that $X$ is covered by its charts. That is $X=\bigcup_{f\in{\mathcal{A}}} f({\mathbb{A}})$.

\[MainThm\] For a space $(X,{\mathcal{A}})$ modeled on ${\mathbb{A}}={\mathbb{A}}({\Phi},\Lambda)$ which satisfies axioms (A1)-(A3), the following are equivalent:

1.  \[i\] $(X,{\mathcal{A}})$ is a generalized affine building, that is axioms (A4), (A5) and (A6) are satisfied.

2.  \[ii\] Axioms (A4), (A5) and (EC) hold.

3.  \[iii\] Axioms (A4) and (A6) are satisfied.

4.  \[iv\] Properties (GG) and (CO) hold.

5.  \[v\] The pair $(X,{\mathcal{A}})$ has properties (A3’) and (CO).

6.  \[vi\] Axioms (A3”), (A4) and properties (FC”) and (EC) are satisfied.

Obviously if one of the properties (A3’) and (A3”) hold axiom (A3) is superfluous.

We will prove the following implications: $$\xymatrix{
    & (2) \ar@{<=>}[dl]_{\text{\cite{Bennett2}}}    & (6) \ar@{=>}[l]\ar@{<=}[d] & \\
(1) \ar@{=>}[r] & (3)   & (4) \ar@{<=}[l]\ar@{<=>}[r] &  (5) 
}$$

The fact that (A6) and (EC) are equivalent assuming (A1) to (A5) is due to Bennett [@Bennett2]. We obtain (GG) and (CO) as discussed in Section \[Sec\_localStructure\] (compare Corollaries \[Cor\_GG\] and \[Cor\_CO\]). Hence item (\[iii\]) implies (\[iv\]).

Section \[Sec\_A3’\] contains the proof of the fact that (\[iv\]) implies property (A3’) and hence (A3”). Later, in Section \[Sec\_A4\] axiom (A4) is shown assuming (\[iv\]). The exchange condition (EC) holds as outlined Section \[Sec\_A6’\]. 