Prove Theorem 9.2. Theorem 9.2 (Li [2015a]). Suppose that P= {1, 2,...} is a partition of the

Prove Theorem 9.2.

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Theorem 9.2 (Li [2015a]). Suppose that P= {1, 2,...} is a partition of the data, and Pa = {, ,..., k} is the partition obtained by moving point a from to 2. Then Wa(P) - Wa(Pa) = = n1 2(1) n2 Ea (a, ) 2(n + 1) -Ea(a, ), (9.11) where Ea(a, Ti) denotes the two-sample a-energy distance between point a and cluster Ti.