of $X$.

By definition of adjacency the set ${\partial_{\mathcal{A}}}X$ is a chamber complex. The sub-complex consisting of all equivalence classes of Weyl simplices contained in a fixed apartment is isomorphic to a Coxeter complex of type ${\Phi}$ if $X$ is modeled on the root system ${\Phi}$. These sub-complexes are the apartments of ${\partial_{\mathcal{A}}}X$. Axiom (A4) implies that two chambers $\partial S$ and $\partial T$ are contained in a common apartment. Following [@Brown p.76/77] it remains to prove that two apartment of ${\partial_{\mathcal{A}}}X$ which contain a common chamber are isomorphic via an isomorphism fixing their intersection, that is (B2”).

Let $A, A'$ be apartments and $c$ a chamber in $\partial A\cap \partial A'$. Then there exist representatives $S\subset A$ and $S'$ in $A'$ of the equivalence class $c$. Hence $S\cap S'$ contains a sub-Weyl chamber $S''$. Therefore we can find charts $f,f'$ of $A,A'$ such that $$f'\circ f^{-1}\vert_{A\cap A'} = id\vert_{A\cap A'}.$$ The induced map $\partial(f'\circ f^{-1})$ at infinity is an isomorphism fixing $\partial A \cap \partial A'$.

Local structure {#Sec_localStructure}
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Let in the following $(X,{\mathcal{A}})$ be a pair modeled on ${\mathbb{A}}={\mathbb{A}}(\Lambda, {\Phi}, T)$ and satisfying all axioms but (A5). Recall from the previous section that this is enough to 