let n be Element of NAT ; :: thesis: Sum (EvenFibs ((2 * n) + 2)) = (Fib ((2 * n) + 3)) - 1
defpred S₁[ Nat] means Sum (EvenFibs ((2 * $1) + 2)) = (Fib ((2 * $1) + 3)) - 1;
A1: S₁[ 0 ] by Th24, Th57, RVSUM_1:103;
A2: for k being Element of NAT st S₁[k] holds
S₁[k + 1]

proof

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
reconsider EE = EvenFibs (2 * (k + 1)) as FinSequence of REAL ;
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
Sum (EvenFibs ((2 * (k + 1)) + 2)) = Sum ((EvenFibs (2 * (k + 1))) ^ <*(Fib ((2 * (k + 1)) + 2))*>) by Th61
.= (Sum EE) + (Fib ((2 * (k + 1)) + 2)) by RVSUM_1:104
.= ((Fib ((2 * k) + 3)) + (Fib ((2 * k) + 4))) - 1 by A3
.= (Fib ((2 * k) + 5)) - 1 by Th29 ;
hence S₁[k + 1] ; :: thesis: verum

end;

for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A1, A2);
hence Sum (EvenFibs ((2 * n) + 2)) = (Fib ((2 * n) + 3)) - 1 ; :: thesis: verum