Vector Space on Cartesian Product is Vector Space/Proof 1
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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.
Proof
This is a special case of the Vector Space of All Mappings, where $S$ is the set $\closedint 1 n \subset \N^*$.
$\blacksquare$