Fibonacci members and powers of the golden ratio

Hello. I searched for a topic for the same question, but my queries were fruitless, although they returned some similar questions. If i have missed it, please lock this one, and redirect me to the proper thread.

It's not a homework assignment, but a personal inquiry due to some home studies I'm performing. I have a little background in maths, but haven't done anything of the sorts in quite a while ( > 4 years), and was looking for some confirmation on my work. Anyhow, here's the problem.

Prove that th Fibonacci number satisfies the equality

Where is the golden ratio and is its conjugate.

What I've done:

Proof by induction.

Base cases:

Inductive hypothesis:

Suppose that it holds for [LaTeX ERROR:
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The same for .

From that we trivially arrive at

Which is what we were trying to prove. Can you please confirm my reasoning? I'm not sure if I'm allowed such an assumption in my inductive hypothesis, and also my reasoning looks a little circular. If there's a gap, can you please point it out to me, as well as any hints at getting the proper solution?