(related to Proposition: Existence of Integer Zero (Neutral Element of Addition of Integers))
By definition of integers \(x\in\mathbb Z\), the integer number \(x\in\mathbb Z\) can be represented by a pair of natural numbers \(a,b\in\mathbb N\): \[x:=[a,b].\] Since \(0\in\mathbb N\), i.e. the natural zero exists, it is also true that the (integer) zero \(0\in\mathbb Z\) exists, because it can be represented by a pair of the same natural numbers \[0=0_{\in\mathbb Z}:=[h,h],\quad h\in\mathbb N,\] in particular by \(h=0\in\mathbb N\): \[0=0_{\in\mathbb Z}=[0_{\in\mathbb N},0_{\in\mathbb N}].\]
By the definition of addition of integers "\( + \)", and because the "natural number \(0\) is neutral with respect to the definition of addition of natural numbers]bookofproofs$1455 we have \[\begin{array}{rclcl} x + 0&=&[a,b]+[0,0]&&\text{by definition of integers}\\ &=&[a+0,b+0]&&\text{by definition of addition of integers}\\ &=&[a,b]&&\text{natural 0 is neutral with respect to the addition of natural numbers}\\ &=&x&&\text{by definition of integers} \end{array} \]
It remains to be shown that also the equation \(0+x=x\) holds for all \(x\in\mathbb Z\). It follows immediately from the commutativity of adding integers.
