Closure of Subset in Subspace/Corollary 1

Corollary to Closure of Subset in Subspace
Let $T = \struct {S, \tau}$ be a topological space.

Let $H$ be a subset of $S$.

Let $T_H = \struct {H, \tau_H}$ be the topological subspace on $H$.

Let $K \subseteq S$.

Let $\map {\cl_T} K$ denote the closure of $K$ in $T$.

Let $\map {\cl_H} {K \cap H}$ denote the closure of $K \cap H$ in $T_H$.

Then:
 * $\map {\cl_H} {K \cap H} \subseteq \map {\cl_T} K \cap H$