Goldbach's Theorem/Proof 2

Proof
Let $F_m$ and $F_n$ be Fermat numbers such that $m < n$.

Let $d = \gcd \set {F_m, F_n}$.

From the corollary to Product of Sequence of Fermat Numbers plus 2:
 * $F_m \divides F_n - 2$

But then:

But all Fermat numbers are odd, so:
 * $d \ne 2$

It follows that $d = 1$.

The result follows by definition of coprime.