Proof
(related to Proposition: Algebraic Structure of Integers Together with Addition)
The set of integers \(\mathbb Z\), together with the specific addition operation "\(+\)" is a commutative group, because:
 The addition operation is associative, i.e. \((x+y)+z=x+(y+z)\) is valid for all \(x,y,z\in\mathbb Z\).
 We have shown the existence of a neutral element of addition  the number \(0\in\mathbb Z\), i.e. such that \(0+x=x\) for all \(x\in\mathbb Z\).
 For every \(x\in\mathbb Z\), there there exists an inverse integer \(x\in\mathbb Z\), such that \(x+(x)=0\).
 For every \(x\in\mathbb Z\), there there exists an inverse integer \(x\in\mathbb Z\), such that \(x+(x)=0\).
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References
Bibliography
 Kramer Jürg, von Pippich, AnnaMaria: "Von den natürlichen Zahlen zu den Quaternionen", SpringerSpektrum, 2013