reserve x,y,Y,Z for set,
  L for LATTICE,
  l for Element of L;

theorem Th33:
  for L being upper-bounded LATTICE, f being Function of L,
  BoolePoset {{}}, p being prime Element of L st chi((downarrow p)`,the carrier
  of L) = f holds f is meet-preserving join-preserving
proof
  let L be upper-bounded LATTICE, f be Function of L,BoolePoset {{}}, p be
  prime Element of L;
  assume chi((downarrow p)`,the carrier of L) = f;
  then for x being Element of L holds f.x = {} iff x <= p by Th32;
  hence thesis by Th25;
end;
