Differential topology, fundamental theorem of algebra

Question

I am reading Milnor's Topology from the Differentiable Viewpoint, in particular page 8-9 about applying regular values to prove the fundamental theorem of algebra. So he defines stereographic projection as $h_{+}: S^{2}-\{(0,0,1)\}\to \mathbb{R}^{2}\times \{0\}$ and also he defines it from the south pole, namely, $h_{-}:S^{2}-\{(0,0,-1)\}\to \mathbb{R}^{2}\times \{0\}.$ Although he says its elementary geometry, I don't understand why the composition $h_{+}h^{-1}_{-}(z)=\frac{z}{|z|^{2}}$. Any help would be greatly appreciated thank you very much.