
theorem ThScDM6:
  for L being Z_Lattice, v being Vector of DivisibleMod(L) holds
  (ScProductDM(L)).(0.DivisibleMod(L), v) = 0 &
  (ScProductDM(L)).(v, 0.DivisibleMod(L)) = 0
  proof
    let L be Z_Lattice, v be Vector of DivisibleMod(L);
    thus A1: (ScProductDM(L)).(0.DivisibleMod(L), v)
    = (ScProductDM(L)).(v-v, v) by RLVECT_1:15
    .= (ScProductDM(L)).(v, v) + (ScProductDM(L)).(-v, v) by ThSPDM1
    .= (ScProductDM(L)).(v, v) - (ScProductDM(L)).(v, v) by ThScDM1
    .= 0;
    thus (ScProductDM(L)).(v, 0.DivisibleMod(L)) = 0 by A1,ThSPDM1;
  end;
