reserve r, r1, r2, x, y, z,
        x1, x2, x3, y1, y2, y3 for Real;
reserve R, R1, R2, R3 for Element of 3-tuples_on REAL;
reserve p, q, p1, p2, p3, q1, q2 for Element of REAL 3;
reserve f1, f2, f3, g1, g2, g3, h1, h2, h3 for PartFunc of REAL,REAL;
reserve t, t0, t1, t2 for Real;

theorem Th49:
  for r being Real holds r*p = |[ r*p.1, r*p.2, r*p.3 ]|
proof
    let r be Real;
A1: (r*p).1 = r*(p.1) by RVSUM_1:44;
A2: (r*p).2 = r*(p.2) by RVSUM_1:44;
    (r*p).3 = r*(p.3) by RVSUM_1:44;
    hence r*p = |[ r*p.1, r*p.2, r*p.3]| by A1,A2,Th1;
end;
