Corollary 19.5.2968273CCE4042

Proof. We may assume that $\mathrm{dd}^{\mathrm{c}}h$ has positive mass and is $\mathcal{I}$-good. By the $d_ S$-continuity of both sides of (19.5.30) as proved in theorem 19.5.26 and theorem 19.4.30, together with theorem 18.1.2, we may assume that $\mathrm{dd}^{\mathrm{c}}h$ has analytic singularities.

In this case, using the birational invariance of both sides of (19.5.30) as proved in proposition 19.4.32 and theorem 19.5.26, we may assume that $\mathrm{dd}^{\mathrm{c}}h$ has log singularities. Finally, after all these reductions, the equality (19.5.30) holds by construction. $\square$