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

theorem Th10:
  X c= Y implies CnPos(X) c= CnPos(Y)
proof
  assume
A1: X c= Y;
  thus CnPos(X) c= CnPos(Y)
  proof
    let a be object;
    assume
A2: a in CnPos(X);
    then reconsider t = a as Element of HP-WFF;
    for T st T is Hilbert_theory & Y c= T holds t in T
    proof
      let T such that
A3:   T is Hilbert_theory and
A4:   Y c= T;
      X c= T by A1,A4;
      hence thesis by A2,A3,Def11;
    end;
    hence thesis by Def11;
  end;
end;
