Let A be a finite set that is partitioned as follows A=A_{1} U...U A_{n}.
A = A1 U A2 U A3 U……. U An
It is given that |A1| = |A2| = |A3| = …….. = |An| = k [ Suppose ]
Let us assume that all the subsets are disjoint then obviously
|A| = |A1| + |A2| + |A3| + …….. + |An|
= k+k+k+ ……. N times = kn
Now I am sure the proof is quite obvious