predicted by quantum mechanics is by a factor of 2 instead of the factor $\sqrt{2}$ achieved in the CHSH inequality. This fact conforms to the existence of Bell type inequalities which yield a violation increasing exponentially with the number of particles [@Mermin1].\] The point we want to emphasize here is that, as shown by Hardy [@Hardy], a maximum violation of inequality (20) always entails a contradiction of the GHZ type. A simple illustration of this statement is provided by the choice $\sigma(\vec{e}_i)\equiv \sigma(x_i)$ and $\sigma(\vec{e}\,^{\prime}_{i})\equiv \sigma(y_i)$, $i=1,2,3$, where $\sigma(x_i)$ \[$\sigma(y_i)$\] denotes the spin operator along the $x$ axis ($y$ axis) for particle $i$. For this choice of operators we find that $|\langle B_{\text H} \rangle| =4$ whenever the expectation value is evaluated for the state vector (1). As is well known, both the operators $\sigma(x_i)$ and $\sigma(y_i)$, and the state vector (1), form the basis of Mermin’s exposition on the GHZ theorem [@Mermin2].[^2] It is worth noting that the plane containing the measurement directions $\vec{e}_i$ and $\vec{e}\,^{\prime}_{i}$ ($i=1,2,3$) giving the largest violation of inequality (20) is fixed by the quantum state since, in contrast to the two-particle case, the triorthogonal decomposition in Eq. (2) is unique even if the coefficients 