Now I am reading Groupes réductifs sur un corps local : I. Données radicielles valuées written by Bruhat and Tits in 1972. Let $\Phi$ be a root system, $G$ a group with root data $(T,(U_{a},M_{a})_{a\in \Phi})$ of type $\Phi$, and $\varphi=(\varphi_{a}\colon U_{a}\longrightarrow \mathbb{R}\cup \{\infty\})$ a valuation of these root data. We denote by $A$ the equivalence class of $\varphi$. Then the bijective $$ V\longrightarrow A;v\longmapsto \varphi+v $$ provides $A$ with a structure of affine apartment. Bruhat and Tits defined an equivalence relation $\sim$ on $G\times A$ to be $$ (g,x)\sim (h,y) \iff \exists n\in N \text{ such that } y=\nu(n)(x) \text{ and } g^{-1}hn\in P_{x}, $$ where all notations are the same in Bruhat–Tits paper. They called the quotient $X=G\times A/{\sim}$ the building of $G$. My questions are that “Do the following hold?

- For any $x,y\in X$, there exist $g\in G$ such that $x,y\in gA$.
- The action of $G$ on $X$ is strongly transitive.”

Since it is so called **building**, perhaps these are valid, but I would like to know the proof.