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I was wondering is this is just a fact, or if it is possible to prove this statement. I did some googling and the best response I could find was :

##h'(z)= g'(f(z)) f'(z)##, which for ##f'(z)## and ##g'(f(z))## not equal to zero, describes an analytic function.. But I'm afraid this doesn't seem like much of a proof to me.

Many thanks :)