Let ${\cal C}$ be a non-unital $A_{\infty}$-pre-category, $(X_{1},X_{2})\in{\cal C}_{2}^{tr}$. We say that $f\in Hom^{0}(X_{1},X_{2})$ (zero stands for degree) is a quasi-isomorphism if $m_{1}(f)=0$, and for any objects $X_{0}$ and $X_{3}$ such that $(X_{0},X_{1},X_{2})\in{\cal C}_{3}^{tr}$ and $(X_{1},X_{2},X_{3})\in{\cal C}_{3}^{tr}$ one has: $m_{2}(f,\cdot):Hom(X_{0},X_{1})\to Hom(X_{0},X_{2})$ and $m_{2}(\cdot,f):Hom(X_{2},X_{3})\to Hom(X_{1},X_{3})$ are quasi-isomorphisms of complexes.