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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?

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  • $\begingroup$ 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. $\endgroup$ – Will Jagy Feb 21 at 3:25

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