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' 'not' p => q iff X|- p => q
proof
  thus X|- 'not' 'not' p => q implies X|- p => q
  proof
    assume
A1: X|- 'not' 'not' p => q;
    X|- ('not' 'not' p => q) => (p => q) by CQC_THE1:59;
    hence thesis by A1,CQC_THE1:55;
  end;
  assume
A2: X|- p => q;
  X|- 'not' 'not' p => p by CQC_THE1:59;
  hence thesis by A2,Th59;
end;
