(related to Proposition: Addition of Integers Is Associative)
Let \(x,y,z\in\mathbb Z\) be integers, which by definition means that each integer is an equivalence class of ordered pairs of natural numbers represented by some natural numbers \(a,b,c,d,e,f\in\mathbb N\)
\[\begin{array}{rcl}x&:=&[a,b],\\y&:=&[c,d],\\z&:=&[e,f].\\\end{array}\]
In order to show the law \[(x+y)+z=x+(y+z)\] we replace the symbols \(x,y,z\) by their representatives \([a,b],[c,d],[e,f]\), and use the definition of adding integers as well as the associativity law for adding natural numbers to conclude that \[\begin{array}{rcll} (x+y)+z&=&([a,b]+[c,d])+[e,f]&\text{by definition of integers}\\ &=&[a+c,b+d]+[e,f]&\text{by definition of adding integers}\\ &=&[(a+c)+e,(b+d)+f]&\text{by definition of adding integers}\\ &=&[a+(c+e),b+(d+f)]&\text{due to associativity law for adding natural numbers}\\ &=&[a,b]+[c+e,d+f]&\text{by definition of adding integers}\\ &=&[a,b]+([c,d]+[e,f])&\text{by definition of adding integers}\\ &=&x+(y+z)&\text{by definition integers}\\ \end{array}\]