theorem extracted_formal_statement_169.{u_2, u_1} {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E]
  [inst_1 : InnerProductSpace ℝ E] [inst_2 : CompleteSpace E] [inst_3 : NormedAddCommGroup F]
  [inst_4 : InnerProductSpace ℝ F] [inst_5 : CompleteSpace F] {f : E → ℝ} {g : F → ℝ} {H : WithLp 2 (E × F) → ℝ}
  {x0 : E} {y0 : F} {l : NNReal} {alg : BCD f g H l x0 y0} [inst_6 : ProperSpace E] [inst_7 : ProperSpace F] (γ : ℝ)
  (hγ : γ > 1) (ck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l)) (dk : ∀ (k : ℕ), alg.d k = 1 / (γ * ↑l))
  (bd : Bornology.IsBounded (z '' univ)) (lbdψ : BddBelow (ψ '' univ)) (a b : Set (WithLp 2 (E × F)))
  (closea : IsClosed a) (closeb : IsClosed b) (sub_ab : limit_set z ⊆ a ∪ b) (nez_a : (limit_set z ∩ a).Nonempty)
  (nez_b : (limit_set z ∩ b).Nonempty) (nz_ab : limit_set z ∩ (a ∩ b) = ∅) :
  let A := limit_set z ∩ a;
  let B := limit_set z ∩ b;
  A ∪ B = limit_set z →
    A ∩ B = ∅ →
      let ω := fun z => (EMetric.infEdist z A).toReal / ((EMetric.infEdist z A).toReal + (EMetric.infEdist z B).toReal);
      (∀ (z : WithLp 2 (E × F)), (EMetric.infEdist z A).toReal + (EMetric.infEdist z B).toReal ≠ 0) →
        Continuous ω →
          let U := {z | ω z < 1 / 4};
          let V := {z | 3 / 4 < ω z};
          (∀ z_ ∈ A, ω z_ = 0) →
            (∀ z_ ∈ B, ω z_ = 1) →
              ∀ (h : ∀ (k0 : ℕ), ∃ k, k0 ≤ k ∧ z k ∉ U ∧ z k ∉ V),
                let W := fun n => Nat.recOn n (Classical.choose (h 0)) fun n p => Classical.choose (h (p + 1));
                StrictMono W → ∀ ⦃k : WithLp 2 (E × F)⦄ (x : ℕ), z (W x) = k → ∃ y, z y = k := sorry

State before:
E : Type u_1
F : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace ℝ E
inst✝⁵ : CompleteSpace E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace ℝ F
inst✝² : CompleteSpace F
f : E → ℝ
g : F → ℝ
H : WithLp 2 (E × F) → ℝ
x0 : E
y0 : F
l : NNReal
alg : BCD f g H l x0 y0
inst✝¹ : ProperSpace E
inst✝ : ProperSpace F
γ : ℝ
hγ : γ > 1
ck : ∀ (k : ℕ), alg.c k = 1 / (γ * ↑l)
dk : ∀ (k : ℕ), alg.d k = 1 / (γ * ↑l)
bd : Bornology.IsBounded (z '' univ)
lbdψ : BddBelow (ψ '' univ)
a b : Set (WithLp 2 (E × F))
closea : IsClosed a
closeb : IsClosed b
sub_ab : limit_set z ⊆ a ∪ b
nez_a : (limit_set z ∩ a).Nonempty
nez_b : (limit_set z ∩ b).Nonempty
nz_ab : limit_set z ∩ (a ∩ b) = ∅
A : Set (WithLp 2 (E × F)) := limit_set z ∩ a
B : Set (WithLp 2 (E × F)) := limit_set z ∩ b
eqAB : A ∪ B = limit_set z
disjoint_AB : A ∩ B = ∅
ω : WithLp 2 (E × F) → ℝ :=
  fun z => (EMetric.infEdist z A).toReal / ((EMetric.infEdist z A).toReal + (EMetric.infEdist z B).toReal)
sum_ne_zero : ∀ (z : WithLp 2 (E × F)), (EMetric.infEdist z A).toReal + (EMetric.infEdist z B).toReal ≠ 0
contω : Continuous ω
U : Set (WithLp 2 (E × F)) := {z | ω z < 1 / 4}
V : Set (WithLp 2 (E × F)) := {z | 3 / 4 < ω z}
A0 : ∀ z_ ∈ A, ω z_ = 0
B1 : ∀ z_ ∈ B, ω z_ = 1
h : ∀ (k0 : ℕ), ∃ k, k0 ≤ k ∧ z k ∉ U ∧ z k ∉ V
W : ℕ → ℕ := fun n => Nat.recOn n (Classical.choose ⋯) fun n p => Classical.choose ⋯
monoW : StrictMono W
k : WithLp 2 (E × F)
⊢ ∀ (x : ℕ), z (W x) = k → ∃ y, z y = k

Tactic: exact fun x zk ↦ ⟨W x, zk⟩

State after:

Relevant lemmas: