reserve k,t,i,j,m,n for Nat,
  x,y,y1,y2 for object,
  D for non empty set;
reserve K for Field,
  V for VectSp of K,
  a for Element of K,
  W for Element of V;
reserve KL1,KL2,KL3 for Linear_Combination of V,
  X for Subset of V;
reserve s for FinSequence,
  V1,V2,V3 for finite-dimensional VectSp of K,
  f,f1,f2 for Function of V1,V2,
  g for Function of V2,V3,
  b1 for OrdBasis of V1,
  b2 for OrdBasis of V2,
  b3 for OrdBasis of V3,
  v1,v2 for Vector of V2,
  v,w for Element of V1;
reserve p2,F for FinSequence of V1,
  p1,d for FinSequence of K,
  KL for Linear_Combination of V1;

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;
