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 Th23:
  p1 <X> (p2+p3) = ( p1 <X> p2 ) + ( p1 <X> p3 )
proof
A1: ( p1 <X> p2 ) + ( p1 <X> p3 ) = |[ (p1`2 * p2`3 - p1`3 * p2`2) + (p1`2 *
p3`3 - p1`3 * p3`2), (p1`3 * p2`1 - p1`1 * p2`3) + (p1`3 * p3`1 - p1`1 * p3`3),
  (p1`1 * p2`2 - p1`2 * p2`1) + (p1`1 * p3`2 - p1`2 * p3`1) ]| by Th6
    .= |[ p1`2 * p2`3 - p1`3 * p2`2 + p1`2 * p3`3 - p1`3 * p3`2, p1`3 * p2`1
  - p1`1 * p2`3 + p1`3 * p3`1 - p1`1 * p3`3, p1`1 * p2`2 - p1`2 * p2`1 + p1`1 *
  p3`2 - p1`2 * p3`1 ]|;
A2: p2+p3 = |[ p2`1 + p3`1, p2`2 + p3`2, p2`3 + p3`3]| by Th5;
  then
A3: (p2+p3)`3 = p2`3 + p3`3;
  (p2+p3)`1 = p2`1 + p3`1 & (p2+p3)`2 = p2`2 + p3`2 by A2;
  hence thesis by A3,A1;
end;
