The multiplication of real numbers is cancellative, i.e. for all real numbers \(x,y,z\in\mathbb R\), 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 \(x,y,z\in\mathbb R\) with \(z\neq 0\).
Proofs: 1 2
Propositions: 3 4
Note that this proposition would be obviously wrong if we allow \(z\) to equal \(0\), e.g. for \(z=0, x=\sqrt 2, y=\sqrt 3\) we would get \(0\cdot \sqrt 2=0\cdot \sqrt 3\), but \(\sqrt 2\neq \sqrt 3\). ↩