theorem extracted_formal_statement_152.{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) = ∅)
  (h_original_goal : limit_set z ∩ a ∪ limit_set z ∩ b = limit_set z) :
  limit_set z ∩ (a ∪ b) ∩ (a ∪ b) = limit_set z ∩ (a ∪ b) := 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
⊢ limit_set z ∩ a ∪ limit_set z ∩ b = limit_set z

Tactic: rw [← inter_union_distrib_left (limit_set z) a b, left_eq_inter.mpr sub_ab]

State after:
[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
 ⊢ limit_set z ∩ (a ∪ b) ∩ (a ∪ b) = limit_set z ∩ (a ∪ b)]

Relevant lemmas:
Set.inter_union_distrib_left : ∀ {α : Type u} (s t u : Set α), s ∩ (t ∪ u) = s ∩ t ∪ s ∩ u