theorem extracted_formal_statement_331.{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)) [inst_9 : Setting E₁ E₂ F admm admm_kkt]
  (h_original_goal :
    Tendsto (fun x => (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (φ x)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (φ x))) atTop
      (𝓝 ((OptProblem.A₁ E₂) x₁'' + (OptProblem.A₂ E₁) x₂''))) :
  Tendsto (fun x => (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (φ x))) atTop (𝓝 ((OptProblem.A₁ E₂) x₁'')) := sorry

State before:
case h.h
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
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm admm_kkt
⊢ Tendsto (fun x => (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (φ x)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (φ x))) atTop
    (𝓝 ((OptProblem.A₁ E₂) x₁'' + (OptProblem.A₂ E₁) x₂''))

Tactic: apply Tendsto.add

State after:
[case h.h.hf
 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
 fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
 fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
 inst✝ : Setting E₁ E₂ F admm admm_kkt
 ⊢ Tendsto (fun x => (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F (φ x))) atTop (𝓝 ((OptProblem.A₁ E₂) x₁'')),
 case h.h.hg
 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
 fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
 fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
 inst✝ : Setting E₁ E₂ F admm admm_kkt
 ⊢ Tendsto (fun x => (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (φ x))) atTop (𝓝 ((OptProblem.A₂ E₁) x₂''))]

Relevant lemmas:
Filter.Tendsto.add : ∀ {M : Type u_1} [inst : TopologicalSpace M] [inst_10 : Add M] [inst_11 : ContinuousAdd M]
  {α : Type u_2} {f g : α → M} {x : Filter α} {a b : M},
  Tendsto f x (𝓝 a) → Tendsto g x (𝓝 b) → Tendsto (fun x => f x + g x) x (𝓝 (a + b))