Subspace Test Proof (of test not subset)

On my midterm review there was a question asking me to:

State and prove the subspace test.

Now, I know how the subspace test works and how to use it to test whether subsets are subspaces or not...but how would I go about proving that the subspace test works?

Would the reasoning for the subspace test be that the only 3 vector space properties that need to be checked in order to declare a subset S of a vector space V a subspace of V (a subset of V that is a vector space with the same operations as V) are "x + y is defined and in S", "kx is defined and in S", and "there is a zero element"?