
theorem NF950:
  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)) & rng f = Seg k holds
  (for j being Nat st 1 <= j & j <= k div 2 holds
  SumBin (a, f, {2 * j - 1}) + SumBin (a, f, {2 * j}) > 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
    BN00: rng f = Seg k;

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

    let j be Nat;

    assume that
    L202: 1 <= j and
    L203: j <= k div 2;

    k <> 0 by BN00;
    then per cases by NAT_1:23;
    suppose k = 1;
      hence thesis by NAT_D:27,L202,L203;
    end;
    suppose L60: 2 <= k;

      set i = len h;

      L220: len h = len a by defPackHistory;

      1 <= len h by NAT_1:14;
      hence SumBin (a, f, {2 * j - 1}) + SumBin (a, f, {2 * j}) > 1
      by L220,NF830,L60,L202,L203,HL00,BN00;
    end;
  end;
