A many-valued logic version of the Craig-Lyndon interpolation theorem has been given by Gill [1] and Miyama [2]. The former dealt with three-valued logic and the latter generalized it to M-valued logic with M ≥ 3. The purpose of this paper is to improve the form of Miyama's version of the interpolation theorem. The system used in this paper is a many-valued analogue (Takahashi [4], Rousseau [3]) of Gentzen's logical calculus LK. Let T = {1,…, M} be the set of truth values. An M-tuple (Γ 1,…, Γ M ) of sets of formulas is called a sequent, which is regarded as valid if for any valuation (in canonical sense) there is a truth value μ Є T such that the set Γμ contains a formula of the value μ with respect to the valuation. (In the next section, and thereafter, we change the format of the sequent for typographical reasons.) Miyama's result is as follows (in representative form):

• (I) If a sequent ({A}, ∅,…, ∅, {B}) is valid, then there is a formula D such that

• (i) every predicate or propositional variable occurring in D occurs in A and B, and

• (ii) the sequents {{A}, ∅,…, ∅, {D}) and (D}, ∅,…, ∅, {B}) are both valid.

• What shall be proved in this paper is the following (in representative form):

• (II) If a sequent ({A 1}, {A 2}, …, {AM }) is valid, then there is a formula D such that

• (i) every predicate or propositional variable occurring in D occurs in at least two of the formulas A 1,…, AM , and

• (ii) the following M sequents are valid:

• ({A 1},{D},…,{D}),({D},{A 2},…,{D}),…,({D},{D},…,{AM }).

• Clearly the former can be obtained as a corollary of the latter.