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Proof

(related to Lemma: Handshaking Lemma for Planar Graphs)

• By hypothesis, $G(V,E)$ is a biconnected planar graph with a given planar drawing $\mathcal D.$
• Since each edge in $\mathcal D$ has two sides, it contributes exactly $2$ to the sum of the face degrees (including the infinite face).
• Thus, $\sum_{f\in \mathcal D}\deg (f)=2\cdot|E|.$
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References

Bibliography

1. Aldous Joan M., Wilson Robin J.: "Graphs and Applications - An Introductory Approach", Springer, 2000