On a splitting of the parallel postulate (Q2668175)

In the realm of Hilbert's absolute geometry \(\mathcal A\), the Euclidean parallel postulate \textbf{P} is known to be equivalent to the conjunction of the following two axioms: \begin{itemize} \item [\textbf{ML}] Given three parallel lines, there is a line that intersects all three of them. \item [\textbf{S}] Given a line \(a\) and a point \(P\) on \(a\), as well as two intersecting lines \(m\) and \(n\), both parallel to \(a\), there exists a line \(g\) through \(P\) which intersects \(m\) but not \(n\). \end{itemize} It is shown that the above equivalence breaks down once the order axioms or the congruence axioms of \(\mathcal A\) are weakened.