\exp\big( \phi_s(r,u) + \phi_s(r,\psi_r(t,u)) + \scal{\psi_s(r,\psi_r(t,u))}{X_s} \big).\end{aligned}$$ Note that the exponent on the right hand side is continuous in $u$ and that the same holds true for . By the same argument as in the proof of Lemma \[lem:uniqueness\] we conclude that $$\phi_s(t,u) + \scal{\psi_s(t,u)}{x} = \phi_s(r,u) + \phi_s(r,\psi_r(t,u)) + \scal{\psi_s(r,\psi_r(t,u))}{x},$$ for all $x \in D$. Since the linear hull of $D$ is $\R^d$ the semi-flow equations follow. Note that the terminal conditions $\psi_t(t,u) = u$ and $\phi_t(t,u) = 0$ are a simple consequence of $\cond{\exp(\scal{u}{X_t})}{\cF_t} = \exp(\scal{u}{X_t})$ and the uniqueness property from Lemma \[lem:uniqueness\].

Note that $s = 0$ is excluded from the semi-flow equations, since Condition \[def:full\_support\]does not apply to the initial value $X_0$ of $X$. However, as soon as quasi-regularity is imposed, the càdlàg property of $\phi$ and $\psi$ immediately allows to extend the semi-flow equations also to $s= 0$.

\[rem:big\_flow\] To express the semi-flow equations in a more succinct matter, it is sometimes convenient to introduce the following ‘big-flow’-notation. Define the set $\wh{\cU} := \C_{\le 0}\times \cU$ and denote its elements by $\wh{u} = (u_0,u)$. Define $$\Psi_s(t, \wh{u}) := \begin{pmatrix}\phi_s(t,u) + u_0\\ \psi_s(t,u) \end{pmatrix}.$$ Part (i) of Lemma \[lem:flow\_cont\] is equivalent to the claim that $u 