V_g  =  \int_{\phi_{\alpha}(U_{\alpha})} \sqrt{ \det g^{\alpha}_{ij}} \left| \eta_{\alpha} u \right|^{p} \circ \phi^{-1}_{\alpha}(x)\ d \mathcal{L}^n(x) ,$$ hence $$\begin{aligned}
  \label{niermodularfunkc}
    2^{-\frac{n}{2p}}\|\eta_{\alpha} u\|_{L^p (X,V_g)} \leq 
\| (\eta_{\alpha} u) \circ \phi^{-1}_{\alpha}\|_{L^p (\mathbb{R}^n,\mathcal{L}^n)} \leq 2^{\frac{n}{2p}}\|\eta_{\alpha} u\|_{L^p (X,V_g)} .  \end{aligned}$$ On the other hand, we estimate the gradient locally by $$\begin{aligned}
\int_{X}  \left| \nabla (\eta_{\alpha} u) \right|^{p}  d V_{g}
&=& \int_{\phi_{\alpha} (U_{\alpha})} \sqrt{ \det g^{\alpha}_{ij}}\ \left| \sum_{k,j=1}^{n} g_{\alpha}^{kj} D_{k} ( (\eta_{\alpha} u ) \circ \phi^{-1}_{\alpha}) D_{j} ( (\eta_{\alpha} u ) \circ \phi^{-1}_{\alpha}) \right|^{p}  d\mathcal{L}^n \nonumber \\
&\geq& 2^{- \frac{n+p}{2}} \int_{\phi_{\alpha} (U_{\alpha})} \left| \nabla ( (\eta_{\alpha} u ) \circ \phi^{-1}_{\alpha}) \right|^{ p }  d\mathcal{L}^n,\end{aligned}$$ therefore $$\| \nabla ( \eta_{\alpha} u) \circ \phi_{\alpha}^{-1} \|_{L^p(\mathbb{R}^n, \mathcal{L}^n)} \leq 2^{\frac{n+p}{2p}}  \| \nabla( \eta_{\alpha} u) \|_{L^p(X,V_g)}$$ for each $\alpha$ in $A$.\
Set $\displaystyle C_0 := \max_{\alpha \in A} \| \nabla \eta_{\alpha} \|_{\infty}+1 $. Then $$\label{eq:1}
 \| \nabla ( \eta_{\alpha} u) \circ \phi_{\alpha}^{-1} \|_{L^p(\mathbb{R}^n, \mathcal{L}^n)} \leq 2^{\frac{n+p}{2p}}  \left( \|  \nabla u \|_{L^p(X, V_g)} + C_0 \|  u \|_{L^p(X,V_g)} \right) 
 \leq 2^{\frac{n+p}{2p}}C_0 \| u \|_{L^p_1(X,V_g)}   \ .$$

Fix some $\epsilon >0$; then there exists a Hajłasz gradient $h_{\alpha}$ for $(\eta_{\alpha} u) \circ \phi_{\alpha}^{-1} $ in $\phi_{\alpha}(U_{\alpha})$, so that $$\label{eq:2}
\| (\eta_{\alpha} u) \circ \phi_{\alpha}^{-1} \|_{L^p(\mathbb{R}^n,\mathcal{L}^n)} + \| h_{\alpha} \|_{L^p(\mathbb{R}^n,\mathcal{L}^n)} - \epsilon \leq \| (\eta_{\alpha} 