simplices such that $G$ is parallel to $F$ and parallel to $H$. It is to prove that $F$ is parallel to $H$ as well. Since $F$ and $G$ are parallel there exist translates $F'$ and $G'$, respectively, which are contained in a common apartment $A$ in which they are translates of one another. Thus there exists a translation $t\in {W}$ such that $F'= t+G'$. For the same reason there exists an apartment $B$ containing translates $G''$ and $H''$ of $G$, respectively $H$. Furthermore there is a translation $s$ such that $G''= s+ H''$.

For the following reason we may assume that $G'=G''$: To find $A$ and $B$ we need to apply Definition \[Def\_parallelSimplex\] to 