Catalan's Identity/Proof 2

Proof
Proof by induction:

For all $n, r \in \N_{>0}$ where $n > r$, let $\map P {n, r}$ be the proposition:
 * ${F_n}^2 - F_{n - r} F_{n + r} = \paren {-1}^{n - r} {F_r}^2$

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

$\map P {2, 1}$ 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 $r = 2$.

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

$\map P {3, 2}$ 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 $\map P {n, r}$ 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:
 * $\forall r < n : {F_n}^2 - F_{n - r} F_{n + r} = \paren {-1}^{n - r} {F_r}^2$

Then we need to show:
 * $\forall r < n : {F_{n + 1} }^2 - F_{n - r + 1} F_{n + r + 1} = \paren {-1}^{n - r + 1} {F_r}^2$

Induction Step
This is our induction step:

It will again be a proof by induction.

Basis for the Induction
When $r = 1$:

So $\map P {n + 1, 1}$ holds.

This is our basis for the induction.

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

So this is our second induction hypothesis:
 * ${F_{n + 1} }^2 - F_{n - r + 1} F_{n + r + 1} = \paren {-1}^{n - r + 1} {F_r}^2$

Then we need to show:
 * ${F_{n + 1} }^2 - F_{n - r} F_{n + r + 2} = \paren {-1}^{n - r} {F_{r + 1} }^2$

Induction Step
This is our induction step:

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

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

Therefore:
 * $F_n^2 - F_{n - r} F_{n + r} = \paren {-1}^{n - r} {F_r}^2$