Proof. The second equality follows from the definition of the bilinear form $(\ \cdot \ )$ in (22.214.171.124) and Lemma 55.9.6. To see the first equality we distinguish two cases. Namely, if $C_ i \not\subset D$, then $D'$ is the scheme theoretic union of $D$ and $C_ i$ (by Divisors, Lemma 31.13.10) and we get a short exact sequence
by Morphisms, Lemma 29.4.6. Since we also have an exact sequence
by Divisors, Lemma 31.14.3 and we immediately see the lemma holds. $\square$
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