Hilbert's axioms. Betweenness and ordered fields.

[caffeinemachine]

I am reading the book "Geometry: Euclid and Beyond - Robin Hartshorne". Here's the first half of Proposition 15.3 from the book.

If $F$ is a field, and if there is a notion of betweenness in the Cartesian plane $\Pi_F$ satisfying Hilbert's axioms (B1)-(B4), then $F$ must be an ordered field.
The proof in the book reads as follows:

Suppose that $F$ is a field and there is a notion of betweenness in the plane $\Pi_F$ satisfying (B1)-(B4). We define the subset $P \subset F$ to consist of all $a \in F$ such that the point $(a,0)$ of the x-axis is on the same side of $0$ as $1$.
"Now one can easily show that $a,b \in P \Rightarrow a+b \in P$."

... 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 $(0,0)*(1,0)*(a,0) \Rightarrow (0,0)*(0,1)*(0,a)$ .But now I am stuck. Can someone help.

 

[Aaron Bex]

To show that $a,b \in P \Rightarrow a+b \in P$, consider the points $(0,0), (1,0), (a,0)$ and $(b,0)$. By assumption, these four points are all on the same side of the y-axis. By Pasch's axiom, it follows that $(0,0)*(1,0)*(a+b,0)$ is also on the same side of the y-axis. Since $(1,0)$ and $(a+b,0)$ are on the same side of $0$, we have that $a+b \in P$, so $a, b \in P \Rightarrow a+b \in P$ as desired.