Proof
(related to Proposition: Algebraic Structure of Real Numbers Together with Addition)
The set of real numbers \(\mathbb R\), 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 \(x,y,z\in\mathbb R\).
 We have shown the existence of a neutral element of addition  the number \(0\in\mathbb R\), i.e. such that \(0+x=x\) for all \(x\in\mathbb R\).
 For every \(x\in\mathbb R\), there there exists an inverse real number \(x\in\mathbb R\), such that \(x+(x)=0\).
 For every \(x\in\mathbb R\), there there exists an inverse real number \(x\in\mathbb R\), such that \(x+(x)=0\).
∎
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References
Bibliography
 Forster Otto: "Analysis 1, Differential und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
 Kramer Jürg, von Pippich, AnnaMaria: "Von den natürlichen Zahlen zu den Quaternionen", SpringerSpektrum, 2013