Lemma 3.9 . [00KH]

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Lemma 3.9.

Let $\phi_{1}$ and $\phi_{2}$ be two upper-semicontinuous metrics on $L$ , then

\dist(\lVert\mathord{\cdot}\rVert_{\phi_{1}},\lVert\mathord{\cdot}\rVert_{\phi_{2}})\leq\dist(\phi_{1},\phi_{2}).

(see also [BE18, Lemma 6.10])