Let A be a language under any alphabet . Define the following language:

REM(A) = {xz : x, z , xyz A for some y },

quasi, REM(A) is the languagem that contains every words that can be obtained by the removal of exactly one symbol on the word A.

Example. If A = {b,ab,abba}, then REM(A) = {, a, b, bba, aba, abb}:

• is obtained removing one symbol from b;
• a e b are obtained removing one symbol from ab;
• bba, aba, e abb are obtained removing one symbol from abba (only 3 words insteand of 4 because removing the first or second b results in the same word aba).

Prove that the regular language class are closed under REM operation.