
theorem

:: 1.2. LEMMA (iii), p. 143
  for S,T being lower complete TopLattice for f being Function of S,T st
  f is continuous & for X being finite Subset of S holds f preserves_inf_of X
  holds f is infs-preserving
proof
  let S,T be lower complete TopLattice;
  let f be Function of S,T;
  assume f is continuous;
  then f is filtered-infs-preserving by Th11;
  then
  for F being filtered non empty Subset of S holds f preserves_inf_of F;
  hence thesis by WAYBEL_0:71;
end;
