We have two random varibles $\gamma_1$ and $\gamma_2$. The CDF of $\gamma_1$ is $$ F_{\gamma_1}(\gamma_1)=(1-e^{-\frac{\gamma_1}{\gamma}})^N $$ were $\gamma$ is constant.
The PDF of $\gamma_2$ is $$ f_{\gamma_2}(\gamma_2)=\frac{\gamma_2^{M-1}}{(M-1)!\gamma^M}e^{-\frac{\gamma2}{\gamma}} $$ We difine a random varible $Z$ as follow $$Z=\frac{\gamma_1\gamma_2}{\gamma_1+\gamma_2+1}$$
First we want to dirive the CDF of $Z$, so the authors use the following steps
\begin{align} F_Z(z)&=Pr(Z\leq z) \\ &=Pr(\frac{\gamma_1\gamma_2}{\gamma_1+\gamma_2+1}\leq z) \end{align} But in the following steps hi use some rules of probability. Can any one explain to me this steps \begin{align} F_Z(z)&=Pr(\frac{\gamma_1\gamma_2}{\gamma_1+\gamma_2+1}\leq z)\\ &=\int_{0}^{z}Pr(\gamma_1 \geq\frac{z\gamma_2+z}{\gamma_2-z})f_{\gamma_2}(\gamma_2)d\gamma_2+\int_{z}^{\infty}Pr(\gamma_1 \leq\frac{z\gamma_2+z}{\gamma_2-z})f_{\gamma_2}(\gamma_2)d\gamma_2\\ &=1-\int_{z}^{\infty}Pr(\gamma_1 \geq\frac{z\gamma_2+z}{\gamma_2-z})f_{\gamma_2}(\gamma_2)d\gamma_2 \end{align}
