reserve X,Y,Z,x,y,y1,y2 for set,
  D for non empty set,
  k,n,n1,n2,m2,m1 for Nat,

  L,K,M,N for Cardinal,
  f,g for Function;
reserve r for Real;
reserve p,q for FinSequence,
  k,m,n,n1,n2,n3 for Nat;

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;
