theorem
  (X is countable & for Y st Y in X holds Y is countable) implies union
  X is countable
proof
  assume that
A1: card X c= omega and
A2: for Y st Y in X holds Y is countable;
  for Y st Y in X holds card Y c= omega by A2,CARD_3:def 14;
  hence card union X c= omega by A1,Th6,CARD_2:87;
end;
