Proposition 16

Let $x$ be a positive element of A. Let $\alpha \in \mathbf{R}^*_+$. Then $x^{1/\alpha}$ is the unique positive element $y$ of A such that $y^{\alpha}=x$.

It was seen above that $x^{1/\alpha}$ satisfies the required properties. Conversely, let $y$ be a positive element of A such that $y^{\alpha}=x$. By formula (5), one has $y= (y^{\alpha})^{1/\alpha}=x^{1/\alpha}$, which was to be proved.