
theorem PPN:
  for f be Complex_Sequence, n be Nat st f.n = 0 holds
    (Partial_Product f).n = 0
  proof
    let f be Complex_Sequence, n be Nat such that
    A1: f.n = 0;
    per cases;
    suppose n = 0;
      hence thesis by A1,PP;
    end;
    suppose n > 0; then
      reconsider m = n -1 as Nat;
      (Partial_Product f).(m+1) = (Partial_Product f).m *f.(m+1) by PP;
      hence thesis by A1;
    end;
  end;
