00HK Proof. For any f=(f1,…,fn)∈(k×)nf=(f_{1},\dots,f_{n})\in(k^{\times})^{n}, the numbers {|fi|∥ei∥}i∈{1,…,r}\{\lvert f_{i}\rvert\lVert e_{i}\rVert\}_{i\in\{1,\dots,r\}} are distinct, otherwise there exist i,j∈{1,…,n},i≠ji,j\in\{1,\dots,n\},i\neq j such that log∥ei∥−log∥ej∥=log|fifj|∈log|k×|\log\lVert e_{i}\rVert-\log\lVert e_{j}\rVert=\log\Big|\frac{f_{i}}{f_{j}}\Big|\in\log\lvert k^{\times}\rvert which contradicts the assumption of ℚ\mathbb{Q}-independence. Hence ‖∑i∈{1,…,n}fiei‖=max0≤i≤n∥fiei∥\Big\|\sum_{i\in\{1,\dots,n\}}f_{i}e_{i}\Big\|=\max_{0\leq i\leq n}\lVert f_{i}e_{i}\rVert by Lemma 2.13. ∎