
theorem ThDB2:
  for L being RATional positive-definite Z_Lattice,
  I being Basis of EMLat(L), v being Vector of DivisibleMod(L)
  st v in DualBasis(I)
  holds v is Dual of L
  proof
    let L be RATional positive-definite Z_Lattice,
    I be Basis of EMLat(L), v be Vector of DivisibleMod(L) such that
    A1: v in DualBasis(I);
    consider u be Vector of EMLat(L) such that
    A2: u in I & (ScProductDM(L)).(u, v) = 1 &
    (for w being Vector of EMLat(L) st w in I & u <> w
    holds (ScProductDM(L)).(w, v) = 0) by A1,defDualBasis;
    reconsider J = I as Basis of EMbedding(L) by ThELEM1;
    for w being Vector of DivisibleMod(L) st w in J holds
    (ScProductDM(L)).(v, w) in INT.Ring
    proof
      let w be Vector of DivisibleMod(L) such that
      B1: w in J;
      per cases;
      suppose u <> w;
        then (ScProductDM(L)).(w, v) = 0 by A2,B1;
        then (ScProductDM(L)).(v, w) = 0 by ZMODLAT2:6;
        hence thesis;
      end;
      suppose u = w;
        then (ScProductDM(L)).(v, w) = 1 by A2,ZMODLAT2:6;
        hence thesis;
      end;
    end;
    hence thesis by LmDE21;
  end;
