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Proof

(related to Corollary: \((-x)y=-(xy)\))

• As a consequence from the existence of negative numbers it follows that $xy+(-(xy))=0.$
• On the other hand, it follows from the distributivity law and its corollary \(0x=0\) that $xy+(-x)y=(x+(-x))y=0y=0.$
• Comparing both equations, the result \((-x)y=-(xy)\) follows as a corollary from the uniqueness of negative numbers.
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References

Bibliography

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