reserve

  k,n for Nat,
  x,y,X,Y,Z for set;

theorem Th4:
  for f being Function st f is one-to-one & X c= dom f holds card(f
  .:X) = card X
proof
  let f be Function;
  assume f is one-to-one & X c= dom f;
  then card(f.:X) c= card X & card X c= card(f.:X) by Th3,CARD_1:67;
  hence thesis by XBOOLE_0:def 10;
end;
