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
  (x*p1) <X> p2 = x* (p1 <X> p2) & (x*p1) <X> p2 = p1 <X> (x*p2)
proof
A1: (x*p1) <X> p2 = |[ x*p1`1 ,x*p1`2 ,x*p1`3]| <X> p2 by Th7
    .= |[ x*p1`1 ,x*p1`2 ,x*p1`3]| <X> |[ p2`1,p2`2,p2`3]|
    .= |[ (x*p1`2 * p2`3) - (x*p1`3 * p2`2) , (x*p1`3 * p2`1) - (x*p1`1 * p2
  `3) , (x*p1`1 * p2`2) - (x*p1`2 * p2`1) ]|;
  then
A2: (x*p1) <X> p2 = |[ x * ( (p1`2 * p2`3) - (p1`3 * p2`2) ), x * ( (p1`3 *
  p2`1) - (p1`1 * p2`3) ), x * ( (p1`1 * p2`2) - (p1`2 * p2`1) ) ]|
    .= x * (p1 <X> p2) by Th8;
  (x*p1) <X> p2 = |[ p1`2 * (x * p2`3) - p1`3 * (x * p2`2), p1`3 * (x * p2
  `1) - p1`1 * (x * p2`3), p1`1 * (x * p2`2) - p1`2 * (x * p2`1) ]| by A1
    .= |[ p1`1, p1`2, p1`3]| <X> |[ x*p2`1 ,x*p2`2 ,x*p2`3]|
    .= p1 <X> |[ x*p2`1 ,x*p2`2 ,x*p2`3]|
    .= p1 <X> (x*p2) by Th7;
  hence thesis by A2;
end;
