Generalized Sum with Finite 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 $\set{i \in I : g_i \ne 0_G}$ be finite.

Let $\set{i_1, i_2, \cdots, i_n}$ be a finite enumeration of $\set{i \in I : g_i \ne 0_G}$.

Then the generalized sum $\ds \sum_{i \mathop \in I} g_i$ converges and:
 * $\ds \sum_{i \mathop \in I} g_i = \sum_{k \mathop = 1}^n g_{i_k}$

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

From User:Leigh.Samphier/Topology/Finite Generalized Sum Converges to Sum:
 * $\ds \sum_{j \mathop \in J} g_j = \sum_{k \mathop = 1}^n g_{i_k}$

From User:Leigh.Samphier/Topology/Generalized Sum Restricted to Non-zero Summands:
 * $\ds \sum_{i \mathop \in I} g_i$ converges

and
 * $\ds \sum_{i \mathop \in I} g_i = \sum_{j \mathop \in J} g_j$

The result follows.