theorem Th35:
  for n be positive Nat holds
    Sum(Newton_Coeff n) = Sum(((Newton_Coeff n)|n)/^1) + 2
  proof
    let n be positive Nat;
    Sum(Newton_Coeff n) = Sum(((Newton_Coeff n)|n)/^1) +
      (Newton_Coeff n).1 + (Newton_Coeff n).(n+1) by Th34
    .= Sum(((Newton_Coeff n)|n)/^1) + (Newton_Coeff n).1 + 1 by Th32
    .= Sum(((Newton_Coeff n)|n)/^1) + 1 + 1 by Th33;
    hence thesis;
  end;
