These results may all be demonstrated by modifications of the diagonal argument by which Turing (1937) originally demonstrated the undecidability of the classical   Halting Problem.[13]   Nonetheless, Theorem 3.1 already has a number of interesting consequences about the relationships between the complexity classes introduced above. For instance, since the functions \(n^{k}\) and \(n^{k+1}\) satisfy the hypotheses of parts i), we can see that \(\textbf{TIME}(n^k)\) is always a proper subset of \(\textbf{TIME}(n^{k+1})\) . It also follows from part i) that \(\textbf{P} \subsetneq \textbf{TIME}(f(n))\) for any time bound \(f(n)\) which is of super-polynomial order of growth – e.g. \(2^{n^{.0001}}\) or \(2^{\log(n)^2}\). Similarly, parts i) and ii) respectively implies that \(\textbf{P} \subsetneq \textbf{EXP}\) and \(\textbf{NP} \subsetneq \textbf{NEXP}\). And it similarly follows from part iii) that \(\textbf{L} \subsetneq \textbf{PSPACE}\).