 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 Th31:
  for M1, M2 being Matrix of the carrier of V1 st len M1 = len M2
  holds Sum Sum M1 + Sum Sum M2 = Sum Sum(M1 ^^ M2)
  proof
    let M1, M2 be Matrix of the carrier of V1 such that
    A1: len M1 = len M2;
    len Sum M1 = len M1 by MATRLIN:def 6
    .= len Sum M2 by A1,MATRLIN:def 6;
    hence Sum Sum M1 + Sum Sum M2 = Sum (Sum M1 + Sum M2) by Th30
    .= Sum Sum(M1 ^^ M2) by Th29;
  end;
