in (\[ve-la\]) and (\[ve-la-tp\]), respectively.

It is not hard to see that the first sum in (\[Q-nilpotent-apply\]) coincides with the left hand side of (\[UandQ\]). Furthermore, combining all the terms with $p=0$ or $p=k$ in the second sum of (\[Q-nilpotent-apply\]) and all the terms with $p=0$ in the third sum of (\[Q-nilpotent-apply\]) we get $$[m, U(v_1, \dots, v_k)]_G (a_1, \dots, a_n)\,.$$ Combining the remaining terms in the second sum and in the third sum of (\[Q-nilpotent-apply\]) we get $$\sum_{p=1}^{k-1}\sum_{{{\lambda}}\in {{\rm Sh}}_{p,k-p}}
(-1)^{{{\varepsilon}}_{{{\lambda}}} + |v_{{{\lambda}}(1)}| + \dots + |v_{{{\lambda}}(p)}|}
[U(v_{{{\lambda}}(1)}, \dots, v_{{{\lambda}}(p)}) ,
U(v_{{{\lambda}}(p+1)}, \dots, v_{{{\lambda}}(k)}) ]_G\,.$$

Thus we proved that a codifferential ${\hat{Q}}$ on the coalgebra ${{\bf S}}(V,A)$ gives us a ${{\bf Ger}}_{\infty}$-algebra structure on $V$, an $A_{\infty}$-structure on $A$, and an $L_{\infty}$-morphism from $V$ to the ${{\Lambda}}{{\bf Lie}}$-algebra ${C^{\bullet}}(A,A)$.

Furthermore, given a ${{\bf Ger}}_{\infty}$-algebra structure ${{\cal D}}$ on $V$, an $A_{\infty}$-structure $m$ on $A$, and an $L_{\infty}$-morphism $U$ from $V$ to ${C^{\bullet}}(A,A)$ the formulas $$Q^{{{\mathfrak{c}}}} = {{\cal D}}\,,$$ (\[m\]), (\[define-U\]) define a degree $1$ codifferential ${\hat{Q}}$ on ${{\bf S}}(V,A)$.

Theorem \[thm:Ger-plus-alg\] is proved.

${{\bf Ger}}^+_{\infty}$ versus the operad ${{\mathcal{O C}}}_{\infty}$ governing open-closed homotopy algebras (OCHA)
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Inspired by Zwiebach’s open-closed string field theory [@Zwiebach] H. Kajiura and J. Stasheff 