reserve n, k, r, m, i, j for Nat;

theorem Th64:
  for n being Nat holds OddFibs (2 * n + 3) = OddFibs (2 * n + 1)
  ^ <* Fib (2 * n + 3) *>
proof
  defpred P[Nat] means OddFibs (2 * $1 + 3) = OddFibs (2 * $1 + 1) ^ <* Fib (2
  * $1 + 3) *>;
  let n be Nat;
A1: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    reconsider ARR = {[1,FIB.(2 * k + 5)]} as FinSubsequence by Th17;
    assume P[k];
    set LEFTk = OddFibs (2 * (k + 1) + 3);
    set RIGHTk = OddFibs (2 * (k + 1) + 1) ^ <* Fib (2 * (k + 1) + 3)*>;
    reconsider RS = FIB | (OddNAT /\ Seg (2 * k + 3)) as FinSubsequence;
    set RR = Shift(ARR,2*k+4);
A2: 2 * k + 4 > 2 * k + 3 by XREAL_1:6;
    dom RS c= OddNAT /\ Seg (2 * k + 3) & OddNAT /\ Seg (2 * k + 3) c= Seg
    (2 * k + 3) by RELAT_1:58,XBOOLE_1:17;
    then consider p1 being FinSequence such that
A3: RS c= p1 and
A4: dom p1 = Seg (2 * k + 4) by A2,Th19,XBOOLE_1:1;
A5: ex p2 being FinSequence st ARR c= p2 by Th20;
    1 + (2 * k + 4) = 2 * k + 5;
    then
A6: RR = {[2 * k + 5,FIB.(2 * k + 5)]} by Th18;
    len p1 = 2 * k + 4 by A4,FINSEQ_1:def 3;
    then consider RSR being FinSubsequence such that
A7: RSR = RS \/ RR and
A8: (Seq RS) ^ (Seq ARR) = Seq RSR by A3,A5,VALUED_1:64;
    RIGHTk = Seq (FIB | (OddNAT /\ Seg (2 * k + 3))) ^ <* FIB.(2 * k + 5)
    *> by Def2
      .= Seq (RSR) by A8,FINSEQ_3:157
      .= LEFTk by A7,A6,Th63;
    hence thesis;
  end;
A9: P[0] by Th22,Th60,Th61;
  for k being Nat holds P[k] from NAT_1:sch 2(A9,A1);
  hence thesis;
end;
