
theorem Th41:
  for S being non empty RelStr
  for T being lower-bounded non empty reflexive antisymmetric RelStr
  holds S --> Bottom T is sups-preserving
proof
  let S be non empty RelStr;
  let T be lower-bounded non empty reflexive antisymmetric RelStr;
  let X be Subset of S such that ex_sup_of X,S;
  set f = (the carrier of S) --> Bottom T;
A1: f.sup X = Bottom T by FUNCOP_1:7;
  (S --> Bottom T).:X c= {Bottom T} by FUNCOP_1:81;
  then (S --> Bottom T).:X = {Bottom T} or (S --> Bottom T).:
  X = {} by ZFMISC_1:33;
  hence thesis by A1,YELLOW_0:38,39,42;
end;
