How to rigorously prove this very simple fact about percolation?

Question

I speak about the last sentence. It is clear by saying we can model the openning at the probability $p_2$ by an openning at the probability $p_1$ followed by another at the probability $p_2- p_1$. But how to prove this rigorously?

Answer 1

Assume otherwise for contradiction. Then $\eta_{p_1}=1$ and $\eta_{p_2}=0$ i.e. $\chi(e)<p_1$ and $\chi(e)\ge p_2$. Note this contradicts the assumption $p_1\le p_2$.