
theorem Th21:
  for L being continuous complete LATTICE for S being non empty
  Poset st S is_a_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_a_retraction_of L,S;
  reconsider g = f as directed-sups-preserving projection Function of L,L by A1
,Th11,Th13;
A2: Image g is continuous LATTICE by WAYBEL15:15;
  Image g = the RelStr of S by A1,Th12;
  hence thesis by A2,YELLOW12:15;
end;
