Two-Step Vector Subspace Test

Theorem
Let $V$ be a vector space over a division ring $K$.

Let $U \subseteq V$ be a non-empty subset of $V$ such that:

Then $U$ is a subspace of $V$.

Proof
Suppose that $(1)$ and $(2)$ hold.

From $(1)$, we obtain for every $\lambda \in K$ and $u \in U$ that $\lambda u \in U$.

An application of $(2)$ yields the condition of the One-Step Vector Subspace Test.

Hence $U$ is a vector subspace of $V$.

Also see

 * One-Step Vector Subspace Test