Closure of Subset in Subspace/Corollary 1

Theorem
Let $T = \struct{S, \tau}$ be a topological space.

Let $T_H = \struct{H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$.

Let $W \subseteq S$ and let $\map {\operatorname {cl}_T} W$ denote the closure of $W$ in $T$.

Let $\map {\operatorname {cl}_H} {W \cap H}$ denote the closure of $W \cap H$ in $T_H$.

Then:
 * $\map {\operatorname {cl}_H} {W \cap H} \subseteq \map {\operatorname {cl}_T} W \cap H$