
theorem Th04:
  for a,b,c,d,e being Real
  for u,v,w being Element of TOP-REAL 3 st
  u = |[a,b,e]| & v = |[c,d,0]| & w = |[ a + c, b + d, e ]| holds
  |{u,v,w}| = 0
  proof
    let a,b,c,d,e be Real;
    let u,v,w be Element of TOP-REAL 3;
    assume that
A1: u = |[a,b,e]| and
A2: v = |[c,d,0]| and
A3: w = |[ a + c, b + d, e]|;
A4: u`1 = a & u`2 = b & u`3 = e & v`1 = c & v`2 = d & v`3 = 0 &
    w`1 = a + c & w`2 = b + d & w`3 = e by A1,A2,A3,EUCLID_5:2;
    |{ 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
               .= a * d * e - e * d * (a+c) - b * c * e + e * c * (b+d) by A4;
    hence thesis;
  end;
