I am reading the book "Geometry: Euclid and Beyond - Robin Hartshorne". Here's the first half of Proposition 15.3 from the book.
If is a field, and if there is a notion of betweenness in the Cartesian plane satisfying Hilbert's axioms (B1)-(B4), then must be an ordered field.
The proof in the book reads as follows:
Suppose that is a field and there is a notion of betweenness in the plane satisfying (B1)-(B4). We define the subset to consist of all such that the point of the x-axis is on the same side of as .
"Now one can easily show that ."
... Which I am not able to show and so I need your help.
I was able to prove, using Pasch's axiom of betweenness, a.k.a (B4) in the book, that .But now I am stuck. Can someone help.