# Prove that characteristic function $f_w$ in write protected input turing machine behave as a 2FSA

Write protected input turing machine is a single-tape TM that cannot write on the input portion of the tape. I almost prove that these TMs can only recognize regular languages but i have a problem in one of my steps.

In one step of proof i define a characteristic function $$f_w$$ such that for any $$q \in Q, f_w(q)=q’$$ implies that if TM $$M$$ is at state $$q$$ just before going back to input portion, the next time that going out from the input portion will change $$M$$ in satate $$q’$$ unless we halt inside the input portion.

Now what i need to prove is that for any two string $$w_1$$ and $$w_2$$ if for every $$q \in Q$$ we have the same value for $$f_{w_1}(q)$$ and $$f_{w_2}(q)$$ then for every $$z \in \Sigma^*$$ we have $$f_{w_1z}(q)=f_{w_2z}(q)$$

It is sensible for me and i have some idea to prove this , one of my idea is that if i can prove for every $$f_w(q)$$ there is a 2FSA then appending one character to $$w$$ leads us to a same equivalance class.

Am i in the right path ?