reserve e,u for set;
reserve X, Y for non empty TopSpace;

theorem Th20:
  for H being Subset-Family of [:X,Y:] st Pr1(X,Y).:H = {} or Pr2(
  X,Y).:H = {} holds H = {}
proof
  let H be Subset-Family of [:X,Y:];
  dom Pr1(X,Y) = bool the carrier of [:X,Y:] & dom Pr2(X,Y) = bool the
  carrier of [:X,Y:] by FUNCT_2:def 1;
  hence thesis;
end;
