Lemma 2

*Let $n$ be a non-zero rational integer. In the ring of formal power series $K[[t]]$ we write $(1+t)^n = \sum_{j \geq 0} c_{j,n} t^j$. There exists an integer $j \geq 1$ such that $c_{j,n} \neq 0$.

If $n > 0$, then $c_{n,n} = 1$ by the binomial formula. Suppose that $n < 0$ and let $m = -n$. If $c_{j,n} = 0$ for all $j \geq 1$, then $(1+t)^n = 1$, whence, taking the inverse, $(1+t)^m = 1$, which contradicts the formula $c_{m,m} = 1$.*