  Following A. Kuznetsov's outline, we restore Kuznetsov's syntactic proof of
the assertoric equipollence of the intuitionistic propositional calculus and
the proof-intuitionistic calculus KM (Kuznetsov's Theorem). Then, we show that
this property is true for a broad class of modal logics on an intuitionistic
basis, which includes, e.g., the modalized Heyting calculus mHC. The last fact
is one of two key properties necessary for the commutativity of a diagram
involving the lattices of normal extensions of four well-known logics. Also, we
give an algebraic interpretation of the assertoric equipollence for subsystems
of KM.
