I don't quite understand your question, so let me give you a general answer.

First, the correlation coefficient is an indicator of how "similar" two users are, so as long as two users share a rating history (i.e. they rated on the same items), we can calculate a correlation coefficient between them. Note that the coefficient is calculated based on the items they both rated and has nothing to do with the items that they didn't rate.

Second, the predicted rating of an item X by a user Y is based on the ratings of the users who have rated this item. In the example in the notes, Nan didn't rate Item 6, so when we predict Ken's rating on Item 6, we only consider Lee and Meg.

alomo wrote:

I understand that for the prediction we must use only history of those users who rated item 6.

My question was only about the correlation coefficient. In the lecture you said that we don't need to calculate the similarity for Ken-Nan based on the fact that Nan didn't rate item 6. Even so, Ken and Nan do "share" history for other four items and it makes sense seeing their similarity.

You can calculate the coefficient of Ken-Nan, but it is unnecessary if you just want to predict the rating of item 6 by Ken.