
theorem
  for a being non empty positive at_most_one FinSequence of REAL,
  f being non empty FinSequence of NAT, k being Nat st
  f = OnlinePacking(a, NextFit(a)) & k = card rng f holds
  k <= 2 * Opt a - 1
  proof
    let a be non empty positive at_most_one FinSequence of REAL,
    f be non empty FinSequence of NAT, k be Nat;

    assume that
    HL00: f = OnlinePacking(a, NextFit(a)) and
    BK00: k = card rng f;

    F200: k <= 2 * [/ Sum a \] - 1 by HL00,BK00,NF980;

    2 * [/ Sum a \] <= 2 * Opt a by NF330,XREAL_1:64;
    then 2 * [/ Sum a \] - 1 <= 2 * Opt a - 1 by XREAL_1:9;
    hence k <= 2 * Opt a - 1 by F200,XXREAL_0:2;
  end;
