English
Let x,y,z be elements of Part α. If x ≤ z and y ≤ z, then either x ≤ y or y ≤ x. This expresses comparability under a common upper bound.
Русский
Пусть x,y,z ∈ Part α. Если x ≤ z и y ≤ z, то либо x ≤ y, либо y ≤ x. Это выражает сравнимость при общем верхнем пределe.
LaTeX
$$$\\forall x,y,z:\\ Part α, \\ \\bigl(x \\le z \\wedge y \\le z \\bigr) \\Rightarrow x \\le y \\lor y \\le x$$$
Lean4
theorem le_total_of_le_of_le {x y : Part α} (z : Part α) (hx : x ≤ z) (hy : y ≤ z) : x ≤ y ∨ y ≤ x :=
by
rcases Part.eq_none_or_eq_some x with (h | ⟨b, h₀⟩)
· rw [h]
left
apply OrderBot.bot_le _
right; intro b' h₁
rw [Part.eq_some_iff] at h₀
have hx := hx _ h₀; have hy := hy _ h₁
have hx := Part.mem_unique hx hy; subst hx
exact h₀
