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 Th60:
  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 l c=
  still_not-bound_in Subst(l,a .--> 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 l;
    then reconsider y9 = y as Element of still_not-bound_in l;
A3: 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;
    consider n being Nat such that
A4: y9 = l.n and
A5: 1 <= n and
A6: n <= len l and
A7: l.n in bound_QC-variables A by A1,A2;
    l.n <> a by A7,QC_LANG3:34;
    then
A8: l.n = Subst(l,a .--> x).n by A5,A6,CQC_LANG:3;
    n <= len Subst(l,a .--> x) by A6,CQC_LANG:def 1;
    hence y in still_not-bound_in Subst(l,a .--> x) by A3,A4,A5,A7,A8;
end;
