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;
