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

theorem Th7:
  p in CnPos(X) & p => q in CnPos(X) implies q in CnPos(X)
proof
  assume that
A1: p in CnPos(X) and
A2: p => q in CnPos(X);
  T is Hilbert_theory & X c= T implies q in T
  proof
    assume that
A3: T is Hilbert_theory and
A4: X c= T;
A5: p => q in T by A2,A3,A4,Def11;
    p in T by A1,A3,A4,Def11;
    hence thesis by A3,A5;
  end;
  hence thesis by Def11;
end;
