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

theorem Th32:
  for L being non empty RelStr, p,x being Element of L holds chi((
  downarrow p)`,the carrier of L).x = {} iff x <= p
proof
  let L be non empty RelStr, p,x be Element of L;
  not x in (downarrow p)` iff x in downarrow p by XBOOLE_0:def 5;
  hence thesis by FUNCT_3:def 3,WAYBEL_0:17;
end;
