any details. We simply assume throughout the following that there exists some ${W}$-invariant $\Lambda$-metric on ${\mathbb{A}}$.

The question arising is the following: Let us assume that $X$ is an affine building with metric $d$, which is induced by a metric $d_{\mathbb{A}}$ on the model space. Let $d'_{\mathbb{A}}$ be a metric on the model space, which differs from $d_{\mathbb{A}}$. Hence $d'_{\mathbb{A}}$ induces a second distance function $d'$ on $X$. Does $d'$ satisfy the triangle inequality? And is $(X,d')$ an affine building? To be able to answer these questions one has to understand whether the retractions appearing in (A5) do exist and are distance diminishing. The answer to these questions is “yes”, and using Theorem \[MainThm\] we do not need to prove (A5) directly.

Let $(X,{\mathcal{A}})$ be an affine building. Then every metric on the model space extends to a metric on $X$.

Since $(X,{\mathcal{A}})$ is a building axioms (A6) and (A1) to (A4) are satisfied. These axioms do not contain conditions on the metric and are, by \[MainThm\] equivalent to the ones listed in Definition \[Def\_LambdaBuilding\]. Hence every distance function on $X$ which is induced by a metric on the model space satisfies the triangle inequality.

Thus whether or not a 