
theorem Th5:
  for L being up-complete non empty Poset for f being Function of
  L, L st f is idempotent directed-sups-preserving holds Image f is
  directed-sups-inheriting
proof
  let L be up-complete non empty Poset;
  let f be Function of L, L;
  set S = subrelstr(rng f);
  set a = the Element of L;
  reconsider S9 = S as non empty full SubRelStr of L;
  assume
A1: f is idempotent directed-sups-preserving;
  S is directed-sups-inheriting
  proof
    let X be directed Subset of S;
    X c= the carrier of S;
    then
A2: X c= rng f by YELLOW_0:def 15;
    assume X <> {};
    then X is non empty directed Subset of S9;
    then reconsider X9= X as non empty directed Subset of L by YELLOW_2:7;
    assume
A3: ex_sup_of X,L;
    f preserves_sup_of X9 by A1;
    then sup(f.:X9) = f.sup X9 by A3;
    then sup X9 = f.sup X9 by A1,A2,YELLOW_2:20;
    then "\/"(X, L) in rng f by FUNCT_2:4;
    hence thesis by YELLOW_0:def 15;
  end;
  hence thesis;
end;
