Lemma: The two preliminary products exist

Both $U$ and $V$ in (eq:UV) are defined distributions on the retained neighborhood.

The relevant conormal directions of $U$ have the form

Those of $V$ in the selected sector have the form

Proof. For $U$, a forbidden product cancellation between the retarded factor and the attachment delta would have zero $z$ covector. The attachment covector would therefore be zero. The retarded covector would then also be zero. Thus the product criterion passes.

For $V$, the first line depends on $(y,a)$ and the second on $(y,b)$. In a forbidden cancellation, the $a$ component forces the first nonzero line selection to vanish, and the $b$ component forces the second to vanish. The criterion again passes.

Differentiation preserves the relevant wavefront inclusions. The pullback and product rules therefore give the stated directions Hormander (Theorems 8.2.4 and 8.2.10).