\(
\begin{align}
\prod_{i=0}^{m-1} \prod_{j=i+1}^m (x_iy_j - x_jy_i) =
&+ x_{0}^m \prod_{\substack{i=0 \\ i \ne 0}}^m y_i \prod_{\substack{k = 0 \\ k \neq 0}}^{m-1} \prod_{\substack{l = k+1 \\ l \neq 0}}^m (x_ky_l-x_ly_k) \\ \nonumber
&- x_{1}^m \prod_{\substack{i=0 \\ i \ne 1}}^m y_i \prod_{\substack{k = 0 \\ k \neq 1}}^{m-1} \prod_{\substack{l = k+1 \\ l \neq 1}}^m (x_ky_l-x_ly_k) \\ \nonumber
&+ \dots \\ \nonumber
&+ x_{m}^m \prod_{\substack{i=0 \\ i \ne m}}^m y_i \prod_{\substack{k = 0 \\ k \neq m}}^{m-1} \prod_{\substack{l = k+1 \\ l \neq m}}^m (x_ky_l-x_ly_k)
\)
Ik heb geverifieerd dat ze klopt voor m = 2 en m = 3. Het lukt me echter niet om deze in het algemeen te bewijzen. Kan iemand helpen?\begin{align}
\prod_{i=0}^{m-1} \prod_{j=i+1}^m (x_iy_j - x_jy_i) =
&+ x_{0}^m \prod_{\substack{i=0 \\ i \ne 0}}^m y_i \prod_{\substack{k = 0 \\ k \neq 0}}^{m-1} \prod_{\substack{l = k+1 \\ l \neq 0}}^m (x_ky_l-x_ly_k) \\ \nonumber
&- x_{1}^m \prod_{\substack{i=0 \\ i \ne 1}}^m y_i \prod_{\substack{k = 0 \\ k \neq 1}}^{m-1} \prod_{\substack{l = k+1 \\ l \neq 1}}^m (x_ky_l-x_ly_k) \\ \nonumber
&+ \dots \\ \nonumber
&+ x_{m}^m \prod_{\substack{i=0 \\ i \ne m}}^m y_i \prod_{\substack{k = 0 \\ k \neq m}}^{m-1} \prod_{\substack{l = k+1 \\ l \neq m}}^m (x_ky_l-x_ly_k)
\)
