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

theorem Th20:
  for m being Matrix of REAL holds len Sum m = len m & for i st i
  in Seg len m holds (Sum m).i=Sum Line(m,i)
proof
  let m be Matrix of REAL;
  thus len Sum m = len m by Def1;
  thus for k st k in Seg len m holds (Sum m).k=Sum Line(m,k)
  proof
    let k such that
A1: k in Seg len m;
A2: k in dom m by A1,FINSEQ_1:def 3;
    k in Seg len Sum m by A1,Def1;
    then k in dom Sum m by FINSEQ_1:def 3;
    hence (Sum m).k = Sum(m.k) by Def1
      .= Sum(Line(m,k)) by A2,MATRIX_0:60;
  end;
end;
