Equivalence of Definitions of Square Function

Theorem
Let $\F$ denote one of the standard classes of numbers: $\N$, $\Z$, $\Q$, $\R$, $\C$.

Proof
By definition of $n$th power (for positive $n$):


 * $x^n = \begin{cases}

1 & : n = 0 \\ x \times x^{n - 1} & : n > 0 \end{cases}$

Thus:

Hence the result.