
theorem LmSumScDM11:
  for L being Z_Lattice, v being Vector of DivisibleMod(L) holds
  SumSc(v, ZeroLC(DivisibleMod(L))) = 0.F_Real
  proof
    let L be Z_Lattice, v be Vector of DivisibleMod(L);
    consider F be FinSequence of DivisibleMod(L) such that
    A1: F is one-to-one & rng F = Carrier(ZeroLC(DivisibleMod(L))) &
    SumSc(v,ZeroLC(DivisibleMod(L)))
    = Sum(ScFS(v, ZeroLC(DivisibleMod(L)),F)) by defSumScDM;
    F = {} by A1,VECTSP_6:def 3;
    then len(ScFS(v, ZeroLC(DivisibleMod(L)),F)) = 0 by defScFSDM;
    hence thesis by A1,RLVECT_1:75;
  end;
