∀ {P : Type u_2} {inst : LE P} {toLowerSet : LowerSet P} {nonempty' : toLowerSet.carrier.Nonempty}
  {directed' : DirectedOn (fun x1 x2 => x1 ≤ x2) toLowerSet.carrier} {toLowerSet_1 : LowerSet P}
  {nonempty'_1 : toLowerSet_1.carrier.Nonempty} {directed'_1 : DirectedOn (fun x1 x2 => x1 ≤ x2) toLowerSet_1.carrier},
  { toLowerSet := toLowerSet, nonempty' := nonempty', directed' := directed' } =
      { toLowerSet := toLowerSet_1, nonempty' := nonempty'_1, directed' := directed'_1 } →
    toLowerSet = toLowerSet_1