Prove that the inverse image $\displaystyle g^{-1}[A]$ of an analytic pointset $\displaystyle A$ by a continuous function $\displaystyle g : \mathcal{N} \rightarrow \mathcal{N}$ is analytic.

Hint. Aim for an equivalence of the form $\displaystyle y \in g^{-1}[A] \text{ } \Leftrightarrow \text{ }(\exists x) \text{ } [y=f(\rho_1(x))=g(\rho_2(x))]$ where $\displaystyle f$ is continuous and $\displaystyle \rho_n$ are defined by $\displaystyle \rho_n(z) = (i \mapsto z(\rho(n, i)))$, and then use the result that says:

If $\displaystyle f, g : \mathcal{N} \rightarrow \mathcal{N}$ are continuous functions, then the set $\displaystyle E=\{ x | f(x)=g(x) \}$ of points on which they agree is closed.

I am not sure how to prove this. I would appreciate a few hints or suggestions. In our book, $\displaystyle \mathcal{N}$ denotes the Baire space. Also, subsets of the Baire space are called pointsets. Thanks in advance.