Consistency on cubic lattices for determinants of arbitrary orders

We consider a special class of two-dimensional discrete equations defined by relations on elementary N ×N squares, N > 2, of the square lattice Z, and propose a new type of consistency conditions on cubic lattices for such discrete equations that is connected to bending elementary N × N squares, N > 2, in the cubic lattice Z. For an arbitrary N we prove such consistency on cubic lattices for two-dimensional discrete equations defined by the condition that the determinants of values of the field at the points of the square lattice Z that are contained in elementary N × N squares vanish.