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Spontaneous mirror symmetry violation is carried out in nature as thetransition between the usual left (right)-handed and the mirror right(left)-handed spaces, in each of which the usual and mirror particles have thedifferent lifetimes. As a consequence, all equations of motion in a unifiedfield theory of elementary particles include the mass, energy and momentum asthe matrices expressing the ideas of the left- and right-handed neutrinos areof long- and short-lived objects, respectively. These ideas require inprinciple to go away from the chiral definitions of the structure of matterfields taking into account that the internally disclosed Dirac matrices are, inthe Weyl presentation, reduced to the matrices having an internal undisclosure.We discuss a theory in which a set comes forward at the new level, namely, atthe level of set-matrix duality principle as a criterion for matrices. Thisconnection gives the exact mathematical definitions of internally disclosed andundisclosed matrices, allowing to formulate four more definitions, three lemmasand one pair of axioms. Thereby, it involves that there is no single matrix,for which an absolutely empty matrix would not exist. The sets of matrixelements and the matrices of set elements thus found unite all of matrix andset operations necessary for deciding the problems in a unified whole.

A Set-Matrix Duality Principle for the Dirac Equation