 reserve x,y,z for object,
   i,j,k,l,n,m for Nat,
   D,E for non empty set;
 reserve M for Matrix of D;
 reserve L for Matrix of E;
 reserve k,t,i,j,m,n for Nat,
   D for non empty set;
 reserve V for free Z_Module;
 reserve a for Element of INT.Ring,
   W for Element of V;
 reserve KL1,KL2,KL3 for Linear_Combination of V,
   X for Subset of V;
 reserve V for finite-rank free Z_Module,
   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-rank free Z_Module,
   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 INT.Ring,
   KL for Linear_Combination of V1;

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;
