Let $x \in \Omega_p(\mathbb{R}^d)$ be a $p$-rough path controlled by $\omega$, and let $q \geq p$. Then there exists a unique extension \begin{align*} S^{\lfloor q \rfloor}(x) : \Delta_T \to T^{\lfloor q \rfloor}(\mathbb{R}^d) \end{align*} satisfying: 1. $S^{\lfloor q \rfloor}(x)_{s,u} \otimes S^{\lfloor q \rfloor}(x)_{u,t} = S^{\lfloor q \rfloor}(x)_{s,t}$ for all $0 \leq s \leq u \leq t \leq T$; 2. $\|\pi_i S^{\lfloor q \rfloor}(x)_{s,t}\|^i_{(\mathbb{R}^d)^{\otimes i}} \leq C\,\omega(s,t)^{i/p}$ for some $C > 0$, all $(s,t) \in \Delta_T$, and all $i \leq \lfloor q \rfloor$; 3. $\pi_{\leq \lfloor p \rfloor}x_{s,t} = \pi_{\leq \lfloor p \rfloor} S^{\lfloor q \rfloor}(x)_{s,t}$ for all $(s,t) \in \Delta_T$. (Lyons–Qian 2002, Theorem 3.7)