Subclass of Subclass is Subclass
Let $A$, $B$ and $C$ be classes.
Let $A$ be a subclass of $B$.
Let $B$ be a subclass of $C$.
Then $A$ is a subclass of $C$.
Let $x \in A$ be arbitrary.
It follows by definition of subclass that $x \in B$.
It further follows by definition of subclass that $x \in C$.
So we have that $x \in A$ implies that $x \in C$.
As $x$ is arbitrary, the result follows.