Can you give me the mathematical proof with derivation for the above ?

Problem 2. Suppose (Gen, Enc, Dec) is a PKE scheme which supports encryptions of one-bit messages. Prove that the following "bit-by-bit" encryption is a PKE scheme (Gen', Enc', Dec') which supports encryptions of two-bit messages. This extends naturally also to longer messages, but you do not have to show this. - Gen' (): Draw and output (pk, sk) Gen(). - Enc(msg,pk): Parse msg=msg(1)msg(2) for msg(1),msg(2){0,1}. For i=1,2, draw ct(i)Enc(msg(i),pk) and output ct=ct(1)ct(2). - Dec (ct,sk): Parse ct=ct(1)ct(2), compute for i=1,2,msg(i)=Dec(ct(i),sk) and output Remark. This shows that, at least conceptually, it is not more difficult to build encryptions long messages than it is to build encryptions for one-bit messages