
theorem
  for L being Z_Lattice, v, u being Vector of DivisibleMod(L),
  a being Element of INT.Ring
  holds (ScProductDM(L)).(v, a*u) = a * (ScProductDM(L)).(v, u)
  proof
    let L be Z_Lattice, v, u be Vector of DivisibleMod(L),
    a be Element of INT.Ring;
    thus (ScProductDM(L)).(v, a*u) = (ScProductDM(L)).(a*u, v) by ThSPDM1
    .= a * (ScProductDM(L)).(u, v) by ThSPDM1
    .= a * (ScProductDM(L)).(v, u) by ThSPDM1;
  end;
