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 Th33:
  |{ p1, p2, p3 }| = |{ p2, p3, p1 }|
proof
  |{ p1, p2, p3 }| = |(|[ p1`1, p1`2, p1`3 ]|, |[ (p2`2*p3`3) - (p2`3*p3`2
  ), (p2`3*p3`1) - (p2`1*p3`3), (p2`1*p3`2) - (p2`2*p3`1) ]|)| by Th3
    .= p1`1*((p2`2*p3`3) - (p2`3*p3`2)) + p1`2*((p2`3*p3`1) - (p2`1*p3`3)) +
  p1`3*((p2`1*p3`2) - (p2`2*p3`1)) by Th30
    .= p2`1*(p3`2*p1`3 - p3`3*p1`2) + p2`2*(p3`3*p1`1 - p3`1*p1`3) + p2`3*(
  p3`1*p1`2 - p3`2*p1`1)
    .= |( |[ p2`1, p2`2, p2`3 ]|, |[ p3`2*p1`3 - p3`3*p1`2, p3`3*p1`1 - p3`1
  *p1`3, p3`1*p1`2 - p3`2*p1`1 ]| )| by Th30
    .= |( p2, p3 <X> p1 )| by Th3;
  hence thesis;
end;
