Honsberger's Identity/Proof 1

Proof
From the initial definition of Fibonacci numbers, we have:
 * $F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3$

Proof by induction:

For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
 * $\forall m \in \Z_{>0} : F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$

Basis for the Induction
$\map P 1$ is the case:

and so $\map P 1$ is seen to hold.

$\map P 2$ is the case:

and so $\map P 2$ is seen to hold.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $\map P k$ and $\map P {k - 1}$ are true, where $k > 1$, then it logically follows that $\map P {k + 1}$ is true.

So this is our induction hypothesis:
 * $F_{m + k} = F_{m - 1} F_k + F_m F_{k + 1}$

and:
 * $F_{m + k - 1} = F_{m - 1} F_{k - 1} + F_m F_k$

from which it is to be shown:
 * $F_{m + k + 1} = F_{m - 1} F_{k + 1} + F_m F_{k + 2}$

Induction Step
This is our induction step:

So $\map P k \land \map P {k - 1} \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\forall m, n \in \Z_{>0} : F_{m + n} = F_{m - 1} F_n + F_m F_{n + 1}$