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

theorem
  for n being Element of NAT holds Sum EvenFibs (2 * n + 2) = Fib (2 * n
  + 3) - 1
proof
  defpred P[Nat] means Sum EvenFibs (2 * $1 + 2) = Fib (2 * $1 + 3) - 1;
  let n be Element of NAT;
A1: for k being Nat st P[k] holds P[k+1]
  proof
    let k be Nat;
    reconsider EE = EvenFibs (2 * ( k + 1)) as FinSequence of REAL
                 by FINSEQ_2:24,NUMBERS:19;
    assume
A2: P[k];
    Sum EvenFibs (2 * (k + 1) + 2) = Sum ((EvenFibs (2 * ( k + 1)) qua
    FinSequence of NAT) ^ <*Fib (2 * (k + 1) + 2)*>) by Th59
      .= Sum EE + Fib (2 * (k + 1) + 2) by RVSUM_1:74
      .= Fib (2 * k + 3) + Fib (2 * k + 4) - 1 by A2
      .= Fib (2 * k + 5) - 1 by Th27;
    hence thesis;
  end;
A3: P[0] by Th22,Th55,RVSUM_1:73;
  for n being Nat holds P[n] from NAT_1:sch 2(A3, A1);
  hence thesis;
end;
