# Proof

(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 $\begin{array}{rcll} (x+y)+z&=&([a,b]+[c,d])+[e,f]&\text{by definition of integers}\\ &=&[a+c,b+d]+[e,f]&\text{by definition of adding integers}\\ &=&[(a+c)+e,(b+d)+f]&\text{by definition of adding integers}\\ &=&[a+(c+e),b+(d+f)]&\text{due to associativity law for adding natural numbers}\\ &=&[a,b]+[c+e,d+f]&\text{by definition of adding integers}\\ &=&[a,b]+([c,d]+[e,f])&\text{by definition of adding integers}\\ &=&x+(y+z)&\text{by definition integers}\\ \end{array}$

Thank you to the contributors under CC BY-SA 4.0!

Github:

### References

#### Bibliography

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