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 Th61:
  for k being Nat, l being QC-variable_list of k,A, a
  being free_QC-variable of A,
  x being bound_QC-variable of A holds still_not-bound_in
  Subst(l,a .--> x) c= still_not-bound_in l \/ {x}
proof
  let k be Nat, l be QC-variable_list of k,A,
a be free_QC-variable of A,
  x be bound_QC-variable of A;
  let y be object;
A1: still_not-bound_in l = { l.n where n is Nat:
 1 <= n & n <= len l & l.n in
  bound_QC-variables A} by QC_LANG1:def 29;
  assume
A2: y in still_not-bound_in Subst(l,a .--> x);
  then reconsider y9 = y as Element of still_not-bound_in Subst(l,a .--> x);
  still_not-bound_in Subst(l,a .--> x)
   = { Subst(l,a .--> x).n  where n is Nat:
1 <= n &
  n <= len Subst(l,a .--> x) & Subst(l,a .--> x).n in bound_QC-variables A}
by QC_LANG1:def 29;
  then consider n being Nat such that
A3: y9 = Subst(l,a .--> x).n and
A4: 1 <= n and
A5: n <= len Subst(l,a .--> x) and
A6: Subst(l,a .--> x).n in bound_QC-variables A by A2;
A7: n <= len l by A5,CQC_LANG:def 1;
  per cases;
  suppose
    l.n = a;
    then y9 = x by A3,A4,A7,CQC_LANG:3;
    then y9 in {x} by TARSKI:def 1;
    hence thesis by XBOOLE_0:def 3;
  end;
  suppose
    l.n <> a;
    then l.n = Subst(l,a .--> x).n by A4,A7,CQC_LANG:3;
    then y9 in still_not-bound_in l by A1,A3,A4,A6,A7;
    hence thesis by XBOOLE_0:def 3;
  end;
end;
