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 $\sequence{i_1, i_2, \cdots, i_n}$ be a finite sequence of distinct terms in $I$:
 * $\set{i_1, i_2, \cdots, i_n} = \set{i \in I : g_i \ne 0_G}$

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