I have been studying the book **Some Nonlinear Problems In Riemannian Geometry - Thierry Aubin**. On page $46$ he begins the proof of the Sobolev imbedding theorem to manifolds. The proof is divided in two steps. In the second step, he defines the function $h(x)=f(\mbox{exp}_{O}(x))$ in a ball $\tilde{B}$. On page $47$ he uses the theorem $1.53$ to obatin the estimate
$$
\left(\int_{\tilde{B}}|\nabla_{E}h(x)|^{q}dE\right)^{1/q}\leq\frac{\sinh(a\delta)}{a\delta}\left(\frac{\pi}{2}\right)^{(n-1)q}\|\nabla f\|_{q}
$$
I don't understand how to use this theorem to obtain the inequality above. Has anyone studied this demonstration of this book? Thank you.