Corollary: No contact enters the first-order identity

Let $N$ be sheet-compatible as in Corollary cor:contact(i) and ${\widehat{A_1}}^N[h]={\mathcal{O}}^{(1)}[h]+{\mathsf{R}}^N(F_f)$. Then ${\widehat{A_1}}^N[h]={\hat{A}}_1[h]$ in ${\mathfrak{A}}_0$ for every $h$, so X-B's Theorem thm:ward holds for ${\widehat{A_1}}^N$ exactly when it holds for ${\hat{A}}_1$. No contact term is selected to repair the identity, and none could break it without violating (eq:sheet).

Proof. Corollary cor:contact(i) with $c=f$.