I am studying "Ordinary Differential Equations and Dynamical Systems" written by Gerald Teschl and I found a below proposition in this book.
(proposition) Consider a first order autonomous equation $x' = f(x)$ with $x(0) = x_0$, assume that $f$ is in $C(\Bbb R)$ (i.e., $f$ is continuously differentiable on $\Bbb R$)
If we have a solution $Φ(t)$ with $Φ(0)=x_0$, then the solution $Ψ(t)$ with $Ψ(t_0)=x_0$ is given by $Ψ(t)= Φ(t - t_0)$.
I can't prove this proposition. Can you give me an answer for this?