Vector Space on Cartesian Product is Vector Space

Theorem
Let $\struct {K, +, \circ}$ be a division ring.

Let $n \in \N_{>0}$.

Let $\struct {K^n, +, \times}_K$ be the $K$-vector space $K^n$.

Then $\struct {K^n, +, \times}_K$ is a $K$-vector space.