have h_sInf := hs.isGLB_sInf <| 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 : NormedAddGroup E
inst✝ : SeminormedAddGroup F
hf : ContinuousOn C s
hC : ∀ i ∈ s, C i ≠ 0
l : Filter α
x : ι × α
hx : x ∈ s ×ˢ Set.univ
hs : IsCompact ((fun a => ‖a‖) '' (C '' s))
⊢ ‖c‖ ≤
    ‖c‖ *
      (‖match x with
          | (i, _x) => C i‖ /
        sInf (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 : NormedAddGroup E
inst✝ : SeminormedAddGroup F
hf : ContinuousOn C s
hC : ∀ i ∈ s, C i ≠ 0
l : Filter α
x : ι × α
hx : x ∈ s ×ˢ Set.univ
hs : IsCompact ((fun a => ‖a‖) '' (C '' s))
h_sInf : IsGLB ((fun a => ‖a‖) '' (C '' s)) (sInf ((fun a => ‖a‖) '' (C '' s)))
⊢ ‖c‖ ≤
    ‖c‖ *
      (‖match x with
          | (i, _x) => C i‖ /
        sInf (norm '' (C '' s)))