The deduction theorem in a functional calculus of first order based on strict implication

In a previous paper, a functional calculus based on strict implication was developed. That system will be referred to as S2. The system resulting from the addition of Becker's axiom will be referred to as S4. In the present paper we will shw that a restricted deduction theorem is provable in S4 or more precisely in a system equivalent to S4. We will also show that such a deduction theorem is not provable in S2.The following theorems not derived in Symbolic logic will be required for the fundamental theorems XXVIII* and XXIX* of this paper. We will state most of them without proofs.