Proof
(related to Corollary: Even Number of Vertices with an Odd Degree in Finite Graphs)
 Let \(G=(V,E,\gamma)\) be a finite graph and let \(O\subseteq V\) be the subset of all vertices with odd degree.
 Then \(V\setminus O\) is the subset of all vertices with even degree.
 According to the handshaking lemma we obtain \(\sum_{v\in V}\deg_G(v)=2E.\)
 It follows \[\sum_{v\in O}\deg_G(v)=2E\sum_{v\in V\setminus O}\deg_G(v).\]
 Because the right side of the equation is even, so must be the left side.
∎
Thank you to the contributors under CC BYSA 4.0!
 Github:

References
Bibliography
 Krumke S. O., Noltemeier H.: "Graphentheoretische Konzepte und Algorithmen", Teubner, 2005, 1st Edition