Summary: A strong representation of the conditional product-limit estimator, uniformly over the covariate values, is shown to hold for arbitrary distributions. Similar results, but valid only for continuous distributions, were established by M.G. Akritas [Ann. Stat. 22, 1299-1327 (1994; Zbl 0819.62028)] (uniformly over the observed covariate values), and by W. Gonzalez-Manteiga and C. Cadarso-Suarez [J. Nonparametric Stat. 4, 65-78 (1994)], and I. Van Keilegom and N. Veraverbeke [Ann. Inst. Stat. Math. 49, No. 3, 467-491 (1997; Zbl 0935.62051)] (for each fixed covariate value). A critical step for achieving the extension to arbitrary distributions is showing that a process indexed by the upper endpoint of a certain integral converges to zero at the same rate as it does in the case of continuous survival distributions. The main tool for establishing this rate is a construction of associated continuous survival times, and associated censoring times (which are not necessarily continuous) in such a way that a time transformation of the associate process equals the original process.