Fundamental metric expressions of a generic quadruple divided random quantity

After establishing a metric over the vector space of the bivariate random quantities which are the components of a generic quadruple divided random quantity I establish a metric over the vector space of the quadruple divided random quantities in order to show that a coherent prevision of a generic bivariate random quantity coincides with the notion of a-product. Therefore, metric properties of the notion of a-product mathematically characterize the notion of coherent prevision of a generic bivariate random quantity. I accept the principles of the theory of concordance into the domain of subjective probability for this reason. This acceptance is well-founded because the definition of concordance is implicit as well as the one of prevision of a random quantity and in particular of probability of an event. By considering quadruple divided random quantities I realize that the notion of coherent prevision of a generic bivariate random quantity can be used in order to obtain fundamental metric expressions of quadruple divided random quantities.