Let $\pi:E \rightarrow M$ be a complex vector bundle over a complex manifold $M$. Prove that if there exists a complex structure on $E$ as a manifold, such that the projection $\pi$ is holomorphic then $E$ has a holomorphic structure. This condition means that there is an open cover of $M$ such that if $U$ is in this open cover then there is a diffeomorphism $\psi_U:\pi^{-1}(U) \rightarrow U \times \mathbf{C}^k$ such that $\pi|_{\pi^{-1}(U)} = pr_U \circ \psi_U$ and the transition functions $g_{UV}:U \cap V \rightarrow CL(n,\mathbf{C})$ are holomorphic (where $\psi_U \psi_V^{-1}(x,v)=(x,g_{UV}(x)v)$.
I can easily prove the converse but I am struggling to proof this statement.