
theorem
  for L being Z_Lattice, v, u, w being Vector of DivisibleMod(L) holds
  (ScProductDM(L)).(v, u + w)
  = (ScProductDM(L)).(v, u) + (ScProductDM(L)).(v, w)
  proof
    let L be Z_Lattice, v, u, w be Vector of DivisibleMod(L);
    thus (ScProductDM(L)).(v, u + w)
    = (ScProductDM(L)).(u + w, v) by ThSPDM1
    .= (ScProductDM(L)).(u, v) + (ScProductDM(L)).(w, v) by ThSPDM1
    .= (ScProductDM(L)).(u, v) + (ScProductDM(L)).(v, w) by ThSPDM1
    .= (ScProductDM(L)).(v, u) + (ScProductDM(L)).(v, w) by ThSPDM1;
  end;
