a germ of $S$ at $x$ and a germ of $T$ at $y$.

By axiom $(A3)$ there exists an apartment $A$ containing $x$ and $y$. We choose an $x$-based Weyl chamber $S_{xy}$ in $A$ that contains $y$ and denote by $S_{yx}$ the Weyl chamber based at $y$ such that $\partial S_{xy}$ and $\partial S_{yx}$ are v in $\partial A$. Then $x$ is contained in $S_{yx}$. If $\Delta_yT$ is not contained in $A$ apply Proposition \[Prop\_tec16\] to obtain an apartment $A'$ containing a germ of $T$ at $y$ and containing $\partial S_{yx}$ at infinity. But then $x$ is also contained in $A'$.

Let us denote by $S'_{xy}$ the unique Weyl chamber contained in $A'$ having the same germ as $S_{xy}$ at $x$. Without loss of generality we may assume that the germ $\Delta_yT$ is contained in $S'_{xy}$. Otherwise $y$ is contained in a face of $S'_{xy}$ and we can replace $S'_{xy}$ by an adjacent Weyl chamber in $A'$ satisfying this condition. A second application of Proposition \[Prop\_tec16\] to $\partial S'_{xy}$ and the germ of $S$ at $x$ yields an apartment $A''$ containing $\Delta_xS$ and $S'_{xy}$ and therefore $\Delta_yT$.

Propositions \[Prop\_A5\] to \[Prop\_liftGallery\] below are due to Linus Kramer.

\[Prop\_A5\]With $X$ as 