Proof
This lemma is known in the case \(Y_{o}=0\) ( see Seibert17-18)
Suppose now, \(Y_{o}\neq 0\), then take \(Z=Y-Y_{o}\),\(g\left(X,Z\right)=\ G\left(X,Z+Y_{o}\right)\),
and apply the known result for the new system
\begin{equation} \left\{\begin{matrix}\dot{X}=F(X),\\ \dot{Z}=G\left(X,Z+Y_{o}\right)\\ \end{matrix}\right.\ \nonumber \\ \end{equation}
Lemma | 3.4 (see for example Farina ; Luenberger19, 20)
The nonnegative octant\(\mathbb{R}_{+}^{n}=\left\{x=\left(x_{1},\ldots,x_{i},\ldots,x_{n}\right)\in\mathbb{R}^{n}/x\geq 0\right\}\)is a positively invariant region (i.e. a trajectory that starts in the nonnegative orthant remains there for t\(\geq 0\)) for the dynamical system
\begin{equation} {\dot{x}}_{i}=f_{i}\left(x_{1},\ldots,x_{i},\ldots{,x}_{n}\right),\ i=1,2,\ldots,n\nonumber \\ \end{equation}
if and only if:
\(f_{i}(x_{1}\geq 0,\ldots x_{i}=0,\ldots,x_{n}\geq 0)\) \(\geq\) 0,\(\forall i\in\left[1,n\right]\)