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Let $A_m =(A,0,m_2,m_3,\dots)$ and $A_n=(A,0,n_2,n_3,\dots)$ be two $A_\infty$-structures on a vector space $A$. Assume that i) $A_m$ and $A_n$ are isomorphic, and ii) $A_m$ and $A_n$ have the same ...