Herbrand expansions of some formulas of modal logic S 4

We shall consider formulas of quantified modal logic. G.Mints described in [1] a reduction of an arbitrary formula F of quantified modal logic to a finite set of such formulas G1, ..., Gm that F is derivable in S4 if and only if G1, ..., Gm is derivable in S4. Moreover, the formula Gi i = 1, 2, ..., m has one of the following forms: ∀(L1 ∨ L2), ∀(L1 ∨ L2 ∨ L3), ∀∃(L1 ∨ L2), ∀(L1 ∨ L2), ∀(L1 ∨ ♦L2), L, where L1, L2, ... are the literals of classical logic and ∀∗ is a complex of universal qantifiers. In addition, the formulas can contain the constants. We assume for simplicity that the variables bounded by universal quantifiers are denoted by x, x1, x2, ... and the variables bounded by existential quantifiers are denoted by y, z, y1, z1, ... We will consider below the formulas of the more general form. We examine the sequents F1, ..., Fn , in which Fi i = 1, ..., n are the formulas having the following form