I know that given two analytic functions on some domain $D$ of the complex plane, then their Wronskian determinant being $0$ is equivalent to them being linearly dependent. I would like to generalize this to a finite family of analytic functions. Naturally, one should use induction. However, the proof of $n =2$ is horrifically computational (for me, that is). Is there any clever way to avoid the mess in the induction process? Here are some notes that contain the proof of $n =2$ in case one would like to look at it.
Thank you very much in advance. I am much obliged.