A short proof can be constructed using cross-ratio preservation. Projecting tetrad ABCE from D onto line AB, we obtain tetrad ABPX, and projecting tetrad ABCE from F onto line BC, we obtain tetrad QBCY. This therefore means that R(AB; PX) = R(QB; CY), where one of the points in the two tetrads overlap, hence meaning that other lines connecting the other three pairs must coincide to preserve cross ratio.