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

theorem
  for n being Nat holds Sum OddFibs (2 * n + 1) = Fib (2 * n + 2)
proof
  defpred P[Nat] means Sum OddFibs (2 * $1 + 1) = Fib (2 * $1 + 2);
  let n be Nat;
A1: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    reconsider EE = OddFibs (2 * k + 1) as FinSequence of REAL
          by FINSEQ_2:24,NUMBERS:19;
    assume
A2: P[k];
    Sum OddFibs (2 * (k + 1) + 1) = Sum ((OddFibs (2 * k + 1)) ^ <*(Fib (2
    * k + 3) qua Element of NAT)*>) by Th64
      .= Sum EE + Fib (2 * k + 3) by RVSUM_1:74
      .= Fib (2 * k + 4) by A2,Th26;
    hence thesis;
  end;
A3: P[0] by Th21,Th60,RVSUM_1:73;
  for n being Nat holds P[n] from NAT_1:sch 2(A3, A1);
  hence thesis;
end;
