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 VerĂ¤nderlichen", Vieweg Studium, 1983