Interpreting another proposition full of symbols Could someone help me interpret the following proposition full of symbols? I've been struggling to comprehend it. Thanks in advance.
Proposition: Suppose that $f:\mathbb{R^n} \rightarrow \mathbb{R^n}, g:\mathbb{R^n} \rightarrow \mathbb{R}$ is a positive function, and $\phi$ is the flow of the differential equation $\dot{x}=f(x)$. If the family of solutions of the family of initial value problems $$\dot{y} = g(\phi(y,\xi)),$$ $$y(0)=0,$$ with parameter $\xi \in \mathbb{R^n}$, is given by $\rho: \mathbb{R} \times \mathbb{R^n} \rightarrow \mathbb{R}$, then $\psi$, defined by $\psi(t,\xi)=\phi(\rho(t,\xi),\xi)$ is the flow of the differential equation $\dot{x}=g(x)f(x)$.
 A: A flow for the ODE $\dot{x} = f(x)$ is a function $\phi: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ such that $\phi(0,x_0) = x_0$ for all $x_0$ and the function $t \mapsto \phi(t,x_0)$ is a solution of the ODE (passing through $x_0$, of course).
Different folks use different notations for the flow, but the same idea.
To simplify notation, I use $\dot{x}$ to signify the time derivative of $x$.
In particular, we have $\dot{\phi}(t,x_0) = f(\phi(t,x_0))$.
In addition, since $t \mapsto \rho(t,\xi)$ is the solution to the differential equation $\dot{y} = g(\phi(y,\xi))$ subject to $y(0) = 0$, we see that
$\rho(0,\xi) = 0$ for all $\xi$,and also $\dot{\rho}(t,\xi) = g(\phi(\rho(t,\xi),\xi))$.
The goal is to show that $\psi$ is a flow for the ODE $\dot{x} = g(x)f(x)$. This means we must show $\psi(0,\xi) = \xi$ for all $\xi$ and that $t \mapsto \psi(t,\xi)$ satisfies the ODE.
We have $\psi(0,\xi) = \phi(\rho(0,\xi),\xi)$, and from above we note that $\rho(0,\xi) = 0$, so $\psi(0,\xi) = \phi(0,\xi) = \xi$ and so the first condition is satisfied.
Now using the chain rule note that
\begin{eqnarray}
\dot{\psi}(t,\xi) &=& \dot{\phi}(\rho(t,\xi),\xi) \dot{\rho}(t,\xi) \\
&=& f(\phi(\rho(t,\xi),\xi)) g(\phi(\rho(t,\xi),\xi)) \\
&=& f(\psi(t,\xi)) g(\psi(t,\xi))
\end{eqnarray}
Hence $t \mapsto \psi(t,\xi)$ satisfies the ODE.
