Solution
= Solution
{c}
By <3/b/i> and <3/b/ii>,
$$
r_{ST}+n_{ST}=r_{S'}+n_{S'}+n_T.
$$
The <domain of a function> of $S'$ is $\operatorname{im}T$. Its <rank-nullity theorem> gives $r_{S'}+n_{S'}=\dim\operatorname{im}T=r_T$. Applying the <rank-nullity theorem> for $T$ now gives
$$
\boxed{r_{ST}+n_{ST}=r_T+n_T=\dim U,}
$$
which is exactly the <rank-nullity theorem> for the <function composition> $ST$.