reserve T, X, Y for Subset of HP-WFF;
reserve p, q, r, s for Element of HP-WFF;

theorem Th12:
  T is Hilbert_theory iff CnPos(T) = T
proof
  hereby
    assume
A1: T is Hilbert_theory;
A2: CnPos(T) c= T
    by A1,Def11;
    T c= CnPos(T) by Th9;
    hence CnPos(T) = T by A2,XBOOLE_0:def 10;
  end;
  thus thesis;
end;
