Equidistance is Independent of Betweenness

Theorem
In Tarski's Geometry, equidistance, $\equiv$, cannot be defined in terms betweenness, $\mathsf{B}$.

More precisely:

Let $\mathcal{G}$ be a formal systematic treatment of geometry containing only:


 * The language and axioms of first-order logic, and the disciplines preceding it


 * The undefined terms of Tarski's Geometry (excluding equidistance)


 * Tarski's Axioms of Geometry

In $\mathcal{G}$, $\equiv$ is necessarily an undefined term with respect to $\mathsf{B}$.

Also see

 * Betweenness Not Independent of Equidistance, which states that there are models where one can define $\mathsf{B}$ in terms of $\equiv$.