Consider the matrix as follows
$$ \begin{matrix} 1 & c_1^{(1)} & c_2^{(1)} & ... & c_n^{(1)}\\ 1 & c_1^{(2)} & c_2^{(2)} & ... & c_n^{(2)}\\ ... & ... & ... & ... & ...\\ 1 & c_1^{(n+1)} & c_2^{(n+1)} & ... & c_n^{(n+1)} \end{matrix} $$$$ \begin{pmatrix} 1 & c_1^{(1)} & c_2^{(1)} & ... & c_n^{(1)}\\ 1 & c_1^{(2)} & c_2^{(2)} & ... & c_n^{(2)}\\ ... & ... & ... & ... & ...\\ 1 & c_1^{(n+1)} & c_2^{(n+1)} & ... & c_n^{(n+1)} \end{pmatrix} $$
If all $c_i$ for $i\in[n]$ are determined after $H_Z$ query, we can say that they are generated from $c=H_Z$ which is randomly picked by $\mathcal{B}$. Then we know that the determinant of this matrix $det(M)=0$ with only negligible possibility.
