  Hennings and Kuperberg defined quantum invariants $Z_{Henn}$ and $Z_{Kup}$
for closed oriented $3$-manifolds based on certain Hopf algebras, respectively.
When the Hopf algebras are semisimple, it is shown that $Z_{Kup}=|Z_{Henn}|^2$.
In this paper, we present a new proof of this equality.
