Gamma function proof

Question

I came across this theorem somewhere and it looked really interesting. I don't know how one would go about proving it though? Can anyone give me some pointers? I'd like to understand this statement better.

The statement of the theorem is as follows

Any function $G$, holomorphic in the right half-plane $\mathbb{C}^+ : \Re(z) > 0$, for which the reduction formula holds $$G(z + 1) = zG(z)$$ which decays in any vertical strip as the indeterminate tends to infinity, and such that $G(1) = 1$, coincides with the $\Gamma$-function: $$G(z) \equiv \Gamma(z)$$

Answer 1

You statement is exactly: Wielandt's Theorem About the Γ-Function.

Answer 2

At this link a found a pretty in depth proof. I think I cannot give anything more that this, the paper explains all in a very intuitive manner.