$|\raisebox{1.72218pt}{.}|_{h}(x)\leq|\raisebox{1.72218pt}{.}|^{\mathrm{quot}}_{h}(x)$ for all $x\in X^{\mathrm{an}}$.

$\|\raisebox{1.72218pt}{.}\|_{h}=\|\raisebox{1.72218pt}{.}\|_{h}^{\mathrm{quot}}$.

Let $(L^{\prime},h^{\prime})$ be a pair of an invertible sheaf $L^{\prime}$ on $X$ and a continuous metric $h^{\prime}=\{|\raisebox{1.72218pt}{.}|_{h^{\prime}}(x)\}_{x\in X^{\operatorname{an}}}$ of ${L^{\prime}}^{\operatorname{an}}$ such that $L^{\prime}$ is generated by global sections. Then

$$|l\cdot l^{\prime}|_{h\otimes h^{\prime}}^{\mathrm{quot}}(x)\leq|l|_{h}^{\mathrm{quot}}(x)|l^{\prime}|_{h^{\prime}}^{\mathrm{quot}}(x)$$

for $l\in{L(x)}$ and $l^{\prime}\in{L^{\prime}(x)}$.