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I was reading Modern Robotics by Kevin Lynch et al. and this robot was presented. They had introduced Grubler's formula which assumes independent joint which gives $dof = 0$ but this is not the case for the robot and they provide an alternative computation but it isn't clear how they got the values for Grubler's formula for this.
I understand that this is Grubler's formula and this is also from the book.
The part that is unclear to me is what is highlighted in red.
Where they say, "...any one of the three parallel links...has no effect on the motion" that pretty much means you could delete any one of the three links. Doing so leaves N=4 and J=4. Plug them into the formula and get the numbers you highlighted in red.
The conclusion "not independent, as required" is sort of a proof by contradiction. More precisely: The thing can move, with a single dof, only in the specific case where the "parallel" links are indeed parallel, and the same length, making one of the three redundant (i.e. not independent). Otherwise (the general case, with weird lengths and angles) the system would be overconstrained and indeed couldn't move (dof=0) as the formula computes.
