theorem extracted_formal_statement_2.{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} : ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ := 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+
⊢ ContinuousOn (proximal_gradient_method.t f h f' x₀ • h) univ

Tactic: apply ContinuousOn.const_smul alg.h₃ alg.t

State after:

Relevant lemmas:
ContinuousOn.const_smul : ∀ {M : Type u_1} {α : Type u_2} {β : Type u_3} [inst : TopologicalSpace α] [inst_4 : SMul M α]
  [inst_5 : ContinuousConstSMul M α] [inst_6 : TopologicalSpace β] {g : β → α} {s : Set β},
  ContinuousOn g s → ∀ (c : M), ContinuousOn (fun x => c • g x) s