English
If {x, y, z} is collinear, then one of the cyclic weak betweenness relations holds among the three points.
Русский
Если {x, y, z} лежат на одной прямой, тогда одном из циклических отношений слабого между выполняется между.
LaTeX
$$$\text{Collinear}(R, {x,y,z}) \Rightarrow (Wbtw_R x y z \lor Wbtw_R y z x \lor Wbtw_R z x y)$$$
Lean4
theorem wbtw_or_wbtw_or_wbtw {x y z : P} (h : Collinear R ({ x, y, z } : Set P)) :
Wbtw R x y z ∨ Wbtw R y z x ∨ Wbtw R z x y :=
by
rw [collinear_iff_of_mem (Set.mem_insert _ _)] at h
rcases h with ⟨v, h⟩
simp_rw [Set.mem_insert_iff, Set.mem_singleton_iff] at h
have hy := h y (Or.inr (Or.inl rfl))
have hz := h z (Or.inr (Or.inr rfl))
rcases hy with ⟨ty, rfl⟩
rcases hz with ⟨tz, rfl⟩
rcases lt_trichotomy ty 0 with (hy0 | rfl | hy0)
· rcases lt_trichotomy tz 0 with (hz0 | rfl | hz0)
· rw [wbtw_comm (z := x)]
rw [← or_assoc]
exact Or.inl (wbtw_or_wbtw_smul_vadd_of_nonpos _ _ hy0.le hz0.le)
· simp
· exact Or.inr (Or.inr (wbtw_smul_vadd_smul_vadd_of_nonneg_of_nonpos _ _ hz0.le hy0.le))
· simp
· rcases lt_trichotomy tz 0 with (hz0 | rfl | hz0)
· refine Or.inr (Or.inr (wbtw_smul_vadd_smul_vadd_of_nonpos_of_nonneg _ _ hz0.le hy0.le))
· simp
· rw [wbtw_comm (z := x)]
rw [← or_assoc]
exact Or.inl (wbtw_or_wbtw_smul_vadd_of_nonneg _ _ hy0.le hz0.le)