Left panel: the redshift distribution of galaxies in the unlensed catalog, UGC0 (red), which contains approximately 64 million galaxies. Right panel: the redshift distributions of galaxies in the lensed catalog, LGC0. The number of quadruple-image systems (green) is smaller than that of double-image systems (blue) because the lensing cross section for quadruple-image formation is much smaller than that for double-image formation.