theorem Th9:
  A <> {} & B <> {} iff A + B <> {}
proof
  thus A <> {} & B <> {} implies A + B <> {}
  proof
    assume
A1: A <> {};
    then reconsider x = the Element of A as Element of G by TARSKI:def 3;
    assume
A2: B <> {};
    then reconsider y = the Element of B as Element of G by TARSKI:def 3;
    x + y in A + B by A1,A2;
    hence thesis;
  end;
  set x = the Element of A + B;
  assume A + B <> {};
  then ex a,b st x = a + b & a in A & b in B by ThX8;
  hence thesis;
end;
