theorem extracted_formal_statement_35.{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 →
          z ∈ A → B ∈ {t | IsClosed t ∧ limit_set BCD.z ∩ b ⊆ t} → limit_set BCD.z ∩ b ⊆ B := sorry

State before:
case a
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
inA : z ∈ A
⊢ B ∈ {t | IsClosed t ∧ limit_set BCD.z ∩ b ⊆ t}

Tactic: simp
    constructor
    apply IsClosed.inter isClosed_setOf_clusterPt closeb

State after:
[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
 inA : z ∈ A
 ⊢ limit_set BCD.z ∩ b ⊆ B]

Relevant lemmas:
isClosed_setOf_clusterPt : ∀ {X : Type u} [inst : TopologicalSpace X] {f : Filter X}, IsClosed {x | ClusterPt x f}
IsClosed.inter : ∀ {X : Type u} {s₁ s₂ : Set X} [inst : TopologicalSpace X],
  IsClosed s₁ → IsClosed s₂ → IsClosed (s₁ ∩ s₂)