Generalized Sum Restricted to Non-zero Summands/Sufficient Condition

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 $J$.

Let $h \in G$.

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

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

Proof
Let $U \subseteq G$ be an open subset of $G$ such that $h \in U$.

By definition of convergent net:
 * $(SC1) \quad \exists F' \subseteq J : F' \ne \O : \forall E' \subseteq J : E' \supseteq F' : \ds \sum_{j \mathop \in E'} g'_j \in U$

where $\ds \sum_{j \mathop \in E'} g'_j$ is the summation over $E$.

We have:
 * $F' \subseteq J \subseteq I$.

Let $E \subseteq I$:
 * $E \supseteq F'$

Let:
 * $E' = E \cap J$

From Set Intersection Preserves Subsets:
 * $E' \supseteq F'$

From $(SC1)$:
 * $\ds \sum_{j \mathop \in E'} g'_j \in U$

From Set Difference Union Intersection:
 * $E = E' \cup E \setminus J$

From Set Difference and Intersection are Disjoint:
 * $E' \cap E \setminus J = \O$

Case : $E \setminus J = \O$
Let:
 * $E \setminus J = \O$

From Union with Empty Set:
 * $E = E'$

Hence:
 * $\ds \sum_{j \mathop \in E} g'_j \in U$

By definition of restricted mapping:
 * $\ds \sum_{i \mathop \in E} g_i = \sum_{j \mathop \in E} g'_j \in U$

Case : $E \setminus J \ne \O$
Let:
 * $E \setminus J \ne \O$

We have:

Hence:
 * $\ds \sum_{i \mathop \in E} g_i \in U$

In either case:
 * $\ds \sum_{i \mathop \in E} g_i \in U$

Since $U$ was arbitrary, it follows that $\ds \sum_{i \mathop \in I} g_i$ converges to $h$ by definition.