reserve A for QC-alphabet;
reserve p, q, r, s, t for Element of CQC-WFF(A);
reserve X for Subset of CQC-WFF(A);

theorem
  X|- 'not' p => 'not' q iff X|- q => p
proof
  thus X|- 'not' p => 'not' q implies X|- q => p
  proof
    assume
A1: X|- 'not' p => 'not' q;
    X|- ('not' p => 'not' q) => (q => p) by CQC_THE1:59;
    hence thesis by A1,CQC_THE1:55;
  end;
  assume
A2: X|- q => p;
  X|- (q => p) => ('not' p => 'not' q) by CQC_THE1:59;
  hence thesis by A2,CQC_THE1:55;
end;
