(related to Lemma: Handshaking Lemma for Finite Digraphs)
In any digraph \(D=(V,E,\alpha,\omega)\), each edge \(e\) is incident to two vertices, its terminal vertex \(\omega(e)\) and its initial vertex \(\alpha(e)\). Because of the definitions of inner and outer degrees, it contributes exactly 1 to the sum \(\sum_{v\in V}d_D^+(v)\) and exactly 1 to the sum \(\sum_{v\in V}d_D^-(v)\). The results \[\sum_{v\in V}d_D^+(v)=\sum_{v\in V}d_D^-(v)=|E|\]and \[\sum_{v\in V}d_D(v)=2|E|\]follow immediately.
