Proof
(related to Proposition: Algebraic Structure of Rational Numbers Together with Addition)
The set of rational numbers \(\mathbb Q\), together with the specific addition operation
"\(+\)" is a commutative group. This is because:
 The addition operation is associative, i.e. \((x+y)+z=x+(y+z)\) is valid for all \(x,y,z\in\mathbb Q\).
 We have shown the existence of a neutral element of addition  the number \(0\in\mathbb Q\), i.e. such that \(0+x=x\) for all \(x\in\mathbb Q\).
 For every \(x\in\mathbb Q\), there there exists an inverse rational number \(x\in\mathbb Q\), such that \(x+(x)=0\).
 For every \(x\in\mathbb Q\), there there exists an inverse rational number \(x\in\mathbb Q\), 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