∀ {α : Type u_1} {β : Type u_2} [inst : ConditionallyCompleteLinearOrder α] [inst_1 : TopologicalSpace α]
  [OrderTopology α] [inst_3 : ConditionallyCompleteLinearOrder β] [inst_4 : TopologicalSpace β] [OrderClosedTopology β]
  {f : α → β} {A : Set α},
  ContinuousWithinAt f A (sInf A) →
    AntitoneOn f A →
      A.Nonempty →
        autoParam (BddBelow A) AntitoneOn.map_csInf_of_continuousWithinAt._auto_1 → f (sInf A) = sSup (f '' A)