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 Th31:
  for M1,M2 be 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 Def6
    .= len Sum M2 by A1,Def6;
  hence Sum Sum M1 + Sum Sum M2 = Sum (Sum M1 + Sum M2) by Th30
    .= Sum Sum(M1 ^^ M2) by Th29;
end;
