Let $K$ be a complete valued field and $X$ be an indeterminate $K[X]$ be the ring of polynomials over $K$ in the indeterminate $X$ and $K(X)$ be its quotient field. For $a\in K$, consider the evaluation map $\sigma_a:K(X) \to K$ defined by $\sigma_a(P/Q)=P(a)/Q(a)$ when $Q(a)\ne 0$. Now consider $K(X)$ embeded in $K((X))$. Suppose that $R(X)\in K(X)$, $R(X)=\sum_{n\ge0}a_nX^n$. Consider its evaluation now by $R(a)=\sum_{n\ge0}a_na^n$ if $a$ is in the disk of convergence of the power series. I wonder why both evaluation gives the same result. I thought it would be easy to prove but I did not manage to do it
Thanks in advance for any answer.