It is obviosly, because $\sum_{i}\frac{1}{a_{i}!}=\frac{P}{a_{n}!}$, were $a_{n}$ - maximal of them and $(P,a_{n}!)=1.$

It's hardly so obvious: 1/2+1/6=2/3, so your main lemma is wrong

Yes, it is not trivial. For example $\frac{1}{4!}+\frac{1}{5!}=\frac{1}{20}$. But we can calculate $ord_{2}(P)-ord_{2}(a_{k}!)$ and $ord_{5}(P)-ord_{5}(a_{k}!)$ and proof for $a_{k}>4$ $ord_{2}(P)-ord_{2}(a_{k}!)<ord_{5}(P)-ord_{5}(a_{k}!)$.