rw [LowerSemicontinuousOn, LowerSemicontinuous, SetCoe.forall]
α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
f : α → β
inst✝¹ : LinearOrder β
inst✝ : OrderClosedTopology β
s : Set α
⊢ LowerSemicontinuousOn f s ↔ LowerSemicontinuous (restrict s f)
α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
f : α → β
inst✝¹ : LinearOrder β
inst✝ : OrderClosedTopology β
s : Set α
⊢ (∀ x ∈ s, LowerSemicontinuousWithinAt f s x) ↔
    ∀ (x : α) (h : x ∈ s), LowerSemicontinuousAt (restrict s f) { val := x, property := h }