Proof
(related to Corollary: \((x^{-1})^{-1}=x\))
- According to the existence of reciprocal numbers there exists a reciprocal number \((x^{-1})^{-1}\) of \((x^{-1})\).
- Thus, we have \[(x^{-1})^{-1}\cdot(x^{-1})=1.\]
- With the same argument regarding the number \(x\) we have \[x\cdot(x^{-1})=1.\]
- By the uniqueness of reciprocal numbers, we can compare both equations.
- This comparison shows that \((x^{-1})^{-1}=x\) follows for all real numbers with \(x\neq 0\).
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References
Bibliography
- Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
