theorem extracted_formal_statement_25.{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} [inst_9 : Setting E₁ E₂ F admm admm_kkt]
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) (M : ℝ)
  (h₁ : {x | ∃ «y», (⇑(OptProblem.A₂ E₁) ∘ e₂) «y» = x} ⊆ ball 0 M) (C : ℝ) (C_pos : C > 0)
  (h₂ : ∀ (n : ℕ), ‖e₂ n‖ ≤ C * ‖(OptProblem.A₂ E₁) (e₂ n)‖) ⦃x : E₂⦄ (n : ℕ) (eq : ADMM.x₂ E₁ F n = x)
  (A₂e₂_bd : ‖(OptProblem.A₂ E₁) (e₂ n)‖ < M) : x ∈ ball 0 (C * M + ‖admm_kkt.x₂‖) := sorry

State before:
case h.intro
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
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
M : ℝ
h₁ : {x | ∃ «y», (⇑(OptProblem.A₂ E₁) ∘ e₂) «y» = x} ⊆ ball 0 M
C : ℝ
C_pos : C > 0
h₂ : ∀ (n : ℕ), ‖e₂ n‖ ≤ C * ‖(OptProblem.A₂ E₁) (e₂ n)‖
x : E₂
n : ℕ
eq : ADMM.x₂ E₁ F n = x
A₂e₂_bd : ‖(OptProblem.A₂ E₁) (e₂ n)‖ < M
⊢ x ∈ ball 0 (C * M + ‖admm_kkt.x₂‖)

Tactic: rw [← eq]
    simp
    calc
  _ = ‖(x₂ n - x₂') + x₂'‖ := by rw [add_comm, add_sub, add_comm, add_sub_assoc, sub_self, add_zero]
  _ ≤ ‖x₂ n - x₂'‖ + ‖x₂'‖ := by apply norm_add_le
  _ = ‖e₂ n‖ + ‖x₂'‖ := by rw [e₂]
  _ ≤ C * ‖A₂ (e₂ n)‖ + ‖x₂'‖ := by linarith [h₂ n]
  _ < C * M + ‖x₂'‖ := by linarith [mul_lt_mul_of_pos_left A₂e₂_bd C_pos]

State after:

Relevant lemmas:
add_sub_assoc : ∀ {G : Type u_1} [inst : SubNegMonoid G] (a b c : G), a + b - c = a + (b - c)
add_sub : ∀ {G : Type u_3} [inst : SubNegMonoid G] (a b c : G), a + (b - c) = a + b - c
sub_self : ∀ {G : Type u_1} [inst : AddGroup G] (a : G), a - a = 0
mul_lt_mul_of_pos_left : ∀ {α : Type u_3} {a b c : α} [inst : Mul α] [inst_10 : Zero α] [inst_11 : Preorder α]
  [inst_12 : PosMulStrictMono α], b < c → 0 < a → a * b < a * c
add_zero : ∀ {M : Type u} [inst : AddZeroClass M] (a : M), a + 0 = a
norm_add_le : ∀ {E : Type u_5} [inst : SeminormedAddGroup E] (a b : E), ‖a + b‖ ≤ ‖a‖ + ‖b‖
add_comm : ∀ {G : Type u_1} [inst : AddCommMagma G] (a b : G), a + b = b + a