(related to Proposition: Addition of Integers Is Cancellative)

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 \(a,b,c,d,e,f\in\mathbb N\)


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