Morse theory is used to rigorously obtain counting formulas and lower bounds for the total number of images of a background point source, not on a caustic, undergoing lensing by a single-plane microlens system having compact bodies plus either subcritical or supercritical continuously distributed matter. An image-counting formula is also found for the case when external shear is added. In addition, it is proven that a microlens system consisting of k lens planes will generate N = 2M-+ Πi=1k(1 - gi) images of a background point source not on a caustic, where M-is the total number of critical points of odd index of the time-delay map and giis the number of stars on the ith lens plane. Morse theoretic tools also yield that the smallest value N can have is Πi=1k(1 + gi). © 1992 American Institute of Physics.