Is anyone familiar with this theorem:

If E(X) > 1 then the extinction probability $\displaystyle \pi$ is the unique root of the equation $\displaystyle \pi = f(\pi)$, which lies strictly between 0 and 1.

If E(X) < 1, then $\displaystyle \pi = 1$

Graph is below:

I cannot seem to find any link between the expectations, and the curve f(x) and y=x in the graph. And what is the meaning of $\displaystyle \pi_1, \pi_2$?

Thanks for your attention.