I refer to Cartan's book "Elementary Theory of Analytic Functions of One or Several Complex Variables".
Let $S(X)=\sum_{n\ge0} a_nX^n, T(Y)=\sum_{p\ge0} b_pY^p$ be formal power series over a field $K$ s.t. $b_0\ne0$. Recall the set of such series is an algebra over $K$. Define (pp.12) substitution of series $T$ in $S$ to be the series $S(T(Y))=(S\circ T)(Y)$ given by $\sum_{n\ge0}a_n(T(Y))^n$ regrouping the powers of $Y$.
We say a family of power series $\{S_i(X)=\sum_{n\ge0} a_{n,i}X^n\colon i\in I\}$ is summable iff for every integer $k>0$ the number of indices $i\in I$ s.t. $a_{n,i}\ne0$ for some $0\le n<k$ is finite. In this case, let $S(X)=\sum_{i\in I} S_i(X)=\sum_{n\ge0} [\sum_{i \in I}a_{n,i}]X^n$ be the sum of this family.
At pp.13 the author claims that $(\sum_{i\in I} S_i) \circ T=\sum_{i\in I}(S_i\circ T)$. To prove this, note that $$(\sum_{i\in I} S_i)\circ T=\sum_{n\ge0}(\sum_i a_{n,i})(T(Y))^n,$$ and $$\sum_{i\in I} (S_i\circ T)=\sum_i(\sum_ {n\ge0} a_{n,i}(T(Y))^n),$$.
To show right hand sides are equal, he says that "the coefficient of a given $Y^p$ in each of them involves only a finite number of the coefficients $a_{n,i}$ and we apply the associativity law of (finite) addition in the field $K$".
Question: I tried to make explicit the above argument without success. So I look for an explicit explanation.
Thank you in advance for your help.