# How to prove a statement in regular expression?

I cannot figure out how to go about proving this statement in regular expression.

$$L(R_1) \subseteq L(R_2) \subseteq L(R_3) \implies L(R_1^*+R_3)^* \subseteq L(R_2^*+R_3^*)$$

Here's what I have so far:

Since $$L_1 \subseteq L_2 \implies L_1^* \subseteq L_2^*$$ then $$L(R_1) \subseteq L(R_2) \subseteq L(R_3) \implies L(R_1)^* \subseteq L(R_2)^* \subseteq L(R_3)^*$$. If $$x$$ belongs to $$L(R_1)$$ then $$x$$ also belongs to $$L(R_2)$$ and $$L(R_3)$$ and if $$y$$ belongs to $$L(R_1)^*$$ then $$y$$ also belongs to $$L(R_2)^*$$ and $$L(R_3)^*$$. I'm not sure how to go from here to show that $$(y+x)^* \subseteq (y+y)$$.

• Try showing that both languages on the right-hand side are in fact equal to $L[R_3^*]$. For this, it suffices that $L[R_1],L[R_2] \subseteq L[R_3]$. Nov 2 '20 at 21:06