The multiplication of rational numbers is cancellative, i.e. for all rational numbers \(x,y,z\in\mathbb Q\), with \(z\neq 0\) the following laws (both) are fulfilled^{1}:
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
for all \(x,y,z\in\mathbb Q\) with \(z\neq 0\).
Proofs: 1
Note that this proposition would be obviously wrong if we allow \(z\) to equal \(0\), e.g. for \(z=0, x=\frac 15, y=\frac 23\) we would get \(0\cdot \frac 15=0\cdot \frac 23\), but \(\frac 15\neq \frac 23\). ↩