  In the article we establish the Hardy-Littlewood inequality $ \pi (x + y)
\leq \pi (x) + \pi (y) $. We also prove that the naturally ordered primes
$p_1=2,p_2=3,p_3=5,p_4=7,\dots$ satisfy the inequality $ p_ {a + b}> p_a + p_b
$ for all $a, \ b \geq 2$.
