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\label{thm:betweenness-axioms}
The betweenness relation $B(x,y,z)$
(Definition~\textup{\ref{def:betweenness}}, II.D19),
derived from the ultrametric distance on~$\tau^3$,
satisfies the three Tarski betweenness axioms:
\begin{enumerate}
    \item[\textup{(T1)}] \textbf{Identity.}
          $B(x,y,x) \implies x = y$
          \textup{(Proposition~\ref{prop:ch18-t1})}.
    \item[\textup{(T2)}] \textbf{Transitivity.}
          $B(x,y,z) \wedge B(y,z,w) \implies B(x,y,w)$
          \textup{(Proposition~\ref{prop:ch18-t2})}.
    \item[\textup{(T3)}] \textbf{Connectivity.}
          For any $x, y, z \in \tau^3$,
          at least one of $B(x,y,z)$, $B(y,x,z)$,
          $B(x,z,y)$ holds
          \textup{(Proposition~\ref{prop:ch18-t3})}.
\end{enumerate}

Proved individually in
Sections~\ref{sec:ch18-tarski-t1}--\ref{sec:ch18-tarski-t3}.
All proofs use only the ultrametric distance
(II.D13),
the ultrametric triangle inequality (II.T05),
and the identity of indiscernibles.
No geometric axioms were invoked.