reserve a,b,i,j,k,l,m,n for Nat;

theorem S1:
  for f be complex-valued FinSequence holds Sum (f|(1 qua Nat)) = f.(1 qua Nat)
  proof
    let f be complex-valued FinSequence;
    per cases;
    suppose not f is empty; then
      f | 1 = <*f.1*> by FINSEQ_5:20;
      hence thesis;
    end;
    suppose
      B1: f is empty; then
      Sum (f| 1) = 0
      .= f.1 by B1;
      hence thesis;
    end;
  end;
