Binomial Theorem/Multiindex

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

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

Then:
 * $\displaystyle \left({x + y}\right)^\alpha = \sum_{0 \mathop \le \beta \mathop \le \alpha} {\alpha \choose \beta} x^\beta y^{\alpha - \beta}$

where $\displaystyle {n \choose k}$ is a binomial coefficient.

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

Then:

On the other hand:

This shows that:
 * $\displaystyle \left({x + y}\right)^\alpha = \sum_{0 \mathop \le \beta \mathop \le \alpha} {\alpha \choose \beta} x^\beta y^{\alpha - \beta}$

as required.