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;
