Generalized Sum Restricted to Non-zero Summands

Theorem
Let $G$ be a commutative topological semigroup with identity $0_G$.

Let $\family{g }_{i \in I}$ be an indexed subset of $G$.

Let $J = \set{i \in I : g_i \ne 0_G}$

Let $g'= g \restriction_J$ be the restriction of $g$ to $J$.

Then:
 * the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converges


 * the generalized sum $\ds \sum_{j \mathop \in J} g'_j$ converges
 * the generalized sum $\ds \sum_{j \mathop \in J} g'_j$ converges

In which case:
 * $\ds \sum_{i \mathop \in I} g_i = \sum_{j \mathop \in J} g'_j$

Necessary Condition
Let the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converge to $h \in G$.

Sufficient COndition
Let the generalized sum $\ds \sum_{j \mathop \in J} g'_j$ converge to $h \in G$.