
theorem Th10:
  for S,T being non empty Poset, f being Function st f
  is_an_UPS_retraction_of T,S holds rng f = the carrier of S
proof
  let S,T be non empty Poset, f be Function;
  assume that
A1: f is directed-sups-preserving Function of T,S and
A2: ex g being directed-sups-preserving Function of S,T st f*g = id S;
   reconsider f as Function of T,S by A1;
   f is onto by A2,FUNCT_2:23;
  hence thesis by FUNCT_2:def 3;
end;
