Binomial Theorem/Abel's Generalisation/Proof 3

Proof
From Hurwitz's Generalisation of Binomial Theorem:
 * $(1): \quad \left({x + y}\right)^n = \displaystyle \sum x \left({x + \epsilon_1 z_1 + \cdots + \epsilon_n z_n}\right)^{\epsilon_1 + \cdots + \epsilon_n - 1} y \left({y - \epsilon_1 z_1 - \cdots - \epsilon_n z_n}\right)^{n - \epsilon_1 - \cdots - \epsilon_n}$

Setting $z = z_1 = z_2 = \cdots z_n$ we have:

Hence the result.