theorem extracted_formal_statement_10.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
  [inst_2 : CompleteSpace E] [inst_3 : ProperSpace E] {x₀ : E} {f : E → ℝ} {f' : E → E} {h : E → ℝ}
  {alg : proximal_gradient_method f h f' x₀} (th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ)
  (th' : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)) :
  let Gt := fun x =>
    (1 / proximal_gradient_method.t f h f' x₀) •
      (x -
        prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (x - proximal_gradient_method.t f h f' x₀ • f' x) th
          th');
  ∀ (z : E),
    z - proximal_gradient_method.t f h f' x₀ • Gt z =
        Classical.choose
          (prox_well_define_convex (proximal_gradient_method.t f h f' x₀ • h) th th'
            (z - proximal_gradient_method.t f h f' x₀ • f' z)) →
      prox_prop (proximal_gradient_method.t f h f' x₀ • h) (z - proximal_gradient_method.t f h f' x₀ • f' z)
        (Classical.choose
          (prox_well_define_convex (proximal_gradient_method.t f h f' x₀ • h) th th'
            (z - proximal_gradient_method.t f h f' x₀ • f' z))) := sorry

State before:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : ProperSpace E
x₀ : E
f : E → ℝ
f' : E → E
h : E → ℝ
alg : proximal_gradient_method f h f' x₀
k : ℕ+
th : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ
th' : ConvexOn ℝ univ (proximal_gradient_method.t f h f' x₀ • h)
Gt : E → E :=
  fun x =>
    (1 / proximal_gradient_method.t f h f' x₀) •
      (x -
        prox_point_c' (proximal_gradient_method.t f h f' x₀ • h) (x - proximal_gradient_method.t f h f' x₀ • f' x) th
          th')
φ : E → ℝ := fun x => f x + h x
z : E
eq1 : z - proximal_gradient_method.t f h f' x₀ • Gt z = Classical.choose ⋯
⊢ prox_prop (proximal_gradient_method.t f h f' x₀ • h) (z - proximal_gradient_method.t f h f' x₀ • f' z)
    (Classical.choose ⋯)

Tactic: apply Classical.choose_spec

State after:

Relevant lemmas:
Classical.choose_spec : ∀ {α : Sort u} {p : α → Prop} (h : ∃ x, p x), p (Classical.choose h)