Any answer depends on how strong yourRuler Postulateis and upon the sequence of betweenness theorems.

And we do not know the answer to that.

In general here is a outline of a proof.

Supposeis a coordinate system on line AB.f

Sois the coordinate of pointf(A).A

Ifthen definef(B)-f(A)>0otherwise defineg(x)=f(x)-f(A).g(x)=f(A)-f(x)

By use of the betweenness theorems the result should follow.