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