
theorem Th08:
  for u,v,w being Element of TOP-REAL 3 st v`3 = 0 &
  w = |[u`1 + v`1,u`2 + v`2, u`3]| holds |{u,v,w}| = 0
  proof
    let u,v,w be Element of TOP-REAL 3;
    assume that
A2: v`3 = 0 and
A3: w = |[u`1 + v`1,u`2 + v`2, u`3]|;
A4: |{ u,v,w }| = u`1 * v`2 * w`3 - u`3*v`2*w`1 - u`1*v`3*w`2 + u`2*v`3*w`1 -
      u`2*v`1*w`3 + u`3*v`1*w`2 by ANPROJ_8:27
               .= u`1 * v`2 * w`3 - u`3*v`2*w`1 - u`2*v`1*w`3 + u`3*v`1*w`2
                   by A2;
    w`1 = u`1 + v`1 & w`2 = u`2 + v`2 & w`3 = u`3 by A3,EUCLID_5:2;
    hence thesis by A4;
  end;
