$(\overline{\mathrm{b}}\times \overline{\mathrm{c}})·(\overline{\mathrm{a}}\times \overline{\mathrm{d}})+(\overline{\mathrm{c}}\times \overline{\mathrm{a}})·(\overline{\mathrm{b}}\times \overline{\mathrm{d}})+(\overline{\mathrm{a}}\times \overline{\mathrm{b}})·(\overline{\mathrm{c}}\times \overline{\mathrm{d}})=$

$=(\overline{\mathrm{b}}\times \overline{\mathrm{c}})·(\overline{\mathrm{a}}\times \overline{\mathrm{d}})+(\overline{\mathrm{c}}\times \overline{\mathrm{a}})·(\overline{\mathrm{b}}\times \overline{\mathrm{d}})+(\overline{\mathrm{a}}\times \overline{\mathrm{b}})·(\overline{\mathrm{c}}\times \overline{\mathrm{d}})$

$=\left|\begin{array}{cc}\overline{\mathrm{b}}·\overline{\mathrm{a}}& \overline{\mathrm{b}}·\overline{\mathrm{d}}\\ \overline{\mathrm{c}}·\overline{\mathrm{a}}& \overline{\mathrm{c}}·\overline{\mathrm{d}}\end{array}\right|+\left|\begin{array}{cc}\overline{\mathrm{c}}·\overline{\mathrm{b}}& \overline{\mathrm{c}}·\overline{\mathrm{d}}\\ \overline{\mathrm{a}}·\overline{\mathrm{b}}& \overline{\mathrm{a}}·\overline{\mathrm{d}}\end{array}\right|+\left|\begin{array}{cc}\overline{\mathrm{a}}·\overline{\mathrm{c}}& \overline{\mathrm{a}}·\overline{\mathrm{d}}\\ \overline{\mathrm{b}}·\overline{\mathrm{c}}& \overline{\mathrm{b}}·\overline{\mathrm{d}}\end{array}\right|$

$=\begin{array}{rcl} & & (\overline{\mathrm{a}}·\overline{\mathrm{b}}) (\overline{\mathrm{c}}·\overline{\mathrm{d}})-(\overline{\mathrm{a}}·\overline{\mathrm{c}}) (\overline{\mathrm{b}}·\overline{\mathrm{d}})+(\overline{\mathrm{b}}·\overline{\mathrm{c}}) (\overline{\mathrm{a}}·\overline{\mathrm{d}})\end{array}\begin{array}{rcl} & & -(\overline{\mathrm{a}}·\overline{\mathrm{b}}) (\overline{\mathrm{c}}·\overline{\mathrm{d}})+(\overline{\mathrm{a}}·\overline{\mathrm{c}}) (\overline{\mathrm{b}}·\overline{\mathrm{d}})-(\overline{\mathrm{b}}·\overline{\mathrm{c}}) (\overline{\mathrm{a}}·\overline{\mathrm{d}})\end{array}$

$\begin{array}{rcl} & =& 0\end{array}$