reserve A,B for Ordinal,
  K,M,N for Cardinal,
  x,x1,x2,y,y1,y2,z,u for object,X,Y,Z,X1,X2, Y1,Y2 for set,
  f,g for Function;
reserve m,n for Nat;
reserve x1,x2,x3,x4,x5,x6,x7,x8 for object;
reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve n,k for Nat;

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;
