Proof
(related to Proposition: Algebraic Structure of Real Numbers Together with Addition and Multiplication)
The set of real numbers \(\mathbb R\), together with the specific addition operation
"\(+\)", and the specific multiplication operation
"\(\cdot\)", forms the field the algebraic structure \((\mathbb R, + , \cdot)\). This is because:
 We have shown that the set \((\mathbb R, + )\) is a commutative group.
 We have shown that the set \((\mathbb R^*, \cdot )\) is a commutative group, in which \(\mathbb R^*\) denotes all nonzero real numbers.
 We have shown that the set \((\mathbb R^*, \cdot )\) is a commutative group, in which \(\mathbb R^*\) denotes all nonzero real numbers.
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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