theorem
  for D being non empty set, f being Function of bool D, D st
  (for a being Element of D holds f.{a} = a) &
  for X being Subset-Family of D holds f.(f.:X) = f.(union X)
  ex L being complete strict Lattice st the carrier of L = D &
  for X being Subset of L holds "\/" X = f.X by Lm3;
