pair $(X,{\mathcal{A}})$ modeled on ${\mathbb{A}}$ is an affine building does not depend on the metric imposed on ${\mathbb{A}}$. This consequence of our main result makes use of the fact that (A5) can be omitted in Definition \[Def\_LambdaBuilding\].

The basic idea is to find a purely combinatorial definition of parallelism of Weyl simplices which allows us to prove existence of a spherical building at infinity without using the metric structure of the affine building. Finally this enables us to eliminate axiom (A5) in the definition of an affine building.

Bennett [@Bennett] did prove already that two Weyl chambers, which are contained in the same apartment, are at bounded distance if and only if they are translates of one another. Using this one observes that “being at bounded distance in the building” is the same as being, in a certain sense, “translates of one another”.

This new approach makes the definition of parallelism a bit lengthy but avoids using the metric. The details are carried out in Section \[Sec\_infinity\], where we prove the following theorem.

Let $(X,{\mathcal{A}})$ be a pair satisfying axioms (A1)-(A4). Then $${\partial_{\mathcal{A}}}X {\mathrel{\mathop:}=}\{\partial F : F \text{ is a Weyl simplex in } X\}$$ is a spherical building of 