Corollary 13.25.31ED63BC3072B

Proof. The preceding proposition, with $\beta =\theta $ and $\gamma =\eta $, applies to $u$, $v$ and $m=\max (u,v)$, and their upper volumes in background $\theta $ are at most $v_+(\Omega )=K$. Thus, for $x=u,v,m$, the errors

lie between zero and $K((1+\varepsilon /\kappa _\delta )^ n-1)$. The difference $d_\eta (u,v)-d_\theta (u,v)$ equals $2e_ m-e_ u-e_ v$, whose absolute value is at most twice this bound. Finally, the projection invariance in proposition 13.22.4, applied in background $\eta $, gives $d_\eta (u,v)=d_\eta (P_\eta [u],P_\eta [v])$, which is the comparison appearing in (13.25.4). $\square$