A proof of Pasch's axiom in the absolute theory of oriented parallelity (Q1207028)

Given a set \({\mathbf S}\) (points) and a relation \(\upharpoonleft \upharpoonright\) on \({\mathbf S}\times{\mathbf S}\) (oriented parallelity) such that a system of ten axioms holds true. Examples are obtained from affine ordered spaces where \((a,b)\upharpoonleft \upharpoonright(a,c)\) means that \(a,b,c\) are collinear and \(b\) lies between \(a\) and \(c\), or \(c\) lies between \(a\) and \(b\); generally, \((a,b)\upharpoonleft \upharpoonright(c,d)\) if \(c\varphi=a\) and \((a,b)\upharpoonleft \upharpoonright(a,d\varphi)\) for an appropriate composition of parallel projections \(\varphi\). The author discusses the Pasch-axiom within this frame.