  A new notion of a dual Poisson-presymplectic pair is introduced and its
properties are examined. The procedure of Dirac reduction of Poisson operators
onto submanifolds proposed by Dirac is in this paper embedded in a geometric
procedure of reduction of dual Poisson-presymplectic pairs. The method
presented generalizes those used by Marsden and Ratiu for reductions of Poisson
manifolds. Two examples are given.
