(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 \(a,b,c,d,e,f\in\mathbb N\)
\[\begin{array}{rcl}x&:=&[a,b],\\y&:=&[c,d],\\z&:=&[e,f]\\\end{array}\]
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 \[x=y\Rightarrow x+z=y+z,~~~~~~(x,y,z\in\mathbb Z).\]
