The goal of screening programmes for cancer is early detection and treatment with a consequent reduction in mortality from the disease. Screening programmes need to assess the true benefit of screening, that is, the length of time of extension of survival beyond the time of advancement of diagnosis (lead-time). This paper presents a non-parametric method to estimate the survival function of the post-lead-time survival (or extra survival time) of screen-detected cancer cases based on the observed total life time, namely, the sum of the lead-time and the extra survival time. We apply the method to the well-known data set of the HIP (Health Insurance Plan of Greater New York) breast cancer screening study. We make comparisons with the survival of other groups of cancer cases not detected by screening such as interval cases, cases among individuals who refused screening, and randomized control cases. As compared with Walter and Stitt's model, in which they made parametric assumptions for the extra survival time, our non-parametric method provides a better fit to HIP data in the sense that our estimator for the total survival time has a smaller sum of squares of residuals.