
theorem Th9:
  for S,T being non empty Poset, f being Function st f
  is_a_retraction_of T,S holds rng f = the carrier of S
proof
  let S,T be non empty Poset, f be Function;
  assume
A1: f is_a_retraction_of T,S;
  then reconsider f as Function of T,S;
  f*incl(S,T) = id S by A1,Th7;
  then f is onto by FUNCT_2:23;
  hence thesis by FUNCT_2:def 3;
end;
