(related to Proposition: Rule of Combining Different Sets of Indices)
It is required for the sets of integers \(K\) and \(L\) to be finite, which makes sure that the sums to be added are well-defined.
Using the Iverson-notation of sums, which allows to obtain the values \(0\) or \(1\) from logical statements in the middle of a formula, we can write the sums as
\[\begin{array}{rcl} \sum_{k\in K} a_k + \sum_{k\in L} a_k &=& \sum_{k} a_k[k\in K] + \sum_{k} a_k[k\in L]\\ &=&\sum_{k} a_k([k\in K] + [k\in L]), \end{array}\]
where the last step was obtained by applying the associative rule. Now, the sum of logical statements \([k\in K] + [k\in L]\) can only have the following values:
\[[k\in K] + [k\in L]=\cases{ 0,\quad\text{if }k\not\in (K \cup L)\\ 1,\quad\text{if }k\in (K \cup L) \wedge k\not\in (K \cap L)\\ 2,\quad\text{if }k\in (K \cap L) } \]
Therefore, we can continue our calculations by
\[\begin{array}{rcl} \sum_{k} a_k([k\in K] + [k\in L])&=&\underbrace{0}_{\text{case 0}}+\underbrace{\sum_k a_k[K\cup L]-\sum_k a_k[K\cap L]}_{\text{case 1}}+\underbrace{2\sum_k a_k[K\cap L]}_{\text{case 2}}\\ &=&\sum_k a_k[K\cup L]+\sum_k a_k[K\cap L]\\ &=&\sum_{k\in K\cup L} a_k+\sum_{k\in K\cap L}a_k, \end{array} \]
which completes the proof.
