let n be Nat; :: thesis: ((Fib n) * (Fib (n + 2))) - ((Fib (n + 1)) ^2 ) = (- 1) |^ (n + 1)
defpred S₁[ Nat] means ((Fib $1) * (Fib ($1 + 2))) - ((Fib ($1 + 1)) ^2 ) = (- 1) |^ ($1 + 1);
A1: S₁[ 0 ] by NEWTON:10, PRE_FF:1;
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
A4: Fib k = (Fib (k + 2)) - (Fib (k + 1))

proof

(Fib (k + 2)) - (Fib (k + 1)) = ((Fib (k + 1)) + (Fib k)) - (Fib (k + 1)) by Th26
.= Fib k ;
hence Fib k = (Fib (k + 2)) - (Fib (k + 1)) ; :: thesis: verum

end;

A5: Fib (k + 2) = (Fib (k + 3)) - (Fib (k + 1))

proof

(Fib (k + 3)) - (Fib (k + 1)) = ((Fib (k + 2)) + (Fib (k + 1))) - (Fib (k + 1)) by Th27
.= Fib (k + 2) ;
hence Fib (k + 2) = (Fib (k + 3)) - (Fib (k + 1)) ; :: thesis: verum

end;

(- 1) |^ ((k + 1) + 1) = (- 1) * (((Fib k) * (Fib (k + 2))) - ((Fib (k + 1)) ^2 )) by A3, NEWTON:11
.= ((Fib (k + 1)) * (Fib ((k + 1) + 2))) - ((Fib ((k + 1) + 1)) ^2 ) by A4, A5 ;
hence S₁[k + 1] ; :: thesis: verum

end;

for n being Nat holds S₁[n] from NAT_1:sch 2(A1, A2);
hence ((Fib n) * (Fib (n + 2))) - ((Fib (n + 1)) ^2 ) = (- 1) |^ (n + 1) ; :: thesis: verum