# Proof

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.

Thank you to the contributors under CC BY-SA 4.0!

Github:

### References

#### Bibliography

1. Graham L. Ronald, Knuth E. Donald, Patashnik Oren: "Concrete Mathematics", Addison-Wesley, 1994, 2nd Edition