# Proof

Let $$x,y,z\in\mathbb Q$$ be rational numbers, which by definition means that each rational number is an equivalence class of ordered pairs of integers represented by some integers $$a,b,c,d,e,f\in\mathbb Z$$, with $$b\neq 0,d\neq 0,f\neq 0$$1:

$\begin{array}{ccc}x:=\frac ab,&y:=\frac cd,&z:=\frac ef.\end{array}$

In order to show the law $(x\cdot y)\cdot z=x\cdot (y\cdot z)$ we use the following mathematical definitions and concepts: * definition of rational numbers. * definition of multiplying rational numbers, and * associativity law for multiplying integers. $\begin{array}{rcll} (x\cdot y)\cdot z&=&\left(\frac ab\cdot \frac cd\right)\cdot \frac ef&\text{by definition of rational numbers}\\ &=&\frac {ac}{bd}\cdot \frac ef& \text{by definition of multiplying rational numbers}\\ &=&\frac {(ac)e}{(bd)f}& \text{by definition of multiplying rational numbers}\\ &=&\frac {a(ce)}{b(df)}& \text{by associativity of multiplying integers}\\ &=&\frac ab\cdot \frac{ce}{df}& \text{by definition of multiplying rational numbers}\\ &=&\frac ab\cdot \left(\frac cd\cdot \frac ef\right)& \text{by definition of multiplying rational numbers}\\ &=&x\cdot(y\cdot z)&\text{by definition of rational numbers} \end{array}$

Github: ### References

#### Bibliography

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

#### Footnotes

1. Note that the symbol "$$0$$" denotes the zero defined for integers, and not the zero defined for rational numbers.