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Suppose there are two observers A & B, separated in space and one is moving towards the other. Their clocks were somehow synchronised at the beginning meaning they both started from $0$ or just started noting $\Delta t$. Now their clocks have same reading, now as soon as they both reach the same place in space-time, they note the readings. Now the readings are different. If readings are different, and they write it down on a piece of paper and then show that paper to each other, they will know one of the readings is less, because time is a real number and real numbers have order properties.
So, they will violate the symmetry of both of them being inertial and equal and hence, one will know whose clock was actually running slow because earlier they could both accuse each other of having slower clocks, but now they know one was slower for real.