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 Th11:
  for f st dom f is countable & for x st x in dom f holds f.x is
  countable holds Union f is countable
proof
  let f such that
A1: card dom f c= omega and
A2: for x st x in dom f holds f.x is countable;
  for x being object st x in dom f holds card (f.x) c= omega
   by A2,CARD_3:def 14;
  hence card Union f c= omega by A1,Th6,CARD_2:86;
end;
