Conform observatiei facute, $(n, n - x{~1~}) = 1, (n, n - x{~2~}) = 1, .. (n, n - x{~a~}) = 1$ =>

x{~a~} = n - x{~1~} <=> x{~1~} + x{~a~} = n
x{~a-1~} = n - x{~2~} <=> x{~2~} + x{~a-1~} = n

$x{~a~} = n - x{~1~} <=> x{~1~} + x{~a~} = n$
$x{~a-1~} = n - x{~2~} <=> x{~2~} + x{~a-1~} = n$

...

x{~1~} = n - x{~a~} <=> x{~a~} + x{~1~} = n

$x{~1~} = n - x{~a~} <=> x{~a~} + x{~1~} = n$

Fie $S1$ suma numerelor prime cu $n$ si mai mici ca {$n$}, fie $S2$ suma numerelor prime cu {$n$}, cuprinse intre $n$ si {$2 * n$}.
Adunand cele a egalitati, obtinem $2 * S1 = a * n$ => {$S1 = (a * n) / 2$}.