In the book Number Theory IV from Parshin ( https://www.springer.com/gp/book/9783540614678 ), one can find this (page 215) with the comment ''it is easy to see that...''. Let $a_{ij}$ ($1\le i,j\le m$) be complex numbers with $|a_{i,j}|\le H_i$ ($1\le j\le m$). One assumes that the linear form $L_i(\underline X)=\sum_{i=1}^ma_ix_i$ are linealy independent. Then, for any $\overline w=(w_1,\cdots,w_m)\in\mathbb C^m$, one has $$\sum_{i=1}^m\frac{L_i(\overline w)}{H_i}\ge c_1\frac{\Delta}{H_1\cdots H_m}$$ where $\Delta=\det(a_{i,j})$ and $c_1=\frac1{m!}\max_{1\le j\le m}|w_j|$.

Unfortunately, for me it is not easy !!. Can one have hints to proves that? Thanks in adavce