Lemma 6

Let $I_1$ and $I_2$ be two-sided ideals of A, I a primitive ideal of A. If I contains $I_1I_2($in particular, if I contains $I_1\cap I_2)$, then I contains $I_1$ or $I_2$.

Let $\pi$ be an irreducible representation with kernel I. If $I\not\supset I_1$ and $I\not\supset I_2$, Lemma 5, b) proves that $\pi (I_1I_2)\not= 0$, whence $I\not\supset I_1I_2$.