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) <X> p2 = p1 <X> (-p2)
proof
  (-p1) <X> p2 = |[ -p1`1, -p1`2, -p1`3 ]| <X> p2 by Th10
    .= |[ -p1`1, -p1`2, -p1`3 ]| <X> |[ p2`1, p2`2, p2`3 ]|
    .= |[ ((-p1`2) * p2`3) - ((-p1`3) * p2`2) , ((-p1`3) * p2`1) - ((-p1`1)
  * p2`3) , ((-p1`1) * p2`2) - ((-p1`2) * p2`1) ]|
    .= |[ (p1`2 * (-p2`3)) - (p1`3 * (-p2`2)) , (p1`3 * (-p2`1)) - (p1`1 * (
  -p2`3)) , (p1`1 * (-p2`2)) - (p1`2 * (-p2`1)) ]|
    .= |[ p1`1, p1`2, p1`3 ]| <X> |[ -p2`1, -p2`2, -p2`3 ]|
    .= |[ p1`1, p1`2, p1`3 ]| <X> -p2 by Th10;
  hence thesis;
end;
