How to Prove the Lagrange Inversion Theorem?

[AdrianZ]

I encountered this beautiful theorem and then I tried hard to prove it using ordinary algebraic methods and my understanding of calculus without involving real analysis in it but I didn't succeed. The theorem states that if f is an analytical function at some point x=a then f_-1 has the following Taylor series:

How can I prove this formula?

 

[AdrianZ]

No one knows?