do not change the framing of the original interval. The intervals $J_B$ are framed depending on whether we have a $(+)$ or a $(-)$–slide. In the case of a $(+)$–slide we use Lemma \[lem:arc\].1. Note that the sign of the second coordinate changes, while the first coordinate remains $\varepsilon_B$. But during the rotation described in the proof of Lemma \[lem:arc\], the second vector equals $-{\overline}{e}$. By the choice of a coherent system of paths as done in [@MR3252965] or [@JLSmorse], this is a standard path if and only if $\varepsilon_B=1$. Hence $fr(J_B)=1+\varepsilon_B$. Passing to ${\mathscr{C}}_S$ gives intervals $I_{B,A}$ for each $A\in {\mathcal{M}}(a,x)$ with the same framing value as $J_B$ by [@JLSmorse Proposition 3.7.6].

If we perform a $(-)$–slide, we use Lemma \[lem:arc\].2, and the sign of the first coordinate changes, while the second coordinate remains. In the proof of Lemma \[lem:arc\] we note that there is a rotation around the $e_2$–axis chosen such that the first standard vector points in the positive direction of ${\mathbb{R}}_+$. Comparing with the choice of coherent system of paths we get a standard path if and only if the first vector of the framing points in the positive direction, that is, when $\varepsilon_B=0$. This implies 