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