transition functions on vector bundles

Question

If we consider a vectorbundle $\pi:V\rightarrow M$ on a smooth manifold and two open subsets $U_1,U_2$ along with isomorphisms $h^{U_i}:\pi^{-1}(U_i)\rightarrow U_i\times \mathbb{R}^n$ s.t. $pr_1 \circ h^{U_i}=\pi$ and $h^{U_i}:\pi^{-1}(y)\rightarrow y\times\mathbb{R}^n$ is a vector space iso. My question is why the transition functions must be given by $h^{U_1}\circ (h^{U_2})^{-1}:U_1\cap U_2\times \mathbb{R}^n\rightarrow U_1\cap U_2\times \mathbb{R}^n$,$h^{U_1}\circ (h^{U_2})^{-1}(x,v)=(x,A(x)v)$ . I think my problem is that i dont understand why $h^{U_i}|_{\pi{^-1}(U_1\cap U_2)}$ is still an isomorphism (is it?).