have h_sSup := hs.isLUB_sSup <| Set.image_nonempty.mpr <| Set.image_nonempty.mpr ⟨x.1, hx.1⟩
case refine_2
α : Type u_1
ι : Type u_2
E : Type u_3
F : Type u_4
s : Set ι
inst✝² : TopologicalSpace ι
C : ι → E
c : F
inst✝¹ : SeminormedAddGroup E
inst✝ : Norm F
hf : ContinuousOn C s
hc : ‖c‖ ≠ 0
l : Filter α
x : ι × α
hx : x ∈ s ×ˢ Set.univ
hs : IsCompact ((fun a => ‖a‖) '' (C '' s))
⊢ ‖match x with
      | (i, _x) => C i‖ ≤
    sSup (norm '' (C '' s))
case refine_2
α : Type u_1
ι : Type u_2
E : Type u_3
F : Type u_4
s : Set ι
inst✝² : TopologicalSpace ι
C : ι → E
c : F
inst✝¹ : SeminormedAddGroup E
inst✝ : Norm F
hf : ContinuousOn C s
hc : ‖c‖ ≠ 0
l : Filter α
x : ι × α
hx : x ∈ s ×ˢ Set.univ
hs : IsCompact ((fun a => ‖a‖) '' (C '' s))
h_sSup : IsLUB ((fun a => ‖a‖) '' (C '' s)) (sSup ((fun a => ‖a‖) '' (C '' s)))
⊢ ‖match x with
      | (i, _x) => C i‖ ≤
    sSup (norm '' (C '' s))