reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;

theorem
  u = |[a,b,1]| & v = |[c,d,1]| & w = |[ (a + c)/2, (b + d)/2, 1]| implies
  |{u,v,w}| = 0
  proof
    assume that
A1: u = |[a,b,1]| and
A2: v = |[c,d,1]| and
A3: w = |[ (a + c)/2, (b + d)/2, 1]|;
A4: u`1 = a & u`2 = b & u`3 = 1 & v`1 = c & v`2 = d & v`3 = 1 &
    w`1 = (a + c)/2 & w`2 = (b + d)/2 & w`3 = 1 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 * 1 - 1 * d * ((a+c)/2) - a * 1 * (b + d)/2
                 + b * 1 * (a + c)/2 - b * c * 1 + 1 * c * (b+d)/2 by A4;
    hence thesis;
  end;
