there exist sub-Weyl chambers of both which are contained in a common apartment.

2.  For any apartment $A$ and all $x\in A$ there exists a *retraction* $r_{A,x}:X\to A$ such that $r_{A,x}$ does not increase distances and $r^{-1}_{A,x}(x)=\{x\}$.

3.  Let $f, g$ and $h$ be charts such that the associated apartments pairwise intersect in half-apartments. Then $f({\mathbb{A}})\cap g({\mathbb{A}})\cap h({\mathbb{A}})\neq \emptyset$.

By $(A5)$ the distance function $d$ on $X$ is well defined and satisfies the triangle inequality.

The main goal of the present paper is to prove equivalence of certain sets of axioms. Let us therefore collect all properties which are necessary to state the main result.

-   Given two apartments $A$ and $B$ intersecting in a half-apartment $M$ with boundary wall $H$, then $(A\bigoplus B)\cup H$ is also an apartment, where $\bigoplus$ denotes the symmetric difference.

We say that two Weyl simplices $S$ and $T$ *share the same germ* if both are based at the same vertex and if $S\cap T$ is a neighborhood of $x$ in $S$ and in $T$. It is easy to see that this is an equivalence relation on the set of Weyl simplices based at a given vertex. The equivalence class of $S$, based at $x$, 