defpred S₁[ Nat] means Sum (OddFibs ((2 * $1) + 1)) = Fib ((2 * $1) + 2);
let n be Nat; :: thesis: Sum (OddFibs ((2 * n) + 1)) = Fib ((2 * n) + 2)
A1: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
reconsider EE = OddFibs ((2 * k) + 1) as FinSequence of REAL by FINSEQ_2:24, NUMBERS:19;
assume A2: S₁[k] ; :: thesis: S₁[k + 1]
Sum (OddFibs ((2 * (k + 1)) + 1)) = Sum ((OddFibs ((2 * k) + 1)) ^ <*(Fib ((2 * k) + 3))*>) by Th64
.= (Sum EE) + (Fib ((2 * k) + 3)) by RVSUM_1:74
.= Fib ((2 * k) + 4) by A2, Th26 ;
hence S₁[k + 1] ; :: thesis: verum

end;

A3: S₁[ 0 ] by Th21, Th60, RVSUM_1:73;
for n being Nat holds S₁[n] from NAT_1:sch 2(A3, A1);
hence Sum (OddFibs ((2 * n) + 1)) = Fib ((2 * n) + 2) ; :: thesis: verum