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

theorem Th59:
  for n being Element of NAT holds EvenFibs (2 * n + 2) = EvenFibs
  (2 * n) ^ <* Fib (2 * n + 2) *>
proof
  defpred P[Nat] means
EvenFibs (2 * $1 + 2) = EvenFibs (2 * $1) ^
  <* Fib (2 * $1 + 2) *>;
  let n be Element of NAT;
A1: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    reconsider ARR = {[1,FIB.(2*k+4)]} as FinSubsequence by Th17;
    assume P[k];
    set LEFTk = EvenFibs (2 * (k+1) + 2);
    set RIGHTk = EvenFibs (2 * (k+1)) ^ <*Fib (2 * (k+1) + 2)*>;
    reconsider RS = FIB | (EvenNAT /\ Seg (2 * k + 2)) as FinSubsequence;
    set RR = Shift(ARR,2*k+3);
A2: 2 * k + 3 > 2 * k + 2 by XREAL_1:6;
    dom RS c= EvenNAT /\ Seg (2 * k + 2) & EvenNAT /\ Seg (2 * k + 2) c=
    Seg (2 * k + 2) 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+3) by A2,Th19,XBOOLE_1:1;
A5: ex p2 being FinSequence st ARR c= p2 by Th20;
    1 + (2 * k + 3) = 2*k+4;
    then
A6: RR = {[2*k+4,FIB.(2*k+4)]} by Th18;
    len p1 = 2*k + 3 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 | (EvenNAT /\ Seg (2 * k + 2))) ^ <* FIB.(2 * k + 4
    ) *> by Def2
      .= Seq (RSR) by A8,FINSEQ_3:157
      .= LEFTk by A7,A6,Th58;
    hence thesis;
  end;
A9: P[0] by Th21,Th53,Th55,FINSEQ_1:34;
  for k being Nat holds P[k] from NAT_1:sch 2(A9,A1);
  hence thesis;
end;
