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Einstein proved that moving clocks slow down.
Einstein postulated that motion slowed time in the early 1900s.
Just to point something out that creates a lot of confusion. "Moving clocks slow down" and "motion slows time" aren't the best phrases to use. No inertial frame is "moving" any more than another. They are all moving relative to each other.
Time dilation in its simplest form refers to the fact that if two events occur at the same location in one inertial frame then any frame moving relative to it will measure a longer time between those two events. The clocks are operating perfectly normal in each frame and time isn't slowing down in either frame. They just measure different spatial and temporal values for the events that occur.