A class of identities in the Grassmann Cayley algebra was found by M. J. Hawrylycz (1994, ``Geometric Identities in Invariant Theory,'' Ph.D. thesis, Massachusetts Institute of Technology) which yields a large number of geometric theorems on the incidence of subspaces of projective spaces. In a previous paper we established a link between such identities in the Grassmann Cayley algebra and a family of inequalities in the class of linear lattices, i.e., the lattices of commuting equivalence relations. We proved that a subclass of identities found by Hawrylycz, namely, the Arguesian identities of order 2, can be systematically translated into inequalities holding in linear lattices. However, it is not known whether the Arguesian identities of higher orders have such latticial extensions. In this paper, we give an affirmative answer to the above question in the congruence variety of Abelian groups. We prove that every Arguesian identity, regardless of the order, can be systematically translated into a lattice inequality holding in the congruence variety of Abelian groups. In particular, such a lattice inequality holds in the lattices of subspaces of vector spaces, which are characteristic-free and independent of dimensions. As a consequence, many classical theorems of projective geometry, including Desargues, Bricard, Fontene , and their higher dimensional generalizations, can be extended to lattice inequalities in the general projective spaces, with the variables representing subspaces of arbitrary dimensions.