reserve D for non empty set,
  i,j,k for Nat,
  n,m for Nat,
  r for Real,
  e for real-valued FinSequence;

theorem Th24:
  for M be Matrix of m,0,REAL holds SumAll M = 0
proof
  let M be Matrix of m,0,REAL;
  per cases;
  suppose
    m = 0;
    then len M = 0 by MATRIX_0:def 2;
    hence thesis by Th23;
  end;
  suppose
A1: m > 0;
    len Sum M > 0 & for k be Nat st k in dom Sum M holds (Sum M).k = 0
    proof
      len M > 0 by A1,MATRIX_0:def 2;
      hence len Sum M > 0 by Def1;
      len M = len Sum M by Def1;
      then
A2:   dom M = dom Sum M by FINSEQ_3:29;
      hereby
        let k be Nat such that
A3:     k in dom Sum M;
        M.k in rng M by A2,A3,FUNCT_1:def 3;
        then len (M.k) = 0 by MATRIX_0:def 2;
        then
A4:     M.k = <*>REAL;
        thus (Sum M).k = Sum (M.k) by A3,Def1
          .= 0 by A4,RVSUM_1:72;
      end;
    end;
    hence SumAll M = Sum ((len Sum M) |-> 0) by Th1,Lm4
      .= (len Sum M) * 0 by RVSUM_1:80
      .= 0;
  end;
end;
