
theorem
  for a being non empty positive at_most_one FinSequence of REAL,
  f being non empty FinSequence of NAT st
  f = OnlinePacking(a, NextFit(a)) holds
  for j being Nat st j in rng f holds SumBin (a, f, {j}) <= 1
  proof
    let a be non empty positive at_most_one FinSequence of REAL,
    f be non empty FinSequence of NAT;

    assume HL00: f = OnlinePacking(a, NextFit(a));

    let j be Nat;

    assume L002: j in rng f;

    set h = OnlinePackingHistory(a, NextFit(a));

    reconsider i = len h as Nat;

    L200: 1 <= i by NAT_1:14;

    i <= len a by defPackHistory;
    hence SumBin (a, f, {j}) <= 1 by L200,L002,HL00,NF890;
  end;
