Does it involve $ L_1L_2 = L_2L_1 $ $ L_1 = L_2?

Leave $ L_1, L_2 subseteq Sigma ^ * $ be two languages, where $ Sigma $ It is a finite alphabet.

Make $ L_1L_2 = L_2L_1 $ to imply $ L_1 = L_2 $?

What if $ L_1 $ Y $ L_2 $ Are they regular languages?

Can you give counterexamples?

demonstrating L1 * ∪ L2 * ⊆ (L1∪L2) *

x∈ L1 * ∪ L2 * ⇔ x∈ L1 * ∨ x ∈L2 * ⇔ x ∈ (L1) * ∨ x∈ (L2) * ⇔ x ∈L1 * ∪ L2 * ⇔ x∈ (L1∪L2) *

Is it enough to try it this way?