On a new proof of completeness of p. i. Propositional Calculus

In this paper, in order to obtain some interesting properties of the implicational algebras, we prove a completeness theorem on positive implicational Propositional Calculus in which we use the axiom schema (( a superset of b) superset of b) superset of (( b superset of a) superset of a) in an essential manner. We demonstrate this theorem without using neither Zorn's Lemma, nor similar properties.