Ger}}^{\vee}(V) \to V$$ and the ${{\mathfrak{o}}}$-component $$\label{Q-ma}
Q^{{{\mathfrak{o}}}} : {{\bf s}}\, \big(\, {{\underline{S}}}({{\bf s}}^{-2} V) \otimes   T({{\bf s}}^{-1}A) \, \big)
\,\oplus \, {{\bf s}}\, {{\underline{T}}}({{\bf s}}^{-1} A)
\to A\,.$$

We claim that $Q^{{{\mathfrak{c}}}}$ (\[Q-mc\]) gives us a ${{\bf Ger}}_{\infty}$-algebra structure on $V$, the restriction $$\label{Q-ma-restr}
Q^{{{\mathfrak{o}}}} \Big|_{ {{\bf s}}{{\underline{T}}}({{\bf s}}^{-1}A) } :  {{\bf s}}{{\underline{T}}}({{\bf s}}^{-1}A) \to A$$ gives us an $A_{\infty}$-structure on $A$ and the restriction $$\label{Q-ma-restr1}
Q^{{{\mathfrak{o}}}} \Big|_{ {{\bf s}}\, \big(\, {{\underline{S}}}({{\bf s}}^{-2} V) \otimes   T({{\bf s}}^{-1}A) \, \big) }
: {{\bf s}}\, \big(\, {{\underline{S}}}({{\bf s}}^{-2} V) \otimes  T({{\bf s}}^{-1}A) \, \big) \to  A$$ gives us an $L_{\infty}$-morphism from the ${{\Lambda}}{{\bf Lie}}_{\infty}$-algebra $V$ to the ${{\Lambda}}{{\bf Lie}}$-algebra $C^{{{\bullet}}}(A,A)$.

The first two statements are obvious. Indeed, the map $Q^{{{\mathfrak{c}}}}$ (\[Q-mc\]) corresponds to the codifferential ${\hat{Q}}^{{{\mathfrak{c}}}}$ on the ${{\bf Ger}}^{\vee}$-coalgebra ${{\bf Ger}}^{\vee}(V)$. This is a ${{\bf Ger}}_{\infty}$-structure on $V$. The map (\[Q-ma-restr\]) corresponds to the codifferential $${\hat{Q}}^{{{\mathfrak{o}}}} \Big|_{ {{\bf s}}{{\underline{T}}}({{\bf s}}^{-1}A) } :  {{\bf s}}{{\underline{T}}}({{\bf s}}^{-1}A) \to
 {{\bf s}}{{\underline{T}}}({{\bf s}}^{-1}A)$$ on the cofree ${{\Lambda}}{{\bf coassoc}}$-coalgebra $${{\Lambda}}{{\bf coassoc}}(A) =  {{\bf s}}{{\underline{T}}}({{\bf s}}^{-1}A)\,.$$ Thus we get an $A_{\infty}$-algebra structure on $A$.

To prove the third statement we replace the ${{\mathfrak{o}}}$-component ${{\bf S}}^{{{\mathfrak{o}}}}(V, A)$ of the coalgebra ${{\bf S}}(V,A)$ by its desuspension $$\label{desusp-bS-ma}
{{\bf 