theorem Th14:
  for M being Matrix of m+1,0,the carrier of V1 holds Sum Sum M = 0.V1
  proof
    let M be Matrix of m+1,0,the carrier of V1;
    for k st k in dom Sum M holds (Sum M)/.k = 0.V1
    proof
      let k such that
      A1: k in dom Sum M;
      reconsider k1 = k as Element of NAT by ORDINAL1:def 12;
      len M = len Sum M by MATRLIN:def 6;
      then dom M = dom Sum M by FINSEQ_3:29;
      then M/.k1 in rng M by A1,PARTFUN2:2;
      then len(M/.k) = 0 by MATRIX_0:def 2;
      then
      A2: M/.k = <*>(the carrier of V1);
      thus (Sum M)/.k = Sum (M/.k) by A1,MATRLIN:def 6
      .= 0.V1 by A2,RLVECT_1:43;
    end;
    hence thesis by MATRLIN:11;
  end;
