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If $L(R_1) \subseteq L(R_2) \subseteq L(R_3)$ then $L(R_1)^* \subseteq L(R_2)^* \subseteq L(R_3)^*$. Does this also imply that $L(R_1)^* \subseteq L(R_3)$?

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Of course not: Consider R1='a', R2='a|b', R3='a|b|c', for instance.

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