Proof of Theorem PSSD

Suppose that $\dimension{U}=m$ and $\dimension{V}=t$. Then $U$ has a basis $B$ of size $m$. If $m>t$, then by Theorem G, $B$ is linearly dependent, which is a contradiction. If $m=t$, then by Theorem G, $B$ spans $V$. Then $U=\spn{B}=V$, also a contradiction. All that remains is that $m<t$, which is the desired conclusion.