It suffices to verify the conditions in Lemma 9.3. In fact, Proposition 9.4, Lemma 9.5 and Proposition 9.6 verify Property (1), Property (2a) and Property (2b) in Lemma 9.3 respectively. So applying the implicit function theorem given by Lemma 9.3, the existence of the hyperkähler triple $\bm{\omega}_{\beta}^{\HK}$ just follows. The Hölder type error estimate (9.135) follows directly from the implicit function theorem and the definition of the weight functions.

Since the hyperkähler triple $\bm{\omega}_{\beta}^{\HK}$ determines a hyperkähler metric on $\mathcal{M}$. By Proposition 6.6, $\chi(\mathcal{M})=24$ and hence $\mathcal{M}$ is diffeomorphic to the $\K 3$ surface.

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