
theorem
  for a,b,c,d,e,f,g,h,i being Real holds
  |{ |[a,b,c]|,
     |[d,e,f]|,
     |[g,h,i]| }| = a * e * i + b * f * g + c * d * h
                    - g * e * c - h * f * a - i * d * b
  proof
    let a,b,c,d,e,f,g,h,i be Real;
    reconsider p = |[a,b,c]|, q = |[d,e,f]|, r = |[g,h,i]| as
      Element of TOP-REAL 3;
    |{ |[a,b,c]|,
       |[d,e,f]|,
       |[g,h,i]| }| = p`1 * q`2 * r`3 - p`3*q`2*r`1 - p`1*q`3*r`2
                      + p`2*q`3*r`1 - p`2*q`1*r`3 + p`3*q`1*r`2 by ANPROJ_8:27;
    hence thesis;
  end;
