Isolated Point of Closure of Subset is Isolated Point of Subset/Proof 2

Proof
Let $x \in \map \cl H$ be isolated in $\map \cl H$.

By definition of isolated point, $x$ is not a limit point of $\map \cl H$.

From Set is Subset of its Topological Closure:
 * $H \subseteq \map \cl H$

We have that Limit Point of Subset is Limit Point of Set.

But $x$ is not a limit point of $\map \cl H$.

So by the Rule of Transposition, $x$ cannot be a limit point of $H$.

As $x$ is not a limit point of $H$, it follows that $x$ must be an isolated point of $H$.