A property of an Archimedean cyclically ordered group

Question

I am trying to prove the following property:

Lemma (?). For any non-negative element $x$ of an Archimedean cyclically ordered group there is a non-negative element $y$ such that $x + y$ is non-positive.

Proof (?):

For a non-negative element $x$ of an Archimedean cyclically ordered group:

1. Case $x = -x$:
   • $-x$ is not negative;
   • $x + (-x) = 0$ is not positive;
2. Case $[0, x, -x]$:
   • Applying Archimedean property: not $[0, nx, -x]$ for some $n \in \mathbb N; n \gt 1$;
     • Case $nx = 0$: $nx$ is not positive;
     • Case $nx = -x$: $nx$ is negative;
     • Case $[0, -x, nx]$:
       • Applying formula $[0, a, b] \iff [0, -b, -a]$: $[0, -x, nx] \iff [0, -nx, x]$;
       • Applying transitivity to $[0, -nx, x]$, $[0, x, -x]$, and $[0, -x, nx]$: $[0, -nx, nx]$;
       • $nx$ is negative.
   • For all the cases from 2: $x$ is positive, but $nx$ is not positive for some $n \in \mathbb N, n > 1$. Therefore, there is $k \in \mathbb N; k \lt n$: $kx$ is positive, but $(1 + k)x = x + kx$ is not positive.

Could somebody verify if it is correct, please?

An element $x$ of a cyclically ordered group is positive (negative) iff $[0, x, -x]$ ($[0, -x, x]$).