I am studying the following theorem from Morris Hirsch's second paper on systems of differential equations which are competitive or cooperative:
Let $V \subset \mathbb{R}^n$ be on open set and $G:\mathbb{R} \times V \rightarrow \mathbb{R}^n$ a continuous map such that $G^i(t,x^1,\cdots,x^m)$ is a non decreasing in $x^k$, for all $k \neq i$. Let $$\xi,\eta : [a,b] \rightarrow V $$ be solutions of $$\frac{dx}{dt}= G(t,x)$$ such that $\xi(a) < \eta(a)$ (resp. $\xi(a) \leq \eta(a)$). Then $\xi(t) < \eta(t)$ ( resp. $\xi(t) \leq \eta(t)$) for all $t \in [a,b]$.
I would love to find a proof but cannot find one anywhere or figure it out. Can anyone suggest where to begin?