
theorem
  for L being INTegral positive-definite Z_Lattice holds
  EMLat(L) is Z_Sublattice of DualLat(L)
  proof
    let L be INTegral positive-definite Z_Lattice;
    A1: EMLat(L) is Submodule of DivisibleMod(L) by ZMODLAT2:20;
    A2: DualLat(L) is Submodule of DivisibleMod(L) by ThDL2;
    for v being Vector of DivisibleMod(L) st v in EMLat(L)
    holds v in DualLat(L)
    proof
      let v be Vector of DivisibleMod(L) such that
      B1: v in EMLat(L);
      set I = the Basis of EMLat(L);
      reconsider J = I as Basis of EMbedding(L) by ThELEM1;
      for u being Vector of DivisibleMod(L) st u in J
      holds (ScProductDM(L)).(v, u) in INT.Ring
      proof
        let u be Vector of DivisibleMod(L) such that
        C1: u in J;
        reconsider vv = v as Vector of EMLat(L) by B1;
        reconsider uu = u as Vector of EMLat(L) by C1;
        the ModuleStr of EMLat(L) = EMbedding(L) by ZMODLAT2:28;
        then (ScProductDM(L)).(v, u) = (ScProductEM(L)).(vv, uu) by ZMODLAT2:8
        .= <; vv, uu ;> by ZMODLAT2:def 4;
        hence thesis by ZMODLAT1:def 5;
      end;
      hence thesis by LmDE21,ThDL1;
    end;
    then EMLat(L) is Submodule of DualLat(L) by A1,A2,ZMODUL01:44;
    then A3: the carrier of EMLat(L) c= the carrier of DualLat(L) &
    0.EMLat(L) = 0.DualLat(L) &
    the addF of EMLat(L) = (the addF of DualLat(L)) ||
    the carrier of EMLat(L) &
    the lmult of EMLat(L) = (the lmult of DualLat(L)) |
    [:the carrier of INT.Ring, the carrier of EMLat(L):] by VECTSP_4:def 2;
    A4: [:the carrier of EMLat(L), the carrier of EMLat(L):]
    c= [:the carrier of DualLat(L), the carrier of DualLat(L):]
    by A3,ZFMISC_1:96;
    (the scalar of DualLat(L)) || the carrier of EMLat(L)
    = ((ScProductDM(L)) || the carrier of DualLat(L)) ||
    the carrier of EMLat(L) by defDualLat
    .= (ScProductDM(L)) || the carrier of EMLat(L) by A4,FUNCT_1:51
    .= (ScProductDM(L)) || (rng MorphsZQ(L)) by ZMODLAT2:def 4
    .= ScProductEM(L) by ZMODLAT2:7
    .= the scalar of EMLat(L) by ZMODLAT2:def 4;
    hence thesis by A3,ZMODLAT1:def 9;
  end;
