reserve A for QC-alphabet;
reserve p, q, r, s, p1, q1 for Element of CQC-WFF(A),
  X, Y, Z, X1, X2 for Subset of CQC-WFF(A),
  h for QC-formula of A,
  x, y for bound_QC-variable of A,
  n for Element of NAT;

theorem Th62:
  for h holds still_not-bound_in h c= still_not-bound_in (h.x)
proof
  defpred P[QC-formula of A] means still_not-bound_in $1
   c= still_not-bound_in ($1.x);
A1: for p being Element of QC-WFF(A) st P[p] holds P['not' p]
  proof
    let p be Element of QC-WFF(A);
    still_not-bound_in (('not' p).x) = still_not-bound_in 'not' (p.x) by
CQC_LANG:19
      .= still_not-bound_in(p.x) by QC_LANG3:7;
    hence thesis by QC_LANG3:7;
  end;
A2: for p, q being Element of QC-WFF(A) st P[p] & P[q] holds P[p '&' q]
  proof
    let p, q be Element of QC-WFF(A) such that
A3: P[p] and
A4: P[q];
A5: still_not-bound_in ((p '&' q).x) = still_not-bound_in ((p.x) '&' (q.x
    )) by CQC_LANG:21
      .= still_not-bound_in (p.x) \/ still_not-bound_in (q.x) by QC_LANG3:10;
    still_not-bound_in (p '&' q) = still_not-bound_in p \/
    still_not-bound_in q by QC_LANG3:10;
    hence thesis by A3,A4,A5,XBOOLE_1:13;
  end;
A6: for x being bound_QC-variable of A, p being Element of QC-WFF(A) st
  P[p] holds
  P[All(x, p)]
  proof
    let y be bound_QC-variable of A, p be Element of QC-WFF(A) such that
A7: P[p];
    per cases;
    suppose
      y = x;
      hence thesis by CQC_LANG:24;
    end;
    suppose
A8:   y <> x;
A9:   still_not-bound_in All(y,p) = still_not-bound_in p \ {y} by QC_LANG3:12;
      still_not-bound_in (All(y,p).x) = still_not-bound_in All(y,p.x) by A8,
CQC_LANG:25
        .= still_not-bound_in (p.x) \ {y} by QC_LANG3:12;
      hence thesis by A7,A9,XBOOLE_1:33;
    end;
  end;
A10: for k being Nat, P being (QC-pred_symbol of k,A), ll being
  QC-variable_list of k,A holds P[P!ll]
  proof
    let k be Nat, P be (QC-pred_symbol of k,A), ll be
    QC-variable_list of k,A;
A11: still_not-bound_in ((P!ll).x) = still_not-bound_in
   (P!Subst(ll,(A)a.0.-->
    x)) by CQC_LANG:17
      .= still_not-bound_in Subst(ll,(A)a.0.-->x) by QC_LANG3:5;
    still_not-bound_in ll c= still_not-bound_in
    Subst(ll,(A)a.0.-->x) by Th60;
    hence thesis by A11,QC_LANG3:5;
  end;
A12: P[VERUM(A)] by CQC_LANG:15;
  thus for h holds P[h] from QC_LANG1:sch 1(A10,A12,A1,A2,A6);
end;
