in the 5th question from the recitation we had today we had to prove that (LR= {<m> | L(m) belong to R} belong to RE\R)

by reduction to Htm.

i understood the reduction from Htm to LR, that usually says as i understood that if i have a word that belong to Htm it will be belong to LR and if it not belong to Htm it won't belong to LR.

but in the recitation they say it if and only if. which means that if w belong to LR i will be belong to Htm too. and from the reduction i can't see how it's works. cause if |x|<k i immediately go to the possible that i won't be at Htm and to the second step. and if |x|<k i have a finite language and i belong to R.

thank you.