(related to Proposition: Existence of Inverse Integers With Respect to Addition)

By definition, any integer \(x\) can be represented by two natural numbers \(a,b\in\mathbb N\) with \(x=[a,b]\). By definition of adding natural numbers, \(a+b\) is the natural numbers we gain if we add the natural number \(a\) to the natural number \(b\). By definition of adding integers and applying the commutativity law for adding natural numbers, we get

\[\begin{array}{rcll}[a,b]+[b,a]&=&[a+b,b+a]&\text{by definition of adding integers}\\ &=&[a+b,a+b]&\text{by commutativity of adding natural numbers}\\ \end{array}\]

Since the integer zero \(0_{\in\mathbb Z}\) can be represented by a pair of equal natural numbers: \[0=0_{\in\mathbb Z}:=[h,h],\quad h\in\mathbb N,\] we can take \(h:=a+b\), which completes the proof.

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  1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013