Axiom:Outer Additivity of Equidistance

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Axiom

Let $\equiv$ be the relation of equidistance.

Let $\mathsf{B}$ be the relation of betweenness.


This axiom asserts:

$\forall x,y,z,x',y',z' : \left({\mathsf{B}xyz \land \mathsf{B}x'y'z' \land xy \equiv x'y' \land yz \equiv y'z'}\right) \implies xz \equiv x'z'$

where $x, y, z, x', y', z'$ are points.


Intuition

AdditivityOfEquidistance.png

Let $xy$ and $x'y'$ be line segments of the same length

Further let $yz$ and $y'z'$ be line segments of the same length.

If you connect $xy$ to $yz$, and $x'y'$ to $y'z'$, segments $xyz$ and $x'y'z'$ will be the same length.


Also see


Sources