Recall that a language is $\omega$-regular if and only if it is recognized by a Bchi automaton. How can I prove that$\qquad (E_1 + E_2).F^\omega$is equivalent to$\qquad {E_1.(F^\omega)+E_2.(F^\omega)}$where  both expressions are omega regular expressions, and  $E_1$, $E_2$ and $F$ are arbitrary regular expressions with  $\epsilon \notin L(F)$.  One way I could think of is to convert expression to a DFA and checkif it is equivalent.Or I will really appreciate a hint on how to do the equivalenceproof, but how to represent $E_1$, $E_2$ and $F$ in DFA.
Distributivity of $\omega$-regular expressions
regular expressions;buchi automata