Sub-Basis for Topological Subspace

Theorem
Let $(X,\tau)$ be a topological space.

Let $K$ be a subbasis for $\tau$.

Let $(S, \tau')$ be a subspace of $(X, \tau)$.

Let $K' = \{ U \cap S: U \in K \}$. That is, $K'$ consists of the open sets in $S$ corresponding to elements of $K$.

Then $K'$ is a subbasis for $\tau'$.

Proof
We need to show that