A plan of the subset is $convex$ if the segment connecting any two of its points is fully contained therein. The simplest examples of $convex$ $sets$ are the plan itself and any half-plane. Show that the intersection of two semi-planes is a convex.
It has an axiom that states that a straight $m$ determines two semi-separate plans whose intersection is the straight $m$. The problem is how to argue.
If the two points to be connected belong to the intersection of two half-planes, they both belong to the two half-planes. By convexity of the half-planes, the line segment that joins them is wholly contained in both half-planes, hence it is contained in their intersection.
$$A,B\in\Pi_0\cap\Pi_1\implies A,B\in\Pi_0,\Pi_1\implies AB\subset\Pi_0,\Pi_1\implies AB\subset\Pi_0\cap\Pi1.$$