# Proof

Because the addition of rational numbers is commutative, it is without loss of generality sufficient to show the right cancellation property, i.e. $x+z=y+z\Longleftrightarrow x=y,~~~~~~(x,y,z\in\mathbb Q).$

By definition of rational numbers, 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:

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

Assume the equation $$x+z=y+z$$ holds. We have to show that $$x=y$$, and we will do it by virtue of the following mathematical definitions and concepts: * definition of adding rational numbers, * cancellation law for multiplying integers, * distributivity law for integers, * associativity law for multiplying integers, * commutativity law for multiplying integers, and * cancellation law for adding integers. $\begin{array}{rcll} x+z=y+z&\Leftrightarrow&\frac ab+\frac ef=\frac cd+\frac ef&\text{by definition of rational numbers}\\ &\Leftrightarrow&\frac {af+eb}{bf}+\frac {cf+ed}{df}& \text{by definition of adding rational numbers}\\ &\Leftrightarrow&(af+eb)df=(cf+ed)bf&\text{by definition of rational numbers}\\ &\Leftrightarrow&(af+eb)d=(cf+ed)b&\text{by cancellation law for multiplying integers}\\ &\Leftrightarrow&(af)d+(eb)d=(cf)b+(ed)b&\text{by distributivity law for integers}\\ &\Leftrightarrow&afd+ebd=cfb+edb&\text{by associativity law for multiplying integers}\\ &\Leftrightarrow&adf+ebd=cbf+ebd&\text{by commutativity law for multiplying integers}\\ &\Leftrightarrow&adf=cbf&\text{by cancellation law for adding integers}\\ &\Leftrightarrow&ad=cb&\text{by cancellation law for multiplying integers}\\ &\Leftrightarrow&\frac ab=\frac cd&\text{by definition of rational numbers}\\ &\Leftrightarrow&x=y&\text{by definition of rational numbers}\\ \end{array}$

Altogether, we have shown that the addition of integers is cancellative $x+z=y+z\Rightarrow x=y,~~~~~~(x,y,z\in\mathbb Q),$ and its conversion $x=y\Rightarrow x+z=y+z,~~~~~~(x,y,z\in\mathbb Q).$

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

#### Footnotes

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