# Proof

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

By definition of integers, the integer numbers $$x,y,z$$ are some equivalence classes of ordered pairs of natural numbers represented by some natural numbers $$a,b,c,d,e,f\in\mathbb N$$

$\begin{array}{rcl}x&:=&[a,b],\\y&:=&[c,d],\\z&:=&[e,f]\\\end{array}$

By definition of adding integers, we have $\begin{array}{rcl} x+z&=&[a+e,b+f],\\ y+z&=&[c+e,d+f]. \end{array}$

Because the addition of natural numbers is cancellative, it follows $\begin{array}{rcll} x+z=y+z&\Leftrightarrow& [a+e,b+f]=[c+e,d+f]&\text{by definition of adding integers}\\ &\Leftrightarrow& [a,b]=[c,d]&\text{because addition of natural numbers is cancellative}\\ &\Leftrightarrow& x=y&\text{by definition of integers} \end{array}$

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

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### References

#### Bibliography

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