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Proof

(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.

"\(\Rightarrow\)"

• 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, $x=0$.

"\(\Leftarrow\)"

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

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References

Bibliography

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