are sub-Weyl chambers of $S'$. By Lemma \[Lem\_subWeyl\] their intersection thus contains a Weyl chamber $C''$.

By the arguments above $C'$ is a translate of $D'$ in every apartment which contains $S'$. The face $F'$ is parallel to $G'$ and the Weyl simplex $G'$ is parallel to $H'$. Therefore $F'$ is a translate of $F''\subset C'$ and $H'$ is a translate of $H''\subset D'$. This implies that $F''$ is a translate of $H''$. Hence $F'$ is parallel to $H'$ in the sense of Definition \[Def\_parallelSimplex\]

We say that $\partial F$ *is a face of* $\partial S$ if there exist representatives $F$ and $S$ such that $F$ is a face of $S$. This defines a simplicial structure on parallel classes of Weyl simplices. We define two parallel classes $\partial F$ and $\partial G$ of Weyl simplices to be *adjacent* if there exist representatives based at the same vertex and having a face in common.

\[Thm\_buildingAtInfinity\] Let $(X,{\mathcal{A}})$ be a pair modeled on ${\mathbb{A}}({\Phi}, \Lambda, T)$ satisfying axioms (A1)-(A4). Then the set $${\partial_{\mathcal{A}}}X {\mathrel{\mathop:}=}\{\partial F : F \text{ is a Weyl simplex in } X\}$$ is a spherical building of type ${\Phi}$ with apartments in one to one correspondence with the apartments 