
theorem ThSLGM1:
  for L being RATional Z_Lattice, LX being Z_Lattice
  st LX is Submodule of DivisibleMod(L) &
  the scalar of LX = (ScProductDM(L)) || the carrier of LX
  holds LX is RATional
  proof
    let L be RATional Z_Lattice, LX be Z_Lattice such that
    A1: LX is Submodule of DivisibleMod(L) &
    the scalar of LX = (ScProductDM(L)) || the carrier of LX;
    for v, u being Vector of LX holds <; v, u ;> in RAT
    proof
      let v, u be Vector of LX;
      reconsider vv = v, uu = u as Vector of DivisibleMod(L) by A1,ZMODUL01:25;
      <; v, u ;> = (ScProductDM(L)).(vv, uu) by A1,FUNCT_1:49;
      hence thesis by RAT_1:def 2;
    end;
    hence thesis by ZMODLAT2:def 1;
  end;
