Claim:  

$\forall x \in \mathbb{R^+} ,$ $ x^n=1 $ $where$ $ n\in \mathbb{N}$ 

Proof by induction on n:

Basis step:  

 $\forall x \in \mathbb{R^+} ,$ $ x^0=1 $

Induction Step:  

Let this holds for all n$\lt n\_0$  

then $x^{n\_0}$=$\frac{x^{n\_0-1}}{x^{n\_0-2}} \times x^{n\_0-1} $=$\frac{1}{1} \times1 $ $\square $