# Proposition: Multiplication of Natural Numbers Is Cancellative

The multiplication of natural numbers is cancellative, if the number we want to "cancel out" does not equal $$0$$1, i.e. for all $$x,y,z\in\mathbb N$$ with $$z\neq 0$$ the following laws (both) are fulfilled:

• Left cancellation property: If the equation $$z \cdot x=z \cdot y$$ holds, then it implies $$x=y$$.

• Right cancellation property: If the equation $$x \cdot z=y \cdot z$$ holds, then it implies $$x=y$$.

Conversely, the equation $$x=y$$ implies

• $$x\cdot z=y\cdot z$$ and
• $$z \cdot x=z \cdot y$$

for all $$x,y,z\in\mathbb N$$ with $$z\neq 0$$.

Proofs: 1

Proofs: 1 2 3 4

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### References

#### Bibliography

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

#### Footnotes

1. Please note that if we allow $$z=0$$, then the proposition is wrong: e.g. $$0\cdot 5=0\cdot 3$$, but $$5\neq 3$$.