
theorem
  for L being Z_Lattice,
  v being Vector of DivisibleMod(L), I being Basis of EMbedding(L)
  st for u being Vector of DivisibleMod(L) st u in I
  holds (ScProductDM(L)).(v, u) = 0
  holds v = 0.DivisibleMod(L)
  proof
    let L be Z_Lattice,
    v be Vector of DivisibleMod(L), I be Basis of EMbedding(L)
    such that
    A1: for u being Vector of DivisibleMod(L) st u in I
    holds (ScProductDM(L)).(v, u) = 0;
    for u being Vector of DivisibleMod(L)
    holds (ScProductDM(L)).(v, u) = 0 by A1,LmDE22;
    hence thesis by ThSPDM1;
  end;
