# formal languages – If \$L\$ is regular then \$L^{|2|}={w_1w_2 mid w_1,w_2in L, |w_1|=|w_2|}\$ is context-free

I have found a problem about proving whether $$L^{|2|}={w_1w_2 mid w_1,w_2in L, |w_1|=|w_2|}$$ is context-free or not, knowing that $$L$$ is regular

So far I know that:

• There are examples where $$L$$ is regular and $$L^{|2|}$$ is regular (for example $$L={a,b}$$)
• There are examples where $$L$$ is regular and $$L^{|2|}$$ is not (for example $$L={w mid w=a^N text{ or } w= b^N , Nge0}$$)

But I am not sure how to prove that it’s context-free regardless of which regular language I use. I have found similar problems with the same language without imposing restrictions on which words to use, but I am not sure if those apply to this one.