reserve a for set;

theorem Th19:
  for L being lower-bounded sup-Semilattice holds
  IdsMap L in the carrier of MonSet L
proof
  let L be lower-bounded sup-Semilattice;
  set s = IdsMap L;
  ex s9 be Function of L, InclPoset Ids L st
  s = s9 & s9 is monotone & for x be Element of L holds s9.x c= downarrow x
  proof
    take s;
    thus thesis by YELLOW_2:def 4;
  end;
  hence thesis by Def13;
end;
