reserve x,y,z for Real,
  x3,y3 for Real,
  p for Point of TOP-REAL 3;
reserve p1,p2,p3,p4 for Point of TOP-REAL 3,
  x1,x2,y1,y2,z1,z2 for Real;

theorem
  |{ p1, p1, p2 }| = 0 & |{ p2, p1, p2 }| = 0
proof
A1: |{ p2, p1, p2 }| = p2`1*(p1<X>p2)`1 + p2`2*(p1<X>p2)`2 + p2`3*(p1<X>p2)
  `3 by Th29
    .= p2`1*((p1`2 * p2`3) - (p1`3 * p2`2)) + p2`2*(p1<X>p2)`2 + p2`3*(p1<X>
  p2)`3
    .= p2`1*( (p1`2 * p2`3) - (p1`3 * p2`2) ) + p2`2*( (p1`3 * p2`1) - (p1`1
  * p2`3) ) + p2`3*(p1<X>p2)`3
    .= ( p2`1*(p1`2 * p2`3) - p2`1*(p1`3 * p2`2) ) + p2`2*( (p1`3 * p2`1) -
  (p1`1 * p2`3) ) + p2`3*( (p1`1 * p2`2) - (p1`2 * p2`1) )
    .= 0 - p2`2*(p1`1 * p2`3) + p2`2*(p1`1 * p2`3);
  |{ p1, p1, p2 }| = p1`1*(p1<X>p2)`1 + p1`2*(p1<X>p2)`2 + p1`3*(p1<X>p2)
  `3 by Th29
    .= p1`1*((p1`2 * p2`3) - (p1`3 * p2`2)) + p1`2*(p1<X>p2)`2 + p1`3*(p1<X>
  p2)`3
    .= p1`1*( (p1`2 * p2`3) - (p1`3 * p2`2) ) + p1`2*( (p1`3 * p2`1) - (p1`1
  * p2`3) ) + p1`3*(p1<X>p2)`3
    .= ( p1`1*(p1`2 * p2`3) - p1`1*(p1`3 * p2`2) ) + p1`2*( (p1`3 * p2`1) -
  (p1`1 * p2`3) ) + p1`3*( (p1`1 * p2`2) - (p1`2 * p2`1) )
    .= 0;
  hence thesis by A1;
end;
