the pairs $F$,$G$ and $G$, $H$. We may use in both cases the same Weyl chamber $S$ having $G$ as a face. Doing so we obtain sub-Weyl chambers $S'$ in $A$ and $S''$ in $B$ having $G'$, respectively $G''$ as a face. Replacing, if necessary, $S'$ and $S''$ by a common sub-Weyl chamber $S'''$ we may assume that $S'=S''$ and that $G'=G''$.

Hence we are in the following situation. The Weyl simplex $F'$ is a face of the Weyl chamber $T'$ which is contained in the same apartment $A$ as the Weyl chamber $S'$ which has $G'$ as a face. Furthermore $F' = t + G'$ in $A$ and $G'=s+H'$ in $B$. The Weyl simplex $H'$ is a face of $U'$, a Weyl chamber contained in $B$ which is an apartment containing $S'$. In particular $S'$ is contained in the intersection of $A$ and $B$.

The translate $C{\mathrel{\mathop:}=}t+S'$ of $S'$ is also a Weyl chamber in $A$ having $F'$ as a face and $D{\mathrel{\mathop:}=}-s+ S'$ is a Weyl chamber in $B$ with face $H'$. The intersection of $D$ and $S'$ contains a Weyl chamber $D'$ and the intersection $S' \cap C$ contains a Weyl chamber $C'$ Both, $C'$ and $D'$, 