Bartlett`s approximation for Wilks Lambda

Question

For two given independent wishart distributed variables A and B, where $$A\sim W(\Sigma,m),~B\sim W(\Sigma,n)$$ it implies for the transformed variable $$\lambda=\frac{det(A)}{det(A)+det(B)}\sim \Lambda(p,m,n)$$ with $m\geq p$, that $$\Rightarrow (\frac{p-n+1}{2}-m)ln(\Lambda(p,m,n))\sim\chi^2_{np}. $$ This approximation is called Bartlett-Approximation and is to be proved.