My question is almost the same of Axioms of order in geometry and ordered fields. I'm reading the book of Robin Hartshorne,Geometry: Euclid and beyond and i'm having hard time to write down a proof for Proposition 15.3 in particular i don't get the first statement of the proof (which is left almost entirely to the reader) which says: "Since addition corresponds to lay out segment consecutively on the x axis, one can show easily that $a,b\in P\implies a+b\in P$" I tried to use Pash (B4) and the line separation by i didn't go any further.
