In Naive Set Theory book of Halmos there is a statement that if $\{I_j\}$ is a family of sets with domain $J$ and $\{A_k\}$ is a family of sets with the domain $K = \bigcup_{j \in J}I_j$ then $$\bigcup_{k \in K}A_k = \bigcup_{j \in J}(\bigcup_{i \in I_j} A_i)$$ which is a generalized version of the associative law for unions.
Now the question is how to see and interpret the generalization? What additional information carries this generalization over the usual associative law $(A \cup B) \cup C = A \cup (B \cup C)$