# Proof

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

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

In order to show the law $x\cdot y=y\cdot x$ we use the following mathematical definitions and concepts: * definition of rational numbers. * definition of multiplying rational numbers, and * commutativity law for multiplying integers. $\begin{array}{rcll} x\cdot y&=&\frac ab\cdot \frac cd&\text{by definition of rational numbers}\\ &=&\frac {ac}{bd}& \text{by definition of multiplying rational numbers}\\ &=&\frac {ca}{db}& \text{by commutativity of multiplying integers}\\ &=&\frac cd\cdot \frac ab& \text{by definition of multiplying rational numbers}\\ &=&y\cdot x&\text{by definition of rational numbers} \end{array}$

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### 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.