Finally, as shown in Section \[Sec\_FC1\], condition (FC”) follows from (A1) to (A3) and (CO). This completes the proof of the fact that (\[iv\]) implies (\[vi\]).

Axiom (A5) is verified in Section \[Sec\_A5\] using (A1), (A2), (A3”) and (FC”). Therefore item (\[vi\]) implies (\[ii\]). Compare Section \[Sec\_A3’\] and \[Sec\_lp\] to obtain that the axioms listed in (\[iv\]) are equivalent to the ones in (\[v\]). See \[Sec\_A3’\] for the fact that (\[iv\]) implies (\[v\]). The converse, that (\[v\]) implies (\[iv\]), is proved in Section \[Sec\_lp\].

Further results
---------------

Let me start this section with a simple yet interesting consequence of Theorem \[MainThm\]. The class of generalized affine building is a generalization of ${\mathbb{R}}$-buildings, which themselves generalize the (geometric realizations of) simplicial affine buildings. The ${\mathbb{R}}$-buildings are the sub-class where $\Lambda={\mathbb{R}}$ and where the translational part $T$ of the affine Weyl group equals the co-root-lattice spanned by a crystallographic root system, or is the full translation group of an apartment in the non-crystallographic case.

For this we are using the metric approach to affine buildings, replacing ${\mathbb{R}}$-metric spaces by $\Lambda$-metric spaces in the following sense.

\[Def\_metric\] A *$\Lambda$-metric* on a space $X$, is a map $d:X\times X \mapsto \Lambda$ such that for 