
theorem
  for S being non empty RelStr, T being lower-bounded antisymmetric
  reflexive non empty RelStr holds Bottom MonMaps(S, T) = S --> Bottom T
proof
  let S be non empty RelStr, T be lower-bounded antisymmetric reflexive non
  empty RelStr;
  set L = MonMaps(S, T);
  reconsider f = S --> Bottom T as Element of L by WAYBEL10:9;
  reconsider f9 = f as Function of S, T;
A1: for b being Element of L st b is_>=_than {} holds f <= b
  proof
    let b be Element of L;
    assume b is_>=_than {};
    reconsider b9 = b as Function of S, T by WAYBEL10:9;
    reconsider b99 = b9, f99 = f as Element of T|^ the carrier of S by
YELLOW_0:58;
    for x being Element of S holds f9.x <= b9.x
    proof
      let x be Element of S;
      f9. x = ((the carrier of S) --> Bottom T). x .= Bottom T by FUNCOP_1:7;
      hence thesis by YELLOW_0:44;
    end;
    then f9 <= b9 by YELLOW_2:9;
    then f99 <= b99 by WAYBEL10:11;
    hence thesis by YELLOW_0:60;
  end;
  f is_>=_than {};
  then f = "\/"({}, L) by A1,YELLOW_0:30;
  hence thesis by YELLOW_0:def 11;
end;
