How to prove $\int_{A\cup F^{-1}(B)} p(x;\xi)dx =\int_{B}Pr (A|y;\xi)q(y;\xi)$?

Question

I am going through section 2.2 of Methods of Information Geometry by Amari & Nagaoka.I was trying to prove

$\int_{A\cup F^{-1}(B)} p(x;\xi)dx =\int_{B}Pr (A|y;\xi)q(y;\xi)dy$.

But I was getting

$\int_{A\cup F^{-1}(B)} p(x;\xi)dx =\int_{B}Pr (A\cup F^{-1}(B)|y;\xi)q(y;\xi)dy$

Here is my approach, I will be thankful if someone tells me what I am doing wrong or missing something