Corollary

*Let $D_1$ and $D_2$ be two distinct half-lines of E. For every half-line $D_0$ of E such that the sequence $D_0, D_1, D_2$ is direct, the set of half-lines $D$ of E such that $D_1 < D < D_2$ (for the total ordering relation defined by $D_0$) is equal to the open angular sector with origin $D_1$ and extremity $D_2$.

Indeed, given a half-line $D_3$, it is a question of proving that the relations “the sequence $D_1, D_3, D_2$ is direct” and “the sequences $D_0, D_1, D_3$ and $D_0, D_3, D_2$ are direct” are equivalent. For brevity let us denote by $(ijk)$ the relation “the sequence $(D_i, D_j, D_k)$ is direct”. By prop. 11, the conjunction of (132) and (120) implies (130); analogously the conjunction of (201) and (213) implies (203); whence one half of our assertion. Conversely suppose (012), (013), and (032); since the conjunction of (312) and (320) implies (310) (prop. 11), and since (310) and (013) are incompatible, (312) is false; this proves (132) and completes the proof.

By virtue of the preceding corollary, the open (resp. closed) angular sector with origin $D_1$ and extremity $D_2$ is denoted by $\{D_1, D_2\}$ (resp. $[D_1, D_2]$), it being understood that these are intervals for the order structure defined by any half-line $D_0$ such that the sequence $D_0, D_1, D_2$ is direct.