Going Down Theorem, AM

Question

I'm trying to understand the proof of the going down theorem in Introduction to Commutative Algebra by Atiyah and Macdonald. My main confusion is when they say it suffices to show that $B_{\mathfrak q_1}\mathfrak p_2\cap A = \mathfrak p_2$. Isn't it true that $x\in B_{\mathfrak q_1}\mathfrak p_2\cap A$ iff $x \in A$ and $x/1 \in B_{\mathfrak q_1}\mathfrak p_2$? I don't understand this notation.