
theorem Th8:
  for S being non empty Poset, T being up-complete non empty
  Poset for f being Function st f is_a_retraction_of T,S holds f
  is_an_UPS_retraction_of T,S
proof
  let S be non empty Poset;
  let T be up-complete non empty Poset, f be Function;
  assume
A1: f is_a_retraction_of T,S;
  hence f is directed-sups-preserving Function of T,S;
  S is directed-sups-inheriting full SubRelStr of T by A1;
  then reconsider
  g = incl(S,T) as directed-sups-preserving Function of S,T by WAYBEL21:10;
  take g;
  thus thesis by A1,Th7;
end;
