In flow group laws, why say that $\forall s \in D^{(p)}, \forall t \in D^{(\theta(s,p))}$ s.t. $s+t \in D^{(p)}$?

Question

In flow group laws, why say that $\forall s \in D^{(p)}, \forall t \in D^{(\theta(s,p))}$ s.t. $s+t \in D^{(p)}$?

Because this is what my notes say.

Isn't it then equivalent to say:

$\forall s \in D^{(p)}, \forall t \in D^{(p)}$

?

https://en.wikipedia.org/wiki/Flow_(mathematics)#Formal_definition

Also is the notion of $\in D^{\theta(s,p)}$ vs $\in D^{(p)}$ significant? Since the wikipedia page only says $s,t \in \mathbb{R}$.

$\theta:D \rightarrow M$ is the flow (continuous map).

$D$ is a flow domain. $D^{(p)}=\{ t \in \mathbb{R} : (t,p) \in D\}$, $\forall p \in M$.

Answer 1

It may be that $s,t\in D^{(p)}$ but $s+t\notin D^{(p)}$. Consider on $\Bbb R$ the flow generated by the differential equation $y'=y^2$. The solution with $y(0)=1$ is $y(t)=1/(1-t)$ and is defined on $(-\infty,1)$; $3/4$ is in the domain, but $3/4+3/4$ is not.