reserve x, X, Y for set;
reserve L for complete LATTICE,
  a for Element of L;

theorem ::Remark 2.5
  for f being Function of L, L st f is idempotent
directed-sups-preserving holds Image f is directed-sups-inheriting & Image f is
  complete LATTICE
proof
  let f be Function of L, L;
  set S = subrelstr(rng f);
  set a = the Element of L;
  dom f = the carrier of L by FUNCT_2:def 1;
  then f.a in rng f by FUNCT_1:def 3;
  then reconsider S9= S as non empty full SubRelStr of L;
  assume
A1: f is idempotent directed-sups-preserving;
  for X being directed Subset of S st X <> {} & ex_sup_of X,L holds "\/"(X
  ,L) in the carrier of S
  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 Th7;
    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,Th20;
    then "\/"(X, L) in rng f by FUNCT_2:4;
    hence thesis by YELLOW_0:def 15;
  end;
  hence Image f is directed-sups-inheriting;
  rng f = {x where x is Element of L: x = f.x} by A1,Th19;
  hence thesis by A1,Th16,Th29;
end;
