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 Th8;
  hence thesis;
end;
