# Proof that L^2 is regular => L is regular

I’m trying to show $$L^2 \in \mathsf{REG} \implies L \in \mathsf{REG}$$ with $$L^2 = \{w = w_1w_2 \mid w_1, w_2 \in L\}$$ but I cant seem to find a proof that feels right.

I first tryed to show $$L \in \mathsf{REG} \implies L^2 \in \mathsf{REG}$$, by constructing an machine $$M$$ that consists of two machines $$A=A’$$ with $$A$$ recognizing $$L$$. $$M$$ has the same start states as $$A$$ but the final states of $$A$$ are put together with the starting states of $$A’$$. Further $$M$$ uses the same accepting states as $$A’$$. Hope that makes sens so far 😀

Now to show $$L^2 \in \mathsf{REG} \implies L \in \mathsf{REG}$$ I’d argue the same way, but:

The machine $$M’$$ that accepts $$L^2$$ has to recognize $$w_i \in L$$ in some way, and because $$L^2$$ is regular, $$M’$$ has to be a NFA/DFA. So the machine has to check if $$w_i \in L$$ and this cant be done by using something else than a NFA/DFA.

This feels wrong and not very mathematical, so maybe somebody knows how to do this?