Continuity of Composite with Inclusion

Theorem
Let $T = \struct {A, \tau}$ and $T' = \struct {A', \tau'}$ be topological spaces.

Let $H \subseteq A$.

Let $T_H = \struct {H, \tau_H}$ be a topological subspace of $T$.

Let $i: H \to A$ be the inclusion mapping.

Let $f: A \to A'$ and $g: A' \to H$ be mappings.

Then the following apply: