reserve i,j,k,n for Nat;
reserve x,y,z for Tuple of n, BOOLEAN;
reserve m,k,k1,k2 for FinSequence of NAT;

theorem Th38:
  for n being non zero Nat,lk being Nat,
Key being Matrix of lk,6,NAT, k being Nat holds IDEA_P_F(Key,n,k) is
  FuncSeq-like FinSequence
proof
  let n be non zero Nat;
  let lk be Nat;
  let Key be Matrix of lk,6,NAT;
  let k be Nat;
  set p = Seg(k+1) --> MESSAGES;
A1: dom p = Seg(k+1) by FUNCOP_1:13;
  reconsider p as FinSequence;
A2: for i being Nat st i in dom IDEA_P_F(Key,n,k)
       holds IDEA_P_F(Key,n,k).i in Funcs(p.i, p.(i+1))
  proof
    let i be Nat;
    assume
A3: i in dom IDEA_P_F(Key,n,k);
    then
A4: i in Seg len IDEA_P_F(Key,n,k) by FINSEQ_1:def 3;
    then i in Seg k by Def17;
    then
A5: i <= k by FINSEQ_1:1;
    then
A6: i <= k+1 by NAT_1:12;
    1 <= (i+1) & (i+1) <= k+1 by A5,NAT_1:12,XREAL_1:6;
    then (i+1) in Seg (k+1) by FINSEQ_1:1;
    then
A7: p.(i+1) = MESSAGES by FUNCOP_1:7;
    1 <= i by A4,FINSEQ_1:1;
    then i in Seg (k+1) by A6,FINSEQ_1:1;
    then
A8: p.i = MESSAGES by FUNCOP_1:7;
    IDEA_P_F(Key,n,k).i = IDEA_P(Line(Key,i),n) by A3,Def17;
    hence thesis by A8,A7,FUNCT_2:9;
  end;
  len p = k+1 by A1,FINSEQ_1:def 3;
  then len p = len IDEA_P_F(Key,n,k)+1 by Def17;
  hence thesis by A2,FUNCT_7:def 8;
end;
