
theorem Th20:
  for L being complete non empty Poset for S being non empty
  Poset st S is_a_retract_of L holds S is complete
proof
  let L be complete non empty Poset;
  let S be non empty Poset;
  given f being Function of L, S such that
A1: f is_a_retraction_of L,S;
  reconsider f as directed-sups-preserving projection Function of L,L by A1
,Th11,Th13;
A2: Image f is complete by YELLOW_2:35;
  the RelStr of S = Image f by A1,Th12;
  hence thesis by A2,YELLOW_0:3;
end;
