Looking for the proof of theorem 5.2.11 of Casella, Berger, Statistical Inference

Theorem 5.2.11 Suppose $X_1,dots, X_n$ is a random sample from a pdf or pmf $f(xmid theta)=h(x)c(theta)exp(sum_{i=1}^kw_i(theta)T_i(x))$ is in exponential family. Define statistics $T_i=sum_it_i(X_j)$ where $i=1,dots, k$. If the set ${(w_1(theta),dots,w_k(theta))}$ contains some open subset of $mathbb{R}^k$, then the distribution of $(T_1,dots, T_k)$ is an exponential family of the form $g(u_1,dots, u_kmidtheta)=H(u_1,dots, u_k)c(theta)^n exp(sum_iw_i(theta)u_i)$

Q: I am looking for a proof of the theorem or reference of proof. How is containing open set for $(w_1(theta),dots,w_k(theta))$ used to derive transformation? All I could see is that somehow Jacobian between $(T_i)$‘s and $(X_i)$‘s is accounted by part of $H$ but this may not be 1-1.