theorem extracted_formal_statement_27.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁))
  (h_original_goal : Bornology.IsBounded (range fun n => (ADMM.x₁ E₂ F n, ADMM.x₂ E₁ F n, ADMM.y E₁ E₂ n))) :
  (range fun n => (ADMM.x₁ E₂ F n, ADMM.x₂ E₁ F n, ADMM.y E₁ E₂ n)) ⊆
    range (ADMM.x₁ E₂ F) ×ˢ range (ADMM.x₂ E₁ F) ×ˢ range (ADMM.y E₁ E₂) := sorry

State before:
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
inst✝ : Setting E₁ E₂ F admm admm_kkt
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
⊢ Bornology.IsBounded (range fun n => (ADMM.x₁ E₂ F n, ADMM.x₂ E₁ F n, ADMM.y E₁ E₂ n))

Tactic: apply IsBounded.subset
    apply IsBounded.prod (x₁_isBounded fullrank₁)
    apply IsBounded.prod (x₂_isBounded fullrank₂) y_isBounded

State after:
[case hs
 E₁ : Type u_1
 E₂ : Type u_2
 F : Type u_3
 inst✝⁹ : NormedAddCommGroup E₁
 inst✝⁸ : InnerProductSpace ℝ E₁
 inst✝⁷ : FiniteDimensional ℝ E₁
 inst✝⁶ : NormedAddCommGroup E₂
 inst✝⁵ : InnerProductSpace ℝ E₂
 inst✝⁴ : FiniteDimensional ℝ E₂
 inst✝³ : NormedAddCommGroup F
 inst✝² : InnerProductSpace ℝ F
 inst✝¹ : FiniteDimensional ℝ F
 admm : ADMM E₁ E₂ F
 admm_kkt : Existance_of_kkt admm
 inst✝ : Setting E₁ E₂ F admm admm_kkt
 fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
 fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
 ⊢ (range fun n => (ADMM.x₁ E₂ F n, ADMM.x₂ E₁ F n, ADMM.y E₁ E₂ n)) ⊆
     range (ADMM.x₁ E₂ F) ×ˢ range (ADMM.x₂ E₁ F) ×ˢ range (ADMM.y E₁ E₂)]

Relevant lemmas:
Bornology.IsBounded.subset : ∀ {α : Type u_2} {x : Bornology α} {s t : Set α},
  Bornology.IsBounded t → s ⊆ t → Bornology.IsBounded s
Bornology.IsBounded.prod : ∀ {α : Type u_1} {β : Type u_2} [inst : Bornology α] [inst_9 : Bornology β] {s : Set α}
  {t : Set β}, Bornology.IsBounded s → Bornology.IsBounded t → Bornology.IsBounded (s ×ˢ t)