
theorem LmSumScDM14:
  for L being Z_Lattice, v, u being Vector of DivisibleMod(L),
      l being Linear_Combination of {u}
  holds SumSc(v, l) = (ScProductDM(L)).(v, l.u * u)
  proof
    let L be Z_Lattice, v, u be Vector of DivisibleMod(L),
        l be Linear_Combination of {u};
    per cases by ZFMISC_1:33,VECTSP_6:def 4;
    suppose Carrier(l) = {};
      then A2: l = ZeroLC(DivisibleMod(L)) by VECTSP_6:def 3;
      hence SumSc(v, l) = 0.F_Real by LmSumScDM11
      .= (ScProductDM(L)).(v, 0.DivisibleMod(L)) by ZMODLAT2:14
      .= (ScProductDM(L)).(v, 0.INT.Ring * u) by VECTSP_1:14
      .= (ScProductDM(L)).(v, l.u * u) by A2,VECTSP_6:3;
    end;
    suppose Carrier(l) = {u};
      then consider F be FinSequence of DivisibleMod(L) such that
      A3: F is one-to-one & rng F = {u} & SumSc(v, l) = Sum(ScFS(v, l, F))
      by defSumScDM;
      F = <* u *> by A3,FINSEQ_3:97;
      then ScFS(v, l, F) = <* (ScProductDM(L)).(v, l.u * u) *> by ThScFSDM3;
      hence thesis by A3,RLVECT_1:44;
    end;
  end;
