
theorem ThSLGM2:
  for L being RATional Z_Lattice, LX being Z_Lattice,
      b being OrdBasis of LX
  st LX is Submodule of DivisibleMod(L) &
  the scalar of LX = (ScProductDM(L)) || the carrier of LX
  holds GramMatrix(InnerProduct(LX), b) is Matrix of dim(LX), F_Rat
  proof
    let L be RATional Z_Lattice, LX be Z_Lattice,
        b be OrdBasis of LX such that
    A1: LX is Submodule of DivisibleMod(L) &
    the scalar of LX = (ScProductDM(L)) || the carrier of LX;
    LX is RATional by A1,ThSLGM1;
    then GramMatrix(b) is Matrix of dim(LX), F_Rat by ZMODLAT2:45;
    hence thesis;
  end;
