Sum of Points of $\mathbb{R}^n$

Let $n$ be a natural number, and let $x=(x_1,\dots,x_n)$ and $y=(y_1,\dots,y_n)$ be points of Euclidean space $\mathbb{R}^n$, so that each $x_i$ and each $y_i$ is a real number.

The sum $x+y$ is the point of $\mathbb{R}^n$ defined by

where in each coordinate the sum is the addition of the field of real numbers.