The multiplication of integers is cancellative, i.e. for all integers \(x,y,z\in\mathbb Z\), with \(z\neq 0\) the following laws (both) are fulfilled1:
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 \(x,y,z\in\mathbb Z\) with \(z\neq 0\).
Proofs: 1 2 3 4 5
Note that this proposition would be obviously wrong if we allow \(z\) to equal \(0\), e.g. for \(z=0, x=5, y=3\) we would get \(0\cdot 5=0\cdot 3\), but \(5\neq 3\). ↩