
theorem LmDE1:
  for L being Z_Lattice, v, u being Dual of L holds
  v + u is Dual of L
  proof
    let L be Z_Lattice, v, u be Dual of L;
    for x being Vector of DivisibleMod(L) st x in EMbedding(L) holds
    (ScProductDM(L)).(v + u, x) in INT.Ring
    proof
      let x be Vector of DivisibleMod(L) such that
      B1: x in EMbedding(L);
      (ScProductDM(L)).(v, x) in INT.Ring by B1,defDualElement;
      then reconsider iv = (ScProductDM(L)).(v, x) as Element of INT.Ring;
      (ScProductDM(L)).(u, x) in INT.Ring by B1,defDualElement;
      then reconsider iu = (ScProductDM(L)).(u, x) as Element of INT.Ring;
      set iiv = iv;
      set iiu = iu;
      (ScProductDM(L)).(v + u, x)
      = (ScProductDM(L)).(v, x) + (ScProductDM(L)).(u, x) by ZMODLAT2:6
      .= iv + iu;
      hence thesis;
    end;
    hence thesis by defDualElement;
  end;
