reserve X for non empty UNITSTR;
reserve a, b for Real;
reserve x, y for Point of X;
reserve X for RealUnitarySpace;
reserve x, y, z, u, v for Point of X;

theorem
  dist(x - y,u - v) <= dist(x,u) + dist(y,v)
proof
  dist(x - y,u - v) = ||.((x - y) - u) + v.|| by RLVECT_1:29
    .= ||.(x - (u + y)) + v.|| by RLVECT_1:27
    .= ||.((x - u) - y) + v.|| by RLVECT_1:27
    .= ||.(x - u) - (y - v).|| by RLVECT_1:29
    .= ||.(x - u) + -(y - v).||;
  then dist(x - y,u - v) <= ||.x - u.|| + ||.-(y - v).|| by Th30;
  hence thesis by Th31;
end;
