Let ${f_x}(m) = \sum\limits_{\left. p \right|m} {{f_x}(p)}$ be a strongly additive function on positive integer number $m$, where $p$ is a prime number. Set $${f_x}(p) = \left\{ {\begin{array}{*{20}{c}} {0,}\\ {1,}\\ 2, \end{array}} \right.\begin{array}{*{20}{c}} {{\rm{ }}p < \ln \ln x{\rm{ }}\ or \ {\rm{ }}p \ge {{(\ln \ln x)}^4}}\\ {\ln \ln x \le p < {{(\ln \ln x)}^2}}\\ {{{(\ln \ln x)}^2} \le p < {{(\ln \ln x)}^4}} \end{array}.$$ Bekelis (1997) say that $$\mathop {\lim }\limits_{x \to \infty } \sum\limits_{\scriptstyle{\rm{ }}p \le x\atop \scriptstyle{f_x}(p) = 1} {\frac{1}{p}} = \ln 2, \mathop {\lim }\limits_{x \to \infty } \sum\limits_{\scriptstyle{\rm{ }}p \le x\atop \scriptstyle{f_x}(p) = 2} {\frac{1}{p}} = \ln 2.$$ But he does not give a detail proof. How to prove it?

[1]Bekelis, D. (1997). Convolutions of the Poisson laws in number theory. In Analytic and Probabilistic Methods in Number Theory: Proceedings of the Second International Conference in Honour of J. Kubilius, Palanga, Lithuania, 23-27 September 1996 (Vol. 4, p. 283). Walter de Gruyter.