Summation is Linear

Theorem
Let $\left({x_1, \ldots, x_n}\right)$ and $\left({y_1, \ldots, y_n}\right)$ be finite sequences of numbers, of equal length.

Let $\lambda$ be a number.

Then:


 * $\displaystyle \sum_{i \mathop = 1}^n x_i + \sum_{i \mathop = 1}^n y_i = \sum_{i \mathop = 1}^n

\left({x_i + y_i}\right)$
 * $\displaystyle \lambda \cdot \sum_{i \mathop = 1}^n x_i = \sum_{i \mathop = 1}^n \lambda \cdot x_i$

Proof
Note that all numbers can naturally be regarded as complex numbers.

Thus it suffices to prove the identities for $\lambda, x_i, y_i \in \C$.