
theorem ThDL2:
  for L being RATional positive-definite Z_Lattice holds
  DualLat(L) is Submodule of DivisibleMod(L)
  proof
    let L be RATional positive-definite Z_Lattice;
A1: the carrier of DualLat(L) c= the carrier of DivisibleMod(L)
    proof
      let x be object such that
      B1: x in the carrier of DualLat(L);
      x in DualSet L by B1,defDualLat;
      then consider v be Dual of L such that
      B2: x = v;
      thus thesis by B2;
    end;
    0.DualLat(L) = 0.DivisibleMod(L) &
    the addF of DualLat(L)
    = (the addF of DivisibleMod(L)) || the carrier of DualLat(L) &
    the lmult of DualLat(L) = (the  lmult of DivisibleMod(L))
    | [:the carrier of INT.Ring, the carrier of DualLat(L):] by defDualLat;
    hence thesis by A1,VECTSP_4:def 2;
  end;
