English
If f is strictly monotone and f x < x, then the iterates f^[n] x form a strictly antitone sequence.
Русский
Если f строго монотонна и f x < x, то f^[n] x образуют строго антитонную последовательность.
LaTeX
$$$\\forall {\\alpha} [\\mathrm{Preorder}\\,\\alpha],\\ {f: \\alpha \\to \\alpha},\\ {x: \\alpha},\\ \\mathrm{StrictMono}\\ f \\to \\operatorname{lt}(f x, x) \\to \\mathrm{StrictAnti}\\ (n \\mapsto \\operatorname{Nat.iterate} f n x).$$$
Lean4
theorem total {s : CompositionSeries X} {x y : X} (hx : x ∈ s) (hy : y ∈ s) : x ≤ y ∨ y ≤ x :=
by
rcases Set.mem_range.1 hx with ⟨i, rfl⟩
rcases Set.mem_range.1 hy with ⟨j, rfl⟩
rw [s.strictMono.le_iff_le, s.strictMono.le_iff_le]
exact le_total i j
