# Proposition: Multiplication of Natural Numbers Is Cancellative With Respect to the Order Relation

If a natural number $$x$$ is smaller than another natural number $$y$$, then the inequality is preserved, if and only if we multiply it by a natural number $$z\neq 0$$1, formally:

$x < y\Longleftrightarrow \begin{cases} z x < z y&\text{or}\\ x z < y z. \end{cases}$

The same can be stated about the order relations smaller or greater "$$\le$$", greater or equal "$$\ge$$", and greater "$$>$$":

$x \le y\Longleftrightarrow \begin{cases} z x\le z y&\text{or}\\ x z\le y z. \end{cases}$

$x \ge y\Longleftrightarrow \begin{cases} z x \ge z y&\text{or}\\ x z \ge y z. \end{cases}$

$x > y\Longleftrightarrow \begin{cases} z x > z y&\text{or}\\ x z > y z. \end{cases}$

Proofs: 1

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### References

#### Bibliography

1. Landau, Edmund: "Grundlagen der Analysis", Heldermann Verlag, 2008

#### Footnotes

1. Please note that the proposition would be wrong, if we allowed $$z=0$$, e.g. $$3 < 5$$, but $$3\cdot 0= 5\cdot 0$$.