# Deriving the F-test for a complex comparison

I would like to clarify a passage from Maxwell & Delaney, Ch.4 Extension in which the authors explain how to perform an F-test for a complex comparison. In what follows, the coefficients $c_j$ come from a complex comparison of group means $\mu_j$; i.e. the null hypothesis takes the form $\psi = \sum_{j=1}^a c_j \mu_j = 0$. The authors write:

Here, $\hat{\psi}$ refers to the empirical contrast $\sum_{j=1}^a c_j \bar{Y}_j = 0$ for group means $\bar{Y}_j$.

I do not understand where equation E.3 comes from, partially because the authors use the index $j$ for both as an index for the fixed predicted score and for each term in the normalizing sum.

Any help in understanding the justification for E.3 would be very welcome. Thank you.