theorem Th14:
  for M be 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 Def6;
    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,Def6
      .= 0.V1 by A2,RLVECT_1:43;
  end;
  hence thesis by Th11;
end;
