reserve A for QC-alphabet;
reserve i,j,k for Nat;
reserve f for Substitution of A;

theorem Th9:
  for p,q being Element of QC-WFF(A) holds
  p '&' q is Element of CQC-WFF(A) iff p is Element of CQC-WFF(A) & q is
  Element of CQC-WFF(A)
proof
  let p,q being Element of QC-WFF(A);
  thus p '&' q is Element of CQC-WFF(A) implies
  p is Element of CQC-WFF(A) & q is
  Element of CQC-WFF(A)
  proof
    assume
A1: p '&' q is Element of CQC-WFF(A);
    then Fixed(p '&' q) = {} by Th4;
    then
A2: Fixed p \/ Fixed q = {} by QC_LANG3:67;
    then
A3: Fixed p = {};
    Free(p '&' q) = {} by A1,Th4;
    then
A4: Free p \/ Free q = {} by QC_LANG3:57;
    then Free p = {};
    hence thesis by A4,A2,A3,Th4;
  end;
  assume p is Element of CQC-WFF(A) & q is Element of CQC-WFF(A);
  then reconsider r = p, s = q as Element of CQC-WFF(A);
  Fixed r = {} by Th4;
  then Fixed r \/ Fixed s = {} by Th4;
  then
A5: Fixed(r '&' s) = {} by QC_LANG3:67;
  Free r = {} by Th4;
  then Free r \/ Free s = {} by Th4;
  then Free (r '&' s) = {} by QC_LANG3:57;
  hence thesis by A5,Th4;
end;
