Optimal predictors and conditional probability

There is a crucial difference between the setting of Theorem 4.4 and setting of Theorems 4.5, 4.6. In Theorems 4.5, 4.6 we consider the intersection of two languages in which case $D\cap D=D$. In Theorem 4.4 we consider the Cartesian product of two languages. $(D\times D,\mu \times \mu )$ is not the same thing as $(D,\mu )$. Moreover, the diagonal embedding of $(D,\mu )$ into $(D\times D,\mu \times \mu )$ is not a valid reduction for the purpose of Theorem 6.1 since condition (ii) is violated (diagonal elements are rare within the product ensemble).