theorem extracted_formal_statement_12.{u_1} {E : Type u_1} {xm : E} {f : E → ℝ} {g : ℕ → E}
  (mono : ∀ (k : ℕ), f (g (k + 1)) ≤ f (g k)) (j : ℕ)
  (a : f (g (j + 1)) - f xm ≤ (∑ k ∈ Finset.range (j + 1), (f (g (k + 1)) - f xm)) / (↑j + 1))
  (h_original_goal :
    (∑ k ∈ Finset.range (j.succ + 1), f (g (k + 1))) / (↑j + 2) - f xm * ((↑j + 2) / (↑j + 2)) + f xm =
      (∑ k ∈ Finset.range (j.succ + 1), f (g (k + 1)) - (↑j + 1 + 1) * f xm) / (↑j + 1 + 1) + f xm) :
  (∑ k ∈ Finset.range (j + 1 + 1), f (g (k + 1))) / (↑j + 2) - f xm * ((↑j + 2) / (↑j + 2)) =
    (∑ k ∈ Finset.range (j + 1 + 1), f (g (k + 1)) - (↑j + 1 + 1) * f xm) / (↑j + 1 + 1) := sorry

State before:
E : Type u_1
xm : E
f : E → ℝ
g : ℕ → E
mono : ∀ (k : ℕ), f (g (k + 1)) ≤ f (g k)
j : ℕ
a✝ : f (g (j + 1)) - f xm ≤ (∑ k ∈ Finset.range (j + 1), (f (g (k + 1)) - f xm)) / (↑j + 1)
⊢ (∑ k ∈ Finset.range (j.succ + 1), f (g (k + 1))) / (↑j + 2) - f xm * ((↑j + 2) / (↑j + 2)) + f xm =
    (∑ k ∈ Finset.range (j.succ + 1), f (g (k + 1)) - (↑j + 1 + 1) * f xm) / (↑j + 1 + 1) + f xm

Tactic: simp

State after:
[E : Type u_1
 xm : E
 f : E → ℝ
 g : ℕ → E
 mono : ∀ (k : ℕ), f (g (k + 1)) ≤ f (g k)
 j : ℕ
 a✝ : f (g (j + 1)) - f xm ≤ (∑ k ∈ Finset.range (j + 1), (f (g (k + 1)) - f xm)) / (↑j + 1)
 ⊢ (∑ k ∈ Finset.range (j + 1 + 1), f (g (k + 1))) / (↑j + 2) - f xm * ((↑j + 2) / (↑j + 2)) =
     (∑ k ∈ Finset.range (j + 1 + 1), f (g (k + 1)) - (↑j + 1 + 1) * f xm) / (↑j + 1 + 1)]

Relevant lemmas: