Summation over Union of Disjoint Finite Index Sets

Theorem
Let $\struct{G, +}$ be a commutative monoid.

Let $I$ and $J$ be disjoint finite indexing sets.

Let $K = I \cup J$.

Let $\family{g_k}_{k \mathop \in K}$ be an indexed family of elements of $G$.

Then:
 * $\ds \sum_{k \mathop \in K} g_k = \paren{\sum_{i \mathop \in I} g_i} + \paren{\sum_{j \mathop \in J} g_j}$

where:
 * $\ds \sum_{k \mathop \in K} g_k$ is the summation of $g$ over $K$


 * $\ds \sum_{i \mathop \in I} g_i$ is the summation of the restriction of $g$ over $I$


 * $\ds \sum_{j \mathop \in J} g_j$ is the summation of the restriction of $g$ over $J$

Proof
Let $\set{i_1, i_2, \ldots, i_n}$ be an enumeration of $I$.

Let $\set{j_1, j_2, \ldots, j_m}$ be an enumeration of $J$.

Let $k: \closedint 1 {n+m}$ be the mapping defined by:
 * $k_l = \begin{cases}

i_l & : \text{ if } 1 \le l \le n \\ j_{l-n} & : \text{ if } l > n \\ \end{cases}$

From Union of Bijections with Disjoint Domains and Codomains is Bijection
 * $\set{k_1, k_2, \ldots, k_{n + m}}$ is an enumeration of $K$

We have: