Distributivity of $\omega$-regular expressions
As suggested by Raphael, here's a proof doing it via the sets represented by those expressions.

$\qquad L_\omega((E_1+E_2) \cdot F^\omega)  \\= L(E_1+E_2) \cdot L_\omega(F^\omega) \\= (L(E_1) \cup L(E_2)) \cdot L_\omega(F^\omega) \\= \{w_1w_2 | w_1 \in (L(E_1) \cup L(E_2)) \wedge w_2 \in L_\omega(F^\omega) \} \\= \{w_1w_2|(w_1 \in L(E_1) \wedge w_2 \in L_\omega(F^\omega)) \lor (w_1 \in L(E_2) \wedge w_2 \in L_\omega(F^\omega))\}\\= \{w_1w_2|(w_1 \in L(E_1) \wedge w_2 \in L_\omega(F^\omega))\} \cup \{w_1w_2|(w_1 \in L(E_2) \wedge w_2 \in L_\omega(F^\omega))\} \\= (L(E_1)\cdot L_\omega(F^\omega)) \cup (L(E_2)\cdot L_\omega(F^\omega))\\=L_\omega(E_1 \cdot F^\omega) \cup L_\omega(E_2 \cdot F^\omega)\\=L_\omega(E_1\cdot F^\omega + E_2 \cdot F^\omega)$

So the only thing that's actual work here is to translate the logical operators into set operators.