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) <X> (p3+p4) = p1<X>p3 + p1<X>p4 + p2<X>p3 + p2<X>p4
proof
  (p1+p2) <X> (p3+p4) = ( p1 <X> (p3+p4) ) + ( p2 <X> (p3+p4) ) by Th24
    .= ( p1 <X> p3 + p1 <X> p4 ) + ( p2 <X> (p3+p4) ) by Th23
    .= ( p1 <X> p3 + p1 <X> p4 ) + (p2 <X> p3 + p2 <X> p4 ) by Th23;
  hence thesis by RLVECT_1:def 3;
end;
