Subspace of Subspace is Subspace

Theorem
Let $T = \left({S, \tau}\right)$ be a topological space.

Let $H \subseteq S$ and $\tau_H$ be the subspace topology on $H$.

Let $K\subseteq H$.

Then the subspace topology on $K$ induced by $\tau$ equals the subspace topology on $K$ induced by $\tau_H$.