theorem extracted_formal_statement_5.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] {x0 : E} {f h : E → ℝ} {f' : E → E} {alg : Nesterov_second f h f' x0} {xm : E}
  (minφ : IsMinOn (f + h) univ xm) :
  let φ := fun z => f z + h z;
  (∀ (z : E), φ z = f z + h z) →
    ∀ (k : ℕ+),
      Nesterov_second.y f h f' x0 (↑k - 1) -
              (Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • f' (Nesterov_second.z f h f' x0 k) -
            Nesterov_second.y f h f' x0 ↑k ∈
          SubderivAt ((Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • h)
            (Nesterov_second.y f h f' x0 ↑k) →
        SubderivAt (Nesterov_second.t f h f' x0 ↑k • h) (Nesterov_second.y f h f' x0 ↑k) =
            Nesterov_second.γ f h f' x0 ↑k •
              SubderivAt ((Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • h)
                (Nesterov_second.y f h f' x0 ↑k) →
          Nesterov_second.γ f h f' x0 ↑k • (Nesterov_second.y f h f' x0 (↑k - 1) - Nesterov_second.y f h f' x0 ↑k) -
                (Nesterov_second.γ f h f' x0 ↑k * (Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k)) •
                  f' (Nesterov_second.z f h f' x0 k) =
              Nesterov_second.γ f h f' x0 ↑k • (Nesterov_second.y f h f' x0 (↑k - 1) - Nesterov_second.y f h f' x0 ↑k) -
                Nesterov_second.t f h f' x0 ↑k • f' (Nesterov_second.z f h f' x0 k) →
            Nesterov_second.γ f h f' x0 ↑k ≠ 0 := sorry

State before:
case h.right
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : CompleteSpace E
x0 : E
f h : E → ℝ
f' : E → E
alg : Nesterov_second f h f' x0
xm : E
minφ : IsMinOn (f + h) univ xm
φ : E → ℝ := fun z => f z + h z
φdef : ∀ (z : E), φ z = f z + h z
k : ℕ+
h1 :
  Nesterov_second.y f h f' x0 (↑k - 1) -
        (Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • f' (Nesterov_second.z f h f' x0 k) -
      Nesterov_second.y f h f' x0 ↑k ∈
    SubderivAt ((Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • h) (Nesterov_second.y f h f' x0 ↑k)
upd2 :
  SubderivAt (Nesterov_second.t f h f' x0 ↑k • h) (Nesterov_second.y f h f' x0 ↑k) =
    Nesterov_second.γ f h f' x0 ↑k •
      SubderivAt ((Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • h)
        (Nesterov_second.y f h f' x0 ↑k)
⊢ Nesterov_second.γ f h f' x0 ↑k • (Nesterov_second.y f h f' x0 (↑k - 1) - Nesterov_second.y f h f' x0 ↑k) -
      (Nesterov_second.γ f h f' x0 ↑k * (Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k)) •
        f' (Nesterov_second.z f h f' x0 k) =
    Nesterov_second.γ f h f' x0 ↑k • (Nesterov_second.y f h f' x0 (↑k - 1) - Nesterov_second.y f h f' x0 ↑k) -
      Nesterov_second.t f h f' x0 ↑k • f' (Nesterov_second.z f h f' x0 k)

Tactic: rw [mul_div, mul_comm, ← mul_div, div_self, mul_one]

State after:
[case h.right
 E : Type u_1
 inst✝² : NormedAddCommGroup E
 inst✝¹ : InnerProductSpace ℝ E
 inst✝ : CompleteSpace E
 x0 : E
 f h : E → ℝ
 f' : E → E
 alg : Nesterov_second f h f' x0
 xm : E
 minφ : IsMinOn (f + h) univ xm
 φ : E → ℝ := fun z => f z + h z
 φdef : ∀ (z : E), φ z = f z + h z
 k : ℕ+
 h1 :
   Nesterov_second.y f h f' x0 (↑k - 1) -
         (Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • f' (Nesterov_second.z f h f' x0 k) -
       Nesterov_second.y f h f' x0 ↑k ∈
     SubderivAt ((Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • h) (Nesterov_second.y f h f' x0 ↑k)
 upd2 :
   SubderivAt (Nesterov_second.t f h f' x0 ↑k • h) (Nesterov_second.y f h f' x0 ↑k) =
     Nesterov_second.γ f h f' x0 ↑k •
       SubderivAt ((Nesterov_second.t f h f' x0 ↑k / Nesterov_second.γ f h f' x0 ↑k) • h)
         (Nesterov_second.y f h f' x0 ↑k)
 ⊢ Nesterov_second.γ f h f' x0 ↑k ≠ 0]

Relevant lemmas:
div_self : ∀ {G₀ : Type u_3} [inst : GroupWithZero G₀] {a : G₀}, a ≠ 0 → a / a = 1
mul_comm : ∀ {G : Type u_1} [inst : CommMagma G] (a b : G), a * b = b * a
mul_div : ∀ {G : Type u_3} [inst : DivInvMonoid G] (a b c : G), a * (b / c) = a * b / c
mul_one : ∀ {M : Type u} [inst : MulOneClass M] (a : M), a * 1 = a