The Special Theory of Relativity (STR) is used to predict the slowing down of clocks (time dilation) as a consequence of their motion relative to an observer. The starting point is the space-time mixing equation of the Lorentz transformation (LT) which is responsible for the longstanding belief in the remote non-simultaneity of events. Since time dilation is based on a strict proportionality between the measured elapsed times of two observers in relative motion, it is argued that this LT prediction is inconsistent with remote non-simultaneity. This is because the latter requires that there can be a null time difference for one of the observers but not for the other, the occurrence of which is rendered impossible in view of the above proportional relationship. The well-known example of two lightning flashes on a moving train is used to illustrate the discrepancy. A straightforward means of eliminating this contradictory feature of Einstein's theory is presented which makes use of a different space-time transformation than the LT.