
theorem
  for S,T being up-complete non empty Poset st S,T are_isomorphic
  holds S is_an_UPS_retract_of T & T is_an_UPS_retract_of S
proof
  let S,T be up-complete non empty Poset;
  assume S,T are_isomorphic;
  then consider
  f be monotone Function of S,T, g be monotone Function of T,S such
  that
A1: f*g = id T and
A2: g*f = id S by Th15;
  g is isomorphic by A1,A2,Th14;
  then
A3: g is sups-preserving by WAYBEL13:20;
  f is isomorphic by A1,A2,Th14;
  then
A4: f is sups-preserving by WAYBEL13:20;
  then
A5: f is_an_UPS_retraction_of S,T by A1,A3;
  g is_an_UPS_retraction_of T,S by A2,A4,A3;
  hence thesis by A5;
end;
