Let and be the spaces of messages, keys, and ciphertexts, respectively.
Proof. Correctness: for any , we have
Perfect indistinguishability: for any and , we have ◻
Proof. Assume for contradiction that there exists an encryption scheme that satisfies both correctness and perfect indistinguishability with . Consider a message , a key , and . Due to the correctness, there is exactly one message in that is mapped to by the key . Therefore, there are at most messages that can be mapped to by any key . This means that there exists a message that is not mapped to by any key , i.e., . Since we know that , we have , which contradicts the perfect indistinguishability. ◻