reserve a for set;
reserve L for lower-bounded sup-Semilattice;
reserve x for Element of L;

theorem Th35:
  for L being Semilattice, I being Ideal of L holds
  DownMap I in the carrier of MonSet L
proof
  let L be Semilattice, I be Ideal of L;
  reconsider mI = DownMap I as Function of L, InclPoset Ids L;
  ex s be Function of L, InclPoset Ids L st mI = s & s is monotone &
  for x be Element of L holds s.x c= downarrow x
  proof
    take mI;
    thus thesis by Lm9,Lm10;
  end;
  hence thesis by Def13;
end;
