$\textbf{Exercise}$: if $\mu:{\mathbb R}^{n-1} \to \mathbb R$ is a Lipschitz function, and we define $f:{\mathbb R}^{n-1} \to {\mathbb R}^{n}$ as $f(z)=(z,\mu(z))$, $z \in {\mathbb R}^{n-1}$, then $\Gamma= f({\mathbb R}^{n-1})$ is locally $\mathcal{H}^{n-1}$-rectifiable and, for a.e.$z \in {\mathbb R}^{n-1}$,
\begin{align*} T_{f(z)}\Gamma =& v(z)^{\bot} & v(z)&=(-\nabla'\mu(z),1) \end{align*}
$\textbf{lemma 10.4}$ : If $M= f(E)$ is a $k$-dimensional regular Lipschitz image in ${\mathbb R}^{n}$ and $z \in E $ then,
\begin{align*}
T_{x}M&= \nabla f(z)({\mathbb R}^{k}) & x&=f(z)
\end{align*}
I can use of the lemma 10.4 above then $T_{f(z)}\Gamma= \nabla f(z)({\mathbb R}^{n-1}) $, how to prove that $\forall y \in \nabla f(z)({\mathbb R}^{k})$ $$ y \cdot v(z) = 0 $$ here I used the F. Maggi's book, and it is very difficult for me to carry out this exercise. If someone could help me, at least with an idea, I would appreciate it.