Can the following lemma be proved?

Lemma (Rokach-Goldstein)

Let $x_i$ be a finite generalized Fibonacci sequence of positive integer numbers, $x_0,x_1,x_2,\ldots,x_m$ such that for every $2\le i\le m$, we have $x_i=x_{i-1}+x_{i-2}$, and where $x_0$ and $x_1$ are coprime to each other.

In this case, there exists some $n$, $0\le n\le m$, such that $x_n$ is coprime to the sum of all numbers in the sequence.