reserve a for set;

theorem Th18:
  for L being lower-bounded sup-Semilattice, AR being auxiliary Relation of L
  holds AR-below in the carrier of MonSet L
proof
  let L be lower-bounded sup-Semilattice, AR be auxiliary Relation of L;
  set s = AR-below;
  ex s be Function of L, InclPoset Ids L st AR-below = s & s is monotone &
  for x be Element of L holds s.x c= downarrow x
  proof
    take s;
    for x be Element of L holds s.x c= downarrow x
    proof
      let x be Element of L;
      s.x = AR-below x by Def12;
      hence thesis by Th12;
    end;
    hence thesis;
  end;
  hence thesis by Def13;
end;
