Proof

By hypothesis, $G$ is a connected planar graph with a planar drawing $\mathcal D,$ and with $|V|$ vertices, $|E|$ edges and $|F|$ faces.
It follows from the construction of the dual graph $G^*_{\mathcal D}$ that $G^*_{\mathcal D}$ has $|F|$ vertices and $|E|$ edges and that it is a planar graph.
From the Euler's formula for planar graphs he have:
- For $G$: $|V|-|E|+|F|=2.$
- For $G^*_{\mathcal D}$: $|F|-|E|+f=2,$ where $f$ is the (unknown) number of face of $G^*_{\mathcal D}.$
Comparing both equations, we get $f=|V|.$
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Aldous Joan M., Wilson Robin J.: "Graphs and Applications - An Introductory Approach", Springer, 2000