Prove of the Cassini-Samson Identity

The Cassini-Samson Identity is an identity that involves the Fibonnaci numbers, it states that Fn1Fn+1F2n=(1)n for nN.
We thus prove the theorem by induction:
The base case n=1 is easily verified. 
Next, assume n=k and prove for n=k+1:
FkFk+2F2k+1
=Fk(Fk+Fk+1)F2k+1
=F2k+Fk+1(FkFk+1)
=F2kFk+1Fk1=(1)k=(1)k+1. Which is thus proved.