  (2) Although the Hilbert space mentioned in the theorem, in order to be suited for QM, must be complex, it is enough, in order to show the inconsistency of claims KS1 and KS2, to consider a real Hilbert space of the same dimension. So, instead of H4 we consider a real Hilbert space R4 and translate VC1′ into the requirement: Of every set of orthogonal rays in R4, exactly one is assigned the number 1 and the others 0. As usual in the literature, we translate all this into the following colouring problem: Of every set of orthogonal rays in R4 exactly one must be coloured white, the others black. This, however, is impossible, as is shown immediately by the following table (Cabello et al. 1996):