defpred S₁[ Nat] means Sum (EvenFibs ((2 * $1) + 2)) = (Fib ((2 * $1) + 3)) - 1;
let n be Element of NAT ; :: thesis: Sum (EvenFibs ((2 * n) + 2)) = (Fib ((2 * n) + 3)) - 1
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 = EvenFibs (2 * (k + 1)) as FinSequence of REAL by FINSEQ_2:24, NUMBERS:19;
assume A2: S₁[k] ; :: thesis: S₁[k + 1]
Sum (EvenFibs ((2 * (k + 1)) + 2)) = Sum ((EvenFibs (2 * (k + 1))) ^ <*(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 S₁[k + 1] ; :: thesis: verum

end;

A3: S₁[ 0 ] by Th22, Th55, RVSUM_1:73;
for n being Nat holds S₁[n] from NAT_1:sch 2(A3, A1);
hence Sum (EvenFibs ((2 * n) + 2)) = (Fib ((2 * n) + 3)) - 1 ; :: thesis: verum