Hausdorff measure, volume form, reference

Question

Could you tell me where I can find a reference to the fourth corollary in this encyclopedia?

Corollary $4$:

Assume that $\Sigma \subset \mathbb{R}^m$ is an $n$-dimensional $C^1$ submanifold. Then the Hausdorff dimension of $\Sigma$ is $n$ and, for any relatively open set $U \subset \Sigma$ $$\mathcal{H}^n(U)= \int_U \mathrm{dvol}$$ where $\mathrm{dvol}$ denotes the usual volume form of $\Sigma$ as Riemannian submanifold of $\mathbb{R}^n$

It is also stated in this article of the encyclopedia.

Answer 1

It seems to me that what you ask is a particular instance of Theorem 2.10.10 (page 176) of

H. Federer "Geometric Measure Theory" (Springer 1969) (here the Zmath reference)