
theorem ThScDM1:
  for L being Z_Lattice, v, u being Vector of DivisibleMod(L) holds
  (ScProductDM(L)).(-v, u) = -(ScProductDM(L)).(v, u) &
  (ScProductDM(L)).(u, -v) = -(ScProductDM(L)).(u, v)
  proof
    let L be Z_Lattice, v, u be Vector of DivisibleMod(L);
    thus A1: (ScProductDM(L)).(-v, u) = (ScProductDM(L)).((-1.(INT.Ring))*v, u)
    by ZMODUL01:2
    .= (-1) * (ScProductDM(L)).(v, u) by ThSPDM1
    .= -(ScProductDM(L)).(v, u);
    thus (ScProductDM(L)).(u, -v) = (ScProductDM(L)).(-v, u) by ThSPDM1
    .= -(ScProductDM(L)).(u, v) by A1,ThSPDM1;
  end;
