theorem extracted_formal_statement_156.{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 = ∅ →
      ∀ (z : WithLp 2 (E × F)),
        (EMetric.infEdist z A).toReal + (EMetric.infEdist z B).toReal = 0 → limit_set BCD.z ∩ a ⊆ A := sorry

State before:
case a.right
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 (BCD.z '' univ)
lbdψ : BddBelow (ψ '' univ)
a b : Set (WithLp 2 (E × F))
closea : IsClosed a
closeb : IsClosed b
sub_ab : limit_set BCD.z ⊆ a ∪ b
nez_a : (limit_set BCD.z ∩ a).Nonempty
nez_b : (limit_set BCD.z ∩ b).Nonempty
nz_ab : limit_set BCD.z ∩ (a ∩ b) = ∅
A : Set (WithLp 2 (E × F)) := limit_set BCD.z ∩ a
B : Set (WithLp 2 (E × F)) := limit_set BCD.z ∩ b
eqAB : A ∪ B = limit_set BCD.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)
z : WithLp 2 (E × F)
eq0 : (EMetric.infEdist z A).toReal + (EMetric.infEdist z B).toReal = 0
⊢ limit_set BCD.z ∩ a ⊆ A

Tactic: simp only [subset_refl, A, B]

State after:

Relevant lemmas:
subset_refl : ∀ {α : Type u} [inst : HasSubset α] [inst_8 : IsRefl α fun x1 x2 => x1 ⊆ x2] (a : α), a ⊆ a