
theorem Th4:
  for a,b,c,d,e being Real holds
  |{ |[1,0,a]|,
     |[0,1,b]|,
     |[c,d,e]| }| = e - c * a - d * b
  proof
    let a,b,c,d,e be Real;
    reconsider p = |[1,0,a]|, q = |[0,1,b]|, r = |[c,d,e]| as
      Element of TOP-REAL 3;
    |{ |[1,0,a]|,
       |[0,1,b]|,
       |[c,d,e]| }| = 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;
