Proof of Theorem EDYES

We give a proof by contradiction (Proof Technique CD). Suppose to the contrary that $U\neq V$. Since $U\subseteq V$, there must be a vector $\vect{v}$ such that $\vect{v}\in V$ and $\vect{v}\not\in U$. Let $B=\set{\vectorlist{u}{t}}$ be a basis for $U$. Then, by Theorem ELIS, the set $C=B\cup\set{\vect{v}}=\set{\vectorlist{u}{t},\,\vect{v}}$ is a linearly independent set of $t+1$ vectors in $V$. However, by hypothesis, $V$ has the same dimension as $U$ (namely $t$) and therefore Theorem G says that $C$ is too big to be linearly independent. This contradiction shows that $U=V$.