
theorem Th5:
  for X be addLoopStr, A,B be Subset of X
  st A is countable & B is countable
  holds A+B is countable
  proof
    let X be addLoopStr, A,B be Subset of X;
    assume
    A1: A is countable & B is countable;
    set D = A+B;
    per cases;
    suppose
      A2: not (A is non empty & B is non empty);
      now
        assume D <> {}; then
        consider x be object such that
        A3: x in D by XBOOLE_0:def 1;
        D = {a+b where a,b is Point of X : a in A & b in B} by IDEAL_1:def 19;
        then ex a,b be Point of X st x = a+b & a in A & b in B by A3;
        hence contradiction by A2;
      end;
      hence thesis;
    end;
    suppose
      A4: A is non empty & B is non empty;
      consider F be Function of D,[:A,B:] such that
      A5: F is one-to-one by Th4;
      dom F = D & rng F c= [:A,B:] by A4,FUNCT_2:def 1; then
      A6: card D c= card [:A,B:] by A5,CARD_1:10;
      [:A,B:] is countable by A1,CARD_4:7; then
      card [:A,B:] c= omega; then
      card D c= omega by A6;
      hence thesis;
    end;
  end;
