Why is the inverse of this chart analytic?

Question

I'm currently going through Differential Topology by Hirsch. While motivating the concept of a chart, the book claims that the map $$\phi : U \to B$$ defined by $\phi (x_1,x_2,x_3) = (x_1,x_2)$ is a homeomorphism where $U \subset S^2$(unit sphere) defined $U=\{ (x_1,x_2,x_3) \in S^2 : x_3>0\}$ and $B$ is the open disk in $\Bbb R^2$. The book further claims that the inverse of $\phi$ is analytic. I don't know how to justify these claims. Any insight into the matter will be appreciated.