
theorem
  for L being continuous complete LATTICE for S being non empty Poset st
  S is_an_UPS_retract_of L holds S is continuous
proof
  let L be continuous complete LATTICE;
  let S be non empty Poset;
  given f being Function of L,S such that
A1: f is_an_UPS_retraction_of L,S;
  consider h being directed-sups-preserving projection Function of L,L such
  that
A2: h is_a_retraction_of L, Image h and
A3: S, Image h are_isomorphic by A1,Th18;
  Image h is_a_retract_of L by A2;
  then Image h is continuous by Th21;
  hence thesis by A3,WAYBEL15:9,WAYBEL_1:6;
end;
