Let K and K' again denote two inertial frames, with K' moving along the x-axis of K in the positive direction, at a constant speed v > 0.
A consequence of time dilation is that:
(Theorem) At any given time t in K, the corresponding coordinate time t' in K' that is read off by an observer in K decreases as the x-coordinate of the observer's location increases. For an increase of Δx, there is a decrease in t' by Δx · (v / c2) / √1 – v2 / c2 .
So two events that are simultaneous in K do not happen at the same (coordinate) time in K', unless they have the very same x-coordinate.