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
  r1*p-r2*p = (r1-r2)*|[ p.1, p.2, p.3 ]|
proof
  r1*R - r2*R = (r1-r2)*R
  proof
    r1*R - r2*R = r1*R + ((-1)*r2)*R by RVSUM_1:49
     .= (r1 + -r2)*R by RVSUM_1:50;
    hence thesis;
  end; then
  r1*p-r2*p = (r1-r2)*p;
  hence thesis by Th1;
end;
