theorem Th84:
  X is countable & Y is countable implies X \/ Y is countable
proof
  assume that
A1: card X c= omega and
A2: card Y c= omega;
A3: card (X \/ Y) c= card X +` card Y by Th33;
A4: omega +` omega = omega by Th74;
  card X +` card Y c= omega +` omega by A1,A2,Th82;
  hence card (X \/ Y) c= omega by A3,A4;
end;
