In the setting of Proposition 4.27, show that:

(i) if the utility function \(u^{i}\) is strictly increasing in its first argument and nondecreasing in the second, then the existence of an optimal portfolio implies the validity of the Law of One Price;

(ii) if the utility function \(u^{i}\) is non-decreasing in its first argument and strictly increasing in the second and there exists a portfolio \(\hat{z}\) with \(D \hat{z}>0\), then the existence of an optimal portfolio implies the validity of the Law of One Price.