I've looked at many texts on rectifiable sets and I continue to see assertions that the following statement is in some sense obvious:
Suppose that $N \subset \mathbf{R}^{n +k }$ is a $n$-dimensional $C^1$ submanifold. Then $N \subset N_0 \cup \cup_{j=1}^\infty F_j(\mathbf{R}^n)$, where $F_j: \mathbf{R}^n \to \mathbf{R}^{n+k}$ are Lipschitz maps and $N_0$ is a set of $\mathcal{H}^n$-measure zero.
I would greatly appreciate an explanation of this statement, directly from the definition of an $n$-dimensional $C^1$-submanifold, i.e.
$N \subset \mathbf{R}^{n +k }$ is a $n$-dimensional $C^1$ submanifold provided that for each $x \in N$, there are opens $V \subset \mathbf{R}^n$ and $W \subset \mathbf{R}^{n+k}$ an injective $C^1$ map $\psi: V \to W$ such that $x \in \psi(V) = W \cap N$. Furthermore, $D\psi$ should be full rank (rank $n$) on $V$, and the inverse map $\varphi: \psi(V) \to V$ should be continuous.