Proof for the (second) Pasch's theorem / Hilbert's geometry axiom II, 4

In Hilbert's Foundations of Geometry, the 4th axiom of order which stays that

Any four points $$A, B, C, D$$ of a straight line can always be so arranged that $$B$$ shall lie between $$A$$ and $$C$$ and also between $$A$$ and $$D$$, and, furthermore, that $$C$$ shall lie between $$A$$ and $$D$$ and also between $$B$$ and $$D$$.

I found in many sites that this proposition is often called as "Pasch's theorem" (although there's another theorem by such name) and that it is a "discarded axiom" since it can be proved using the other four axioms of order, however, I can't see how... Can you give me a formal proof of this statement?

• suggest one of the more recent treatments. The two I know are by Hartshorne, Geometry: Euclid and Beyond. or by Greenberg, Euclidean and Non-Euclidean Geometry. Both are intended for students, and fill in details. – Will Jagy Feb 21 at 3:25