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 Th50:
  for r being Real holds r*|[ x,y,z ]| = |[ r*x, r*y, r*z ]|
proof
  let r be Real;
  set p = |[ x,y,z ]|;
  r*p = |[ r*p.1, r*p.2, r*p.3 ]| by Th49
    .=|[ r*x, r*p.2, r*p.3 ]|
    .=|[ r*x, r*y, r*p.3 ]|
    .=|[ r*x, r*y, r*z ]|;
    hence thesis;
end;
