How to Prove the Lagrange Inversion Theorem?

AdrianZ
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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:

c31894bd772bb55bd1f98d0e0dd770f2.png


How can I prove this formula?
 
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