
theorem Th18:
  for T being up-complete non empty Poset, S being non empty
  Poset for f being Function st f is_an_UPS_retraction_of T,S ex h being
directed-sups-preserving projection Function of T,T st h is_a_retraction_of T,
  Image h & S, Image h are_isomorphic
proof
  let T be up-complete non empty Poset;
  let S be non empty Poset, f be Function such that
A1: f is directed-sups-preserving Function of T,S;
  given g being directed-sups-preserving Function of S,T such that
A2: f*g = id S;
  reconsider f as directed-sups-preserving Function of T,S by A1;
  consider h being projection Function of T,T such that
A3: h = g*f and
A4: h|the carrier of Image h = id Image h and
A5: S, Image h are_isomorphic by A2,Th17;
  reconsider h as directed-sups-preserving projection Function of T,T by A3,
WAYBEL20:28;
  take h;
  reconsider R = Image h as non empty Poset;
  h = corestr h by WAYBEL_1:30;
  then
A6: h is directed-sups-preserving Function of T, R by WAYBEL20:22;
  R is directed-sups-inheriting full SubRelStr of T by Th5;
  hence h is_a_retraction_of T, Image h by A4,A6;
  thus thesis by A5;
end;
