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 Th41:
  for n being non zero Nat,lk being Nat,
  Key being Matrix of lk,6,NAT, k being Nat holds k <> 0 implies
  IDEA_Q_F(Key,n,k) is FuncSequence of MESSAGES,MESSAGES
proof
  let n be non zero Nat, lk be Nat, Key be Matrix of lk
  ,6,NAT, r be Nat;
  assume
A1: r<>0;
  then 0+1 < r+1 by XREAL_1:6;
  then
A2: 1 <= r by NAT_1:13;
  0+1 < r+1 by A1,XREAL_1:6;
  then
A3: 1 <= r by NAT_1:13;
  r = len IDEA_Q_F(Key,n,r) by Def18;
  then 1 in Seg len IDEA_Q_F(Key,n,r) by A3,FINSEQ_1:1;
  then
A4: 1 in dom IDEA_Q_F(Key,n,r) by FINSEQ_1:def 3;
  len (IDEA_Q_F(Key,n,r)) <> 0 by A1,Def18;
  then not IDEA_Q_F(Key,n,r)=<*>MESSAGES;
  then
A5: firstdom IDEA_Q_F(Key,n,r) = proj1 (IDEA_Q_F(Key,n,r).1) by FUNCT_7:def 6
    .= proj1 (IDEA_Q(Line(Key,r-'1+1),n)) by A4,Def18
    .= dom IDEA_Q(Line(Key,r-'1+1),n)
    .= MESSAGES by FUNCT_2:def 1;
  r = len IDEA_Q_F(Key,n,r) by Def18;
  then r in Seg len IDEA_Q_F(Key,n,r) by A2,FINSEQ_1:1;
  then
A6: r in dom IDEA_Q_F(Key,n,r) by FINSEQ_1:def 3;
  len (IDEA_Q_F(Key,n,r)) <> 0 by A1,Def18;
  then not IDEA_Q_F(Key,n,r)=<*>MESSAGES;
  then
  lastrng IDEA_Q_F(Key,n,r) = proj2 (IDEA_Q_F(Key,n,r). len (IDEA_Q_F(Key
  ,n,r))) by FUNCT_7:def 7
    .= proj2 (IDEA_Q_F(Key,n,r).r) by Def18
    .= proj2 (IDEA_Q(Line(Key,r-'r+1),n)) by A6,Def18
    .= rng IDEA_Q(Line(Key,r-'r+1),n);
  then
A7: lastrng IDEA_Q_F(Key,n,r) c= MESSAGES by RELAT_1:def 19;
  IDEA_Q_F(Key,n,r) is FuncSequence by Th39;
  hence thesis by A7,A5,FUNCT_7:def 9;
end;
