In [1, §14E], a sequent calculus formulation (L.-formulation) of type assignment (theory of functionality) is given for a system based either on a system of combinators with strong reduction or on a system of λη-calculus provided that the rule for subject conversion (which says that if X has type α and X cnv Y then Y has type α) is postulated for the system. This sequent calculus formulation does not work for a system based on the λβ-calculus. In [2] I introduced a sequent calculus formulation for a system without the rule of subject conversion based on any of the three systems mentioned above. Further, in [2, §5] I pointed out that if proper inclusions of the form of the statement that λx·x is a function from α to β are postulated, then functions are identified with their restrictions in the λη-calculus but not in the λβ-calculus, and that therefore there is some interest in having a sequent calculus formulation of type assignment with the rule of subject conversion for systems based on the λβ-calculus. In this paper, such a system is presented, the elimination theorem (Gentzen's Hauptsatz) is proved for it, and it is proved equivalent to the natural deduction formulation of [1, §14D].

I shall assume familiarity with the λβ-calculus, and shall use (with minor modifications) the notational conventions of [1]. Hence, the theory of type assignment (theory of functionality) will be based on an atomic constant F such that if α and β are types then Fαβ represents roughly the type of functions from α to β (more exactly it represents the type of functions whose domain includes α and under which the image of α is included in β).