Another equivalent of the Lotschnittaxiom

We prove that Bachmann’s Lotschnittaxiom, which states that perpendiculars raised on the two legs of a right angle always meet, is equivalent, with respect to Hilbert’s plane absolute geometry, to the statement \(\mathbf{P}\): “For any point P, line l, with P not incident with l, and any line g, there exists a point G on g for which the distance to P exceeds the distance to l”.