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It cannot be proven because it is not true.

Suppose S1 = {0,1,2,3} and S2 = {0,5,10} then S1 u S2 = {0,1,2,3,5,10}

then |S1| = n = 4, |S2| = m = 3

but |S1 u S2| = 6 which is NOT n+m = 7

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