(related to Corollary: A product of two real numbers is zero if and only if at least one of these numbers is zero.)

Let \(x,y\) be real numbers.

- Let \(yx=0\), where $0$ denotes the real zero.
- Assume \(y\neq 0\). We have to show that $x=0.$
- It follows from the unique solvability of ax=b (for $a=y$ and $b=0$) that \(x=y^{-1}\cdot 0\) is the unique solution.
- From $0x=x$ for all $x\in\mathbb R$, it follows further that $x=y^{-1}\cdot 0=0.$

- Therefore, $x=0$.

- Let at least on of the two numbers \(x,y\) equal zero.
- Then it follows immediately as a corollary from $0x=0$, that \(xy=0.\)∎

Parts: 1

**Forster Otto**: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983