
theorem NF505:
  for a being non empty FinSequence of REAL,
  Alg being Function of [:REAL, NAT*:],NAT,
  h being non empty FinSequence of NAT* st
  h = OnlinePackingHistory(a, Alg) holds
  (for i being Nat st 1 <= i & i <= len a holds
  h . i is FinSequence of NAT)
  proof
    let a be non empty FinSequence of REAL,
    Alg be Function of [:REAL, NAT*:],NAT,
    h be non empty FinSequence of NAT*;

    assume HC00: h = OnlinePackingHistory(a, Alg);

    let i be Nat;

    assume that
    L000: 1 <= i and
    L001: i <= len a;
    L023: i <= len h by HC00,defPackHistory,L001;

    reconsider i as Element of dom h by L000,L023,FINSEQ_3:25;

    h . i is FinSequence of NAT;
    hence thesis;
  end;
