Theorem 9.21 Suppose $\mathbf{f}$ maps an open set $E\subset R^n$ into $R^m$. Then $\mathbf{f}\in \mathcal{C}^\prime(E)$ if and only if the partial derivatives $D_jf_i$ exist and are continuous on $E$ for $1\leq i\leq m$, $1\leq j\leq n$.
In the proof, it says "for the converse, it suffices to consider the case $m=1$", but I don't know why. Could you help me? Thanks.