How to derive Zamolodchikov's recursion relation of conformal block?

Question

Could any one explain how to derive the recursion relation, equation (7) of Zamolodchikov's paper http://link.springer.com/article/10.1007%2FBF01214585? 

I understand that there is when $F(D,di,C,x)$ will diverge when c becomes closer to $c_{mn}$, so there should be a pole. But how do we know that it's the form of$\frac{1}{c−c_{mn}}$instead of $\frac{1}{(c−c_{mn})^2}$ or other forms?