Proof: By Induction

Because the addition of natural numbers "$$+$$" is commutative, it is without loss of generality sufficient to show the right cancellation property, i.e. $x+z=y+z\Rightarrow x=y,~~~~~~(x,y,z\in\mathbb N).$ The proposition can be proven by induction.

Base case Base case $$z=0$$.

For arbitrary $$x,y\in\mathbb N$$, it follows from the definition of addition that $x+0=y+0\Rightarrow x=y.$

Note that $$x+0=x$$ and $$y+0=y$$ are both equalities being equivalence relations. Thus, we can replace the implication sign "$$\Rightarrow$$" by the equivalence sign "$$\Leftrightarrow$$", and the reasoning is still logically correct:

$x+0=y+0\Leftrightarrow x=y.$

Induction step $$z\to z^+:= z+1$$.

Now, let assume that the inclusion $$x+a_0=y+a_0\Leftrightarrow x=y$$ has been proven for all $$z_0\le z$$, where we use "$$\le$$" as the order relation of natural numbers. Then it follows again from the definition of addition that $x+z^+=y+z^+\Leftrightarrow (x+z)^+=(y+z)^+.$ Because both successors are unique and because $$x+z=y+z$$ is equivalent to $$x=y$$ by assumption, it follows that $(x+z)^+=(y+z)^+\Leftrightarrow x=y.$ Altogether, it follows from $$x + z=y + z$$ that $$x=y$$ for all natural numbers $$x,y,z$$. Thus, the addition of natural numbers is cancellative. $x+z=y+z \Rightarrow x=y,$

and we have proven the conversion of the cancellative law:

$x=y\Rightarrow x+z=y+z$

for all natural numbers $$x,y,z$$.

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References

Bibliography

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