
theorem LmDE2:
  for L being Z_Lattice, v being Dual of L, a being Element of INT.Ring
  holds a * v is Dual of L
  proof
    let L be Z_Lattice, v be Dual of L, a be Element of INT.Ring;
    for x being Vector of DivisibleMod(L) st x in EMbedding(L) holds
    (ScProductDM(L)).(a * v, 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)).(a * v, x) = a*iv by ZMODLAT2:6;
      hence thesis;
    end;
    hence thesis by defDualElement;
  end;
