  We construct an $A_\infty$-structure on the two-sided bar construction
involving homotopy Gerstenhaber algebras (hgas). It extends the non-associative
product defined by Carlson and the author and generalizes the dga structure on
the one-sided bar construction due to Kadeishvili-Saneblidze. As a consequence,
the multiplicative cohomology isomorphism from the Eilenberg-Moore theorem is
promoted to a quasi-isomorphism of $A_\infty$-algebras.
  We also show that the resulting product on the differential torsion product
involving cochain algebras agrees with the one defined by Eilenberg-Moore and
Smith, for all triples of spaces. This is a consequence of the following
result, which is of independent interest: The strongly homotopy commutative
(shc) structure on cochains inductively constructed by Gugenheim-Munkholm
agrees with the one previously defined by the author for all hgas.
