On the representation theorem for quantum logic

The main assumption of Varadarajan's version of Piron's representation theorem for quantum logic, stating that the lattice under any finite element of the logic is a geometry of finite rank, is eliminated by means of more plausible assumptions, concerning the property of symmetry of the transition probability between pure states. It is also proved, that the quantum logic with symmetric transition probability is irreducible iff it is completely irreducible.