Given languages $L_1$ and $L_2$, how do we prove that$$(L_1L_2)^{\mathrm{rev}} = (L_2^{\mathrm{rev}})(L_1^{\mathrm{rev}})\,,$$where ${}^{\mathrm{rev}}$ denotes reversal?I think using mathematical induction, it can be shown true for two languages $L_1$ and $L_2$ of length $1$ each and then do induction using one of the lengths and then repeat with the other length. Is there any other way of proving this in a simpler way without making it too big with induction method?
How to prove that the reversal of the concatenation of two strings is the concatenation of the reversals?
formal languages;proof techniques;induction