
theorem
  for S being non empty RelStr, T being upper-bounded antisymmetric
  reflexive non empty RelStr holds Top MonMaps(S, T) = S --> Top T
proof
  let S be non empty RelStr, T be upper-bounded antisymmetric reflexive non
  empty RelStr;
  set L = MonMaps(S, T);
  reconsider f = S --> Top 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) --> Top T). x .= Top T by FUNCOP_1:7;
      hence thesis by YELLOW_0:45;
    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_0:31;
  hence thesis by YELLOW_0:def 12;
end;
