theorem extracted_formal_statement_4.{u_1} {E : Type u_1} [inst : NormedAddCommGroup E] {s : Set E} (hs : s.Nonempty)
  (x : E) (xs : x ∈ s) (this : EMetric.infEdist 0 s ≤ edist 0 x) : (EMetric.infEdist 0 s).toReal ≤ dist 0 x := sorry

State before:
case neg
E : Type u_1
inst✝ : NormedAddCommGroup E
s : Set E
hs : s.Nonempty
x : E
xs : x ∈ s
this : EMetric.infEdist 0 s ≤ edist 0 x
⊢ (EMetric.infEdist 0 s).toReal ≤ dist 0 x

Tactic: exact ENNReal.toReal_le_of_le_ofReal dist_nonneg (edist_dist 0 x ▸ this)

State after:

Relevant lemmas:
dist_nonneg : ∀ {α : Type u} [inst : PseudoMetricSpace α] {x y : α}, 0 ≤ dist x y
ENNReal.toReal_le_of_le_ofReal : ∀ {a : ENNReal} {b : ℝ}, 0 ≤ b → a ≤ ENNReal.ofReal b → a.toReal ≤ b
edist_dist : ∀ {α : Type u} [inst : PseudoMetricSpace α] (x y : α), edist x y = ENNReal.ofReal (dist x y)