(related to Proposition: Uniqueness Of Rational One)

- By the existence of rational one, we have $x=x\cdot 1$ all $x\in\mathbb Q\quad ( * ).$
- Suppose, \(1^{\ast}\) is any (other) rational number, for which $x=x\cdot 1^{\ast}$ all $x\in\mathbb Q\quad ( * * ).$
- By $( * )$, we have $1^{\ast}=1^{\ast}\cdot 1.$
- Since the multiplication of rational numbers is commutative, we get $1^{\ast}=1\cdot 1^{\ast}.$
- By $( * * )$ we get $1^{\ast}=1.$
- Thus, the rational number one $1$ is unique.∎