We consider the simplest non-trivial extension of the propositional provability logic, denoted by G4, which is based on variables and logical connectives and;_;;:;
, axioms of the classical logic of propositions and
-axioms:
(p  q)  (
p 
q);
p 

p;
(
p  p) 
p;
(p  p)  (p  p):

p; (p  q) _ (q  p); where p denotes (p and
p). Rules of inference of G4 are the rules of the classical logic of propositions and the rule A
A. They say formula F is weak-expressible by formulas of the system  in the logic L if F can be obtained from unary formulas of L and from formulas of  by applying the rule of weak substitution (which allows to pass from the formulas A and B to the result of the substitution of one of them in another one instead of all occurrences of the same variable, say p) and by the rule of replacement by an equivalent formula (which permit to pass from the formula A to an equivalent to it in L formula B). They say the system of formulas  is complete with respect to weak-expressibility in the logic L if any formula of the calculus of L is weak-expressible via  in L. In the present paper we found out the conditions for a system of formulas  to be complete as to weak-expressibility in the logic G4.