Gradient Operator Distributes over Addition

Theorem
Let $\mathbf V$ be a vector space of $n$ dimensions.

Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis of $\mathbf V$.

Let $f \left({x_1, x_2, \ldots, x_n}\right), g \left({x_1, x_2, \ldots, x_n}\right): \mathbf V \to \R$ be real-valued functions on $\mathbf V$.

Let $\nabla f$ denote the gradient of $f$.

Then:
 * $\nabla \left({f + g}\right) = \nabla f + \nabla g$