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

let k be Nat; :: thesis:
reconsider EE = OddFibs ((2 * k) + 1) as FinSequence of REAL by FINSEQ_2:24, NUMBERS:19;
assume A2: S₁[k] ; :: thesis:
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:

end;