The check, re-derived

For the pair (A1)–(A2) define, as in X-B,

Both pairings are of a compactly supported smooth function against a smooth function: $V^{\prime }(\phi )=-\mathsf{L}\phi$ is smooth with support in the compact set carrying the coefficients of $\mathsf{L}$, and ${\Delta }_{\mathrm{F}}c$, ${\Delta }_{\mathrm{F}}{V}^{\prime }$ are smooth by Lemma lem:DF(iii) and because $V^{\prime }$ is itself a compact smooth test for each $\phi$.