Binomial Theorem/Abel's Generalisation/Proof 3

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

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

Hence the result.