Corollary 2

Let $(E_i)_{i \in I}$ be a family of A-modules, F an A-module and, for each $i \in I$, let $f_i : E_i \to F$ be a linear mapping. For $f = \sum_{i \in I} f_i$ to be an isomorphism of $E = \bigoplus_{i \in I} E_i$ onto F, it is necessary and sufficient that there exist for each $i \in I$ a linear mapping $g_i : F \to E_i$ with the following properties: (1) $g_i \circ f_i = 1_{E_i}$ for all $i \in I$. (2) $g_i \circ f_\kappa = 0$ for $i \neq \kappa$. (3) For all $y \in F$, the family $(g_i(y))$ has finite support and

Note that if I is finite, the last condition may also be written as Obviously the conditions are necessary for they are satisfied by the $g_i = \operatorname{pr}_i \circ f_i^{-1}$. Conversely if they hold, for all $y \in F$, $g(y) = \sum_i j_i(g_i(y))$ is defined and it is immediate that $g$ is a linear mapping of F into E. For all $y \in F$, $f(g(y)) = \sum_i f_i(g_i(y)) = y$ by hypothesis. On the other hand, for all $x \in E$, $g_\kappa(f(x)) = g_\kappa \left( \sum_i f_i(\operatorname{pr}_i(x)) \right) = g_\kappa(f_\kappa(\operatorname{pr}_\kappa(x))) = \operatorname{pr}_\kappa(x)$ by hypothesis;

therefore $g(f(x)) = \sum_i j_i(g_i(f(x))) = \sum_i j_i(\mathrm{pr}_i(x)) = x$, which proves the corollary.