Catalan's Identity/Proof 2

Theorem
Let $F_k$ be the $k$th Fibonacci number.

Then:


 * $\displaystyle F_n^2 - F_{n-r} F_{n+r} = \left( -1 \right) ^ {n-r} F_r^2$.

Proof
Proof by induction:

For all $n,r \in \N_{>0}$ where $n > r$, let $P \left({n,r}\right)$ be the proposition:
 * $\displaystyle F_n^2 - F_{n-r} F_{n+r} = \left( -1 \right) ^ {n-r} F_r^2$.

Basis for the Induction
$n = 1$ yields no suitable $r$, so we look at $n = 2$ instead, which only gives us $r = 1$.

$P \left({2,1}\right)$ is true:
 * $F_2^2 - F_3 F_1 = 1^2 - 2 \times 1 = -1 = -1 \times F_1^2$

$n = 3$ gives us only $r = 1$ and $r = 2$.

$P \left({3,1}\right)$ is true:
 * $F_3^2 - F_2 F_4 = 2^2 - 1 \times 3 = 1 = 1 \times F_1^2$

$P \left({3,2}\right)$ is true:
 * $F_3^2 - F_1 F_5 = 2^2 - 1 \times 5 = -1 = -1 \times F_2^2$

This is our basis for the induction.

First Induction Hypothesis
Now we need to show that, if $P \left({n,r}\right)$ is true for all $r$, where $n > 3$, then it logically follows that $P \left({n+1,r}\right)$ is true for all $r$.

So this is our induction hypothesis:
 * $\displaystyle \forall r < n : F_n^2 - F_{n-r} F_{n+r} = \left( -1 \right) ^ {n-r} F_r^2$

Then we need to show:
 * $\displaystyle \forall r < n : F_{n+1}^2 - F_{n-r+1} F_{n+r+1} = \left( -1 \right) ^ {n-r+1} F_r^2$

Induction Step
This is our induction step:

It will again be a proof by induction.

Basis for the Induction
When $r = 1$:

So $P\left({n+1,1}\right)$ holds.

This is our basis for the induction.

Second Induction Hypothesis
Now we need to show that, if $P \left({n+1,r}\right)$ is true, where $2 < r < n$, then it logically follows that $P \left({n+1,r+1}\right)$ is true.

So this is our second induction hypothesis:
 * $\displaystyle F_{n+1}^2 - F_{n-r+1} F_{n+r+1} = \left( -1 \right) ^ {n-r+1} F_r^2$

Then we need to show:
 * $\displaystyle F_{n+1}^2 - F_{n-r} F_{n+r+2} = \left( -1 \right) ^ {n-r} F_{r+1}^2$

Induction Step
This is our induction step:

So $P \left({n+1,r}\right) \implies P \left({n+1,r+1}\right)$ and the result follows by the Principle of Mathematical Induction.

So $P \left({n,r}\right) \implies P \left({n+1,r}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\displaystyle F_n^2 - F_{n-r} F_{n+r} = \left( -1 \right) ^ {n-r} F_r^2$.