reserve A for QC-alphabet;
reserve i,j,k,l,m,n for Nat;
reserve a,b,e for set;
reserve t,u,v,w,z for QC-symbol of A;
reserve p,q,r,s for Element of CQC-WFF(A);
reserve x for Element of bound_QC-variables(A);
reserve ll for CQC-variable_list of k,A;
reserve P for QC-pred_symbol of k,A;
reserve f,h for Element of Funcs(bound_QC-variables(A),bound_QC-variables(A)),
  K,L for Element of Fin bound_QC-variables(A);

theorem
  index p <= index(p '&' q) & index q <= index(p '&' q)
proof
A1: still_not-bound_in(p '&' q) = (still_not-bound_in p) \/ (
  still_not-bound_in q) by QC_LANG3:10;
A2: NBI(p '&' q) c= NBI q
  proof
    let e be object;
    assume e in NBI(p '&' q);
    then consider t such that
A3: t = e and
A4: for u st t<=u holds not x.u in still_not-bound_in p '&' q;
    now
      let u;
      assume
A5:   t<=u;
      still_not-bound_in q c= still_not-bound_in p '&' q by A1,XBOOLE_1:7;
      hence not x.u in still_not-bound_in q by A4,A5;
    end;
    hence thesis by A3;
  end;
  NBI(p '&' q) c= NBI p
  proof
    let e be object;
    assume e in NBI(p '&' q);
    then consider t such that
A6: t = e and
A7: for u st t<=u holds not x.u in still_not-bound_in p '&' q;
    now
      let u;
      assume
A8:   t<=u;
      still_not-bound_in p c= still_not-bound_in p '&' q by A1,XBOOLE_1:7;
      hence not x.u in still_not-bound_in p by A7,A8;
    end;
    hence thesis by A6;
  end;
  hence thesis by A2,QC_LANG1:28;
end;
