Binomial Theorem/Multiindex

Theorem
Let $\alpha$ be a multiindex, indexed by $\set {1, \ldots, n}$ such that $\alpha_j \ge 0$ for $j = 1, \ldots, n$.

Let $x = \tuple {x_1, \ldots, x_n}$ and $y = \tuple {y_1, \ldots, y_n}$ be ordered tuples of real numbers.

Then:
 * $\ds \paren {x + y}^\alpha = \sum_{0 \mathop \le \beta \mathop \le \alpha} \dbinom \alpha \beta x^\beta y^{\alpha - \beta}$

where $\dbinom n k$ is a binomial coefficient.

Proof
First of all, by definition of multiindexed powers:
 * $\ds \paren {x + y}^\alpha = \prod_{k \mathop = 1}^n \paren {x_k + y_k}^{\alpha_k}$

Then:

On the other hand:

This shows that:
 * $\ds \paren {x + y}^\alpha = \sum_{0 \mathop \le \beta \mathop \le \alpha} \dbinom \alpha \beta x^\beta y^{\alpha - \beta}$

as required.