$A$ such that $\Delta_xS$ is contained in $A$ and such that $c$ is a chamber of $\partial A$.

The proof of the proposition above is precisely the same as the proof of Proposition 1.8 in [@Parreau]. Parreau’s proof uses the fact that ${\partial_{\mathcal{A}}}X$ is a spherical building and that axioms (A1) to (A3) as well as (A6) are satisfied. Recall that assuming (A1) to (A4) we where able to prove in Section \[Sec\_infinity\] that ${\partial_{\mathcal{A}}}X$ is a spherical building.

\[Cor\_GG\]Any pair $(X,{\mathcal{A}})$ satisfying all axioms but (A5) has property (GG).

Let $S$ and $T$ be Weyl chambers both based at a point $x$. By Proposition \[Prop\_tec16\] there exists an apartment $A$ of $X$ containing $S$ and a germ of $T$ at $x$.

Notice that, by the previous corollary, such a pair $(X,{\mathcal{A}})$ satisfies the assertion of Theorem \[Thm\_residue\], i.e. the germs at a fixed vertex form a spherical building. Hence the notion of opposite germs as defined in the introduction makes sense.

\[Prop\_A3’\]If $(X,{\mathcal{A}})$ is a pair satisfying all axioms but (A5) then property (A3’) holds.

We need to prove that if $S$ and $T$ are Weyl chambers based at $x$ and $y$, respectively, then there exists an apartment containing 