Number of equivalence classes of weakly equivalent lattices

Two complete lattices, $M$ and $N$, lying in an algebra over the field of rational numbers, are said to be weakly left equivalent if $N=KM$ and $M=\overline KN$, where $K$ is a two-sided invertible lattice and $\overline K$ is the inverse for $K$. In this paper we prove that the number of equivalence classes of lattices contained in a weak equivalence class is finite.