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 Th7:
  x*p = |[ x*p`1, x*p`2, x*p`3]|
proof
  reconsider q = p as Element of REAL 3 by EUCLID:22;
  (x*q).2 = x*(q.2) by RVSUM_1:44; then
A1: (x*p)`2 = x*(p`2);
  (x*q).3 = x*(q.3) by RVSUM_1:44; then
A2: (x*p)`3 = x*(p`3);
  (x*q).1 = x*(q.1) by RVSUM_1:44;
  then (x*p)`1 = x*(p`1);
  hence thesis by A1,A2,Th3;
end;
