What if LSPACE = NP?

P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? This question was first mentioned in a letter written by John Nash to the National Security Agency in 1955. A precise statement of the P versus NP problem was introduced independently in 1971 by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major complexity classes are EXP and LSPACE. Whether LSPACE = P is another fundamental question that it is as important as it is unresolved. However, we already know that P is not equal to EXP. We prove if P = NP and LSPACE = P, then every EXP-complete language can indeed contain a P-complete subset.