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Let $K$ be a finite field of characteristic $p$. Let $K\left(\right(x\left)\right)$ be the field of formal Laurent series $f\left(x\right)$ in $x$ with coefficients in $K$. That is, $$f\left(x\right)=\sum _{n=n_0}^{\infty }{f}_n{x}^n$$ with $n_0\in 𝐙$ and $f_n\in K(n=n_0,n_0+1,\cdots )$. We discuss the distribution of ${\left(\left\{f^m\right\}\right)}_{m=0,1,2,\cdots }$ for $f\in K\left(\right(x\left)\right)$, where $$\left\{f\right\}:=\sum _{n=0}^{\infty }{f}_n{x}^n\in K\left[\left[x\right]\right]$$ denotes the nonnegative part of $f\in K\left(\right(x\left)\right)$. This is a little different from the real number case where the fractional part that excludes constant term (digit of order 0) is considered. We give an alternative proof of a result by De Mathan obtaining the generic distribution...