
theorem Th42:
  for S being non empty RelStr
  for T being upper-bounded non empty reflexive antisymmetric RelStr
  holds S --> Top T is infs-preserving
proof
  let S be non empty RelStr;
  let T be upper-bounded non empty reflexive antisymmetric RelStr;
  let X be Subset of S such that ex_inf_of X,S;
  set t = Top T, f = (the carrier of S) --> t;
A1: f.inf X = t by FUNCOP_1:7;
  (S --> t).:X c= {t} by FUNCOP_1:81;
  then (S --> t).:X = {t} or (S --> t).:X = {} by ZFMISC_1:33;
  hence thesis by A1,YELLOW_0:38,39,43;
end;
