English
If s1 ≃ s2 and there is an Iso between their last elements, then snoc preserves equivalence in a refined setting.
Русский
Если s1 эквивалентна s2 и существует изоморфизм между их последними элементами, то snoc сохраняет эквивалентность в более точных условиях.
LaTeX
$$$\\forall s_1\\ s_2, h: s_1.Equivalent s_2 \\Rightarrow \\exists h\\_iso: Iso (s_1.last, s_1.last') (s_2.last, s_2.last') \\Rightarrow s_1.snoc s_1.last h\\_iso \\text{Equivalent} s_2.snoc s_2.last h\\_iso.$$$
Lean4
/-- The sequence given by Kőnig's lemma as an order embedding -/
theorem exists_orderEmbedding_covby_of_forall_covby_finite (hfin : ∀ (a : α), {x | a ⋖ x}.Finite)
(hb : (Ici b).Infinite) : ∃ f : ℕ ↪o α, f 0 = b ∧ ∀ i, f i ⋖ f (i + 1) :=
by
obtain ⟨f, hf⟩ := exists_seq_covby_of_forall_covby_finite hfin hb
exact ⟨OrderEmbedding.ofStrictMono f (strictMono_nat_of_lt_succ (fun i ↦ (hf.2 i).lt)), hf⟩
