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Proof

(related to Proposition: Uniqueness of Real One)

• By the existence of real one, we have $x=x\cdot 1$ all $x\in\mathbb R\quad ( * ).$
• Suppose, \(1^{\ast}\) is any (other) real number, for which $x=x\cdot 1^{\ast}$ all $x\in\mathbb R\quad ( * * ).$
• By $( * )$, we have $1^{\ast}=1^{\ast}\cdot 1.$
• Applying the commutativity law for multiplying real numbers, we get $1^{\ast}=1\cdot 1^{\ast}.$
• By $( * * )$ we get $1^{\ast}=1.$
• Thus, the real number one $1$ is unique.
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References

Bibliography

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