theorem
  dist(x + y,u + v) <= dist(x,u) + dist(y,v)
proof
  dist(x + y,u + v) = ||.((-u) + (-v)) + (x + y).|| by RLVECT_1:31
    .= ||.x + ((-u) + (-v)) + y.|| by RLVECT_1:def 3
    .= ||.x - u + (-v) + y.|| by RLVECT_1:def 3
    .= ||.x - u + (y - v).|| by RLVECT_1:def 3;
  hence thesis by Th41;
end;
