Pick representative sets K,LandM, respectively (all mutually disjoint). You want to show thatK×(L∪M) ∼ (K×L)∪(K×M).

These are actually equal since if (a,b) is in the LHS then eitherb∈Lor b ∈M. In the first case, (a,b) ∈K×Land in the second (a,b) ∈K×M.

The converse also applies.