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 Th63:
  for h holds still_not-bound_in (h.x) c= still_not-bound_in h \/ {x}
proof
  defpred P[QC-formula of A] means
  still_not-bound_in ($1.x) 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;
A6: still_not-bound_in p \/ {x} \/ (still_not-bound_in q \/ {x}) =
    still_not-bound_in p \/ ({x} \/ still_not-bound_in q) \/ {x} by XBOOLE_1:4
      .= still_not-bound_in p \/ still_not-bound_in q \/ {x} \/ {x} by
XBOOLE_1:4
      .= still_not-bound_in p \/ still_not-bound_in q \/ ({x} \/ {x}) by
XBOOLE_1:4
      .= still_not-bound_in p \/ still_not-bound_in q \/ {x};
    still_not-bound_in (p.x) \/ still_not-bound_in (q.x) c=
    still_not-bound_in p \/ {x} \/ (still_not-bound_in q \/ {x}) by A3,A4,
XBOOLE_1:13;
    hence thesis by A5,A6,QC_LANG3:10;
  end;
A7: 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
A8: P[p];
    per cases;
    suppose
A9:   y = x;
A10:  (still_not-bound_in p) \ {x} c= still_not-bound_in p by XBOOLE_1:36;
A11:  still_not-bound_in All(x,p) = (still_not-bound_in p) \ {x} by QC_LANG3:12
;
      still_not-bound_in p c= still_not-bound_in (p.x) by Th62;
      then
      still_not-bound_in All(x,p) c= still_not-bound_in (p.x) by A11,A10;
      then
A12:  still_not-bound_in All(x,p) c= still_not-bound_in p \/ {x} by A8,
XBOOLE_1:1;
      still_not-bound_in All(y,p) \/ {x} = ((still_not-bound_in p) \ {x})
      \/ {x} by A9,QC_LANG3:12
        .= still_not-bound_in p \/ {x} by XBOOLE_1:39;
      hence thesis by A9,A12,CQC_LANG:24;
    end;
    suppose
A13:  y <> x;
A14:  still_not-bound_in All(y,p) \/ {x} = (still_not-bound_in p \ {y})
      \/ {x} by QC_LANG3:12
        .= (still_not-bound_in p \/ {x}) \ {y} by A13,XBOOLE_1:87,ZFMISC_1:11;
      still_not-bound_in (All(y,p).x) = still_not-bound_in All(y,p.x) by A13,
CQC_LANG:25
        .= still_not-bound_in (p.x) \ {y} by QC_LANG3:12;
      hence thesis by A8,A14,XBOOLE_1:33;
    end;
  end;
A15: 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;
A16: 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 Subst(ll,(A)a.0.-->x) c= still_not-bound_in ll \/ {x}
    by Th61;
    hence thesis by A16,QC_LANG3:5;
  end;
A17: still_not-bound_in VERUM(A) = {} by QC_LANG3:3;
  VERUM(A).x = VERUM(A) by CQC_LANG:15; then
 still_not-bound_in ((VERUM(A)).x) = {} by A17; then
A18: still_not-bound_in ((VERUM(A)).x) c=
    still_not-bound_in (VERUM(A)) \/ {x} by XBOOLE_1:2;
A19: P[VERUM(A)] by A18;
  thus for h holds P[h] from QC_LANG1:sch 1(A15,A19,A1,A2,A7);
end;
