Going back to (5), notice that the genitive NP Pats's and GP eating fast food are combined to become a GP by the constraint of a noun-possessive-ex (7) and that the category of the GP is gerund all the way from bottom to top, satisfying the endocentricity condition.
the constraint (7) states that the category of a noun-possessive-cx is a noun, and the category of the GP Pat's eating fast food is not a noun but a gerund.